FunctionalEquations.nb

(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 12.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[    253326,       6791]
NotebookOptionsPosition[    222474,       6276]
NotebookOutlinePosition[    222811,       6291]
CellTagsIndexPosition[    222768,       6288]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox[ButtonBox["If f(x) + 2f(1/x) + 3f(x/(x-1)) == x, what \
is f(x)?",
 BaseStyle->"Hyperlink",
 ButtonData->{
   URL["https://www.quora.com/If-f-x-2-f-frac-1-x-3f-frac-x-x-1-x-what-is-f-\
x"], None},
 ButtonNote->
  "https://www.quora.com/If-f-x-2-f-frac-1-x-3f-frac-x-x-1-x-what-is-f-x"],
 FontSize->24,
 FontVariations->{"Underline"->True}]], "Text",
 CellChangeTimes->{{3.813506090917964*^9, 3.813506165732568*^9}, {
  3.8135062035260353`*^9, 3.813506219806766*^9}, {3.8135064696534853`*^9, 
  3.813506469653968*^9}, {3.81350789800949*^9, 3.81350791215191*^9}, {
  3.8135102477482862`*^9, 
  3.813510287117432*^9}},ExpressionUUID->"6cd68e5d-db9f-40e0-a667-\
b077a0e31723"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "3"}], "]"}]}], "Input",
 CellChangeTimes->{{3.813504787650751*^9, 3.813504799649858*^9}, {
   3.813504952610477*^9, 3.813504956974752*^9}, {3.813505234791041*^9, 
   3.813505236820531*^9}, {3.8135053737746487`*^9, 3.813505410934164*^9}, 
   3.813506316043571*^9, {3.8135069347414503`*^9, 3.8135069390999613`*^9}},
 CellLabel->
  "In[102]:=",ExpressionUUID->"c32c2ce9-46ef-4340-82c6-89c9a026c5b6"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "3"], " ", 
    RowBox[{"a", "[", "3", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "3"], " ", 
    RowBox[{"b", "[", "3", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.813505237235305*^9, 3.8135054194946957`*^9, 
  3.813506991118051*^9, 3.813507919126927*^9},
 CellLabel->
  "Out[103]=",ExpressionUUID->"532a617b-30b5-4f4f-b55e-57a6c06fe878"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "+", 
          RowBox[{"2", 
           RowBox[{"f", "[", 
            RowBox[{
             FractionBox["1", "x"], ",", "n"}], "]"}]}], "+", 
          RowBox[{"3", 
           RowBox[{"f", "[", 
            RowBox[{
             FractionBox["x", 
              RowBox[{"x", "-", "1"}]], ",", "n"}], "]"}]}]}], "\[Equal]", 
         "x"}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.8135048260594053`*^9, 3.81350482607148*^9}, {
  3.813505056478435*^9, 3.813505130368132*^9}, {3.813505208110484*^9, 
  3.8135052150044518`*^9}, {3.8135052765539713`*^9, 3.813505278807876*^9}, {
  3.813505567858522*^9, 3.813505574542973*^9}},
 CellLabel->
  "In[104]:=",ExpressionUUID->"4e9afd22-5845-4aa8-950e-18bc179db96c"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "3", "]"}]], "Input",
 CellChangeTimes->{{3.813504845513647*^9, 3.8135048474228487`*^9}, {
  3.813505109221377*^9, 3.8135051097890463`*^9}, {3.8135051403430023`*^9, 
  3.813505180961503*^9}, {3.8135052517356997`*^9, 3.8135052895336*^9}, {
  3.81350543908039*^9, 3.81350546915981*^9}, {3.813505503004808*^9, 
  3.81350550336756*^9}, {3.813505579249201*^9, 3.813505593918486*^9}, {
  3.813505713478937*^9, 3.813505743772086*^9}, {3.8135057761986847`*^9, 
  3.813505782208042*^9}},
 CellLabel->
  "In[111]:=",ExpressionUUID->"5675651d-e8c0-4435-b1ea-b03c061dcc69"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "24"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"5", " ", 
           RowBox[{"b", "[", "1", "]"}]}], "24"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "12"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"b", "[", "1", "]"}], "12"]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "1", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "24"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "12"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"5", " ", 
           RowBox[{"b", "[", "1", "]"}]}], "24"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"b", "[", "1", "]"}], "12"]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "1", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "60"}], "}"}]], "Output",
 CellChangeTimes->{
  3.813504848970455*^9, {3.813505110230097*^9, 3.8135051816193647`*^9}, {
   3.813505252471683*^9, 3.8135052900533524`*^9}, {3.813505439415394*^9, 
   3.813505452459654*^9}, 3.8135054864532146`*^9, {3.8135055588749113`*^9, 
   3.813505615771502*^9}, {3.813505737598242*^9, 3.8135058043858423`*^9}, 
   3.81350794287166*^9, 3.81350885120888*^9},
 CellLabel->
  "Out[111]=",ExpressionUUID->"cfebd1c7-961b-490f-9ff5-8798d9cf3a54"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "3"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "3", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.813505657971726*^9, 3.813505702495915*^9}, {
   3.813505856361128*^9, 3.813505951680684*^9}, 3.81350649519394*^9, {
   3.8135067203573227`*^9, 3.813506731714675*^9}, {3.81350676515132*^9, 
   3.813506766156868*^9}, {3.813507939706362*^9, 3.813507939776822*^9}, {
   3.8135086788668013`*^9, 3.813508692527812*^9}, {3.813508839403384*^9, 
   3.8135088413291473`*^9}},
 CellLabel->
  "In[112]:=",ExpressionUUID->"bb63cb6d-3335-41bf-8d2a-b6e21db96c17"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"-", "2"}], "+", 
   RowBox[{"5", " ", "x"}], "+", 
   SuperscriptBox["x", "2"], "+", 
   RowBox[{"2", " ", 
    SuperscriptBox["x", "3"]}]}], 
  RowBox[{"24", " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", "x"}]]], "Output",
 CellChangeTimes->{
  3.81350570363623*^9, {3.8135058599990797`*^9, 3.81350595205895*^9}, 
   3.813506495662269*^9, 3.813506754415502*^9, 3.8135067879737377`*^9, 
   3.813507972442503*^9, 3.813508702631777*^9, 3.8135087593445663`*^9, 
   3.813508852139019*^9},
 CellLabel->
  "Out[112]=",ExpressionUUID->"f585aeb1-c19a-4ad9-94d2-66d13bd7b38b"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", " ", "//", " ", "Apart"}]], "Input",
 CellChangeTimes->{{3.8135087943215837`*^9, 3.81350879575986*^9}},
 CellLabel->
  "In[113]:=",ExpressionUUID->"feb70026-6ae8-4b28-8577-06a902d4cf0c"],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "8"], "+", 
  FractionBox["1", 
   RowBox[{"4", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"-", "1"}], "+", "x"}], ")"}]}]], "+", 
  FractionBox["1", 
   RowBox[{"12", " ", "x"}]], "+", 
  FractionBox["x", "12"]}]], "Output",
 CellChangeTimes->{3.813508796232889*^9, 3.813508855176696*^9},
 CellLabel->
  "Out[113]=",ExpressionUUID->"531da635-f869-4063-9328-00bd36e631ca"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"Another", ":", " ", 
    RowBox[{
     RowBox[{"f", 
      RowBox[{
       RowBox[{"(", "x", ")"}], "/", "x"}]}], " ", "\[Equal]", " ", 
     RowBox[{"f", 
      RowBox[{"(", 
       RowBox[{"2", "x"}], ")"}]}]}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], 
   "\[IndentingNewLine]", 
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.8136050763570127`*^9, 3.813605115822432*^9}, {
   3.813605257365159*^9, 3.813605259654187*^9}, {3.8136053173975143`*^9, 
   3.8136053220401154`*^9}, 3.8136054075429792`*^9, {3.813635337989945*^9, 
   3.8136353531212177`*^9}, {3.813635383314427*^9, 3.813635401244659*^9}, {
   3.813654153543305*^9, 3.813654154983891*^9}},
 CellLabel->"In[7]:=",ExpressionUUID->"e8f44cff-0d38-4287-8574-428bd4fc3930"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{
  3.813605322399168*^9, 3.813605408171554*^9, 3.813605508131374*^9, {
   3.813605908767952*^9, 3.8136059256582527`*^9}, 3.813635140033386*^9, 
   3.813635205963688*^9, 3.813635354691306*^9, {3.8136353944800386`*^9, 
   3.813635401537003*^9}, {3.813654144323443*^9, 3.813654170908374*^9}},
 CellLabel->"Out[8]=",ExpressionUUID->"9bb3c44d-8c53-47d0-ba38-fda7b42a4240"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         FractionBox[
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "x"], "\[Equal]", 
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"2", " ", "x"}], ",", "n"}], "]"}]}], ",", "x"}], "]"}]}], 
     "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.81363581488352*^9, 3.813635820034042*^9}, {
   3.8136369809425373`*^9, 3.813636981737424*^9}, {3.813637395006608*^9, 
   3.8136373954251747`*^9}, 3.813637609296712*^9, {3.813638111069549*^9, 
   3.813638111522077*^9}, 3.813654106101417*^9},
 CellLabel->"In[9]:=",ExpressionUUID->"09ac34d6-f9cd-42dc-83a1-d4ddd87a8717"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.813635549555217*^9, 3.813635552298292*^9}, {
  3.813635666030437*^9, 3.8136356661269083`*^9}, {3.8136357426452923`*^9, 
  3.813635755981214*^9}, {3.813637062425641*^9, 3.8136370625261793`*^9}, {
  3.813637125300922*^9, 3.813637125312858*^9}, {3.813638119888328*^9, 
  3.813638125193652*^9}, {3.813642440593869*^9, 3.813642440656591*^9}, {
  3.8136425045022917`*^9, 3.813642504628644*^9}, {3.8136540331479387`*^9, 
  3.813654060976853*^9}},
 CellLabel->"In[11]:=",ExpressionUUID->"a67b81f1-bfd4-436d-9ee1-40d53b5641c5"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "2"}], "}"}]], "Output",
 CellChangeTimes->{
  3.81364244104872*^9, 3.813642501492628*^9, 3.813654064111664*^9, 
   3.813654128973093*^9, {3.813654175538208*^9, 3.813654187217842*^9}},
 CellLabel->"Out[11]=",ExpressionUUID->"87b8f662-9db2-4964-8257-8cd2d3a00b14"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "+", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", 
           RowBox[{"(", 
            RowBox[{"1", "/", "#"}], ")"}]}]}], "&"}], ",", 
        RowBox[{"n", "-", "1"}], ",", "2"}], "]"}]}]}], ")"}], "/", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"c", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "+", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"d", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", 
           RowBox[{"(", 
            RowBox[{"1", "/", "#"}], ")"}]}]}], "&"}], ",", 
        RowBox[{"n", "-", "1"}], ",", "2"}], "]"}]}]}], 
    ")"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[12]:=",ExpressionUUID->"7047b3c7-5077-4696-85e7-4b6be5dd8186"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}], "+", 
   RowBox[{
    SqrtBox["x"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"c", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"c", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"c", "[", "2", "]"}]}], "+", 
   RowBox[{
    SqrtBox["x"], " ", 
    RowBox[{"d", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.813654189221879*^9},
 CellLabel->"Out[13]=",ExpressionUUID->"82372910-5879-4821-b148-39b30ac896f4"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         FractionBox[
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "x"], "\[Equal]", 
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"2", " ", "x"}], ",", "n"}], "]"}]}], ",", "x"}], "]"}]}], 
     "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellLabel->"In[14]:=",ExpressionUUID->"759a3f5d-ba90-4456-bdd9-b7be4bdefb2a"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.813654198779529*^9, 3.8136541995041323`*^9}},
 CellLabel->"In[15]:=",ExpressionUUID->"83867158-21bb-4231-9580-935830752bec"],

Cell[BoxData["$Aborted"], "Output",
 CellChangeTimes->{3.813655127498576*^9},
 CellLabel->"Out[15]=",ExpressionUUID->"733d9d73-af7d-40f0-ab82-c7e5db6bc96f"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"Another", ":", " ", 
    RowBox[{
     RowBox[{
      RowBox[{"f", 
       RowBox[{"(", "x", ")"}]}], "+", 
      RowBox[{"2", " ", "f", 
       RowBox[{"(", 
        FractionBox["1", 
         RowBox[{"1", "-", "x"}]], ")"}]}]}], "\[Equal]", 
     RowBox[{"x", " ", "\[IndentingNewLine]", 
      ButtonBox[
       RowBox[{
        RowBox[{"https", ":"}], "//", 
        RowBox[{
         RowBox[{
          RowBox[{"www", ".", "quora", ".", "com"}], "/", "How"}], "-", "do", 
         "-", "I", "-", "solve", "-", "f", "-", "x", "-", 
         RowBox[{
          RowBox[{"+", 
           RowBox[{"-", "2"}]}], "f"}], "-", "left", "-", "frac", "-", "1", 
         "-", "1", "-", "x", "-", "right", "-", "x", "-", "for", "-", "f"}]}],
       
       BaseStyle->"Hyperlink",
       ButtonData->{
         URL[
         "https://www.quora.com/How-do-I-solve-f-x-+-2f-left-frac-1-1-x-right-\
x-for-f"], None},
       ButtonNote->
        "https://www.quora.com/How-do-I-solve-f-x-+-2f-left-frac-1-1-x-right-\
x-for-f"]}]}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], 
   "\[IndentingNewLine]", 
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.814629738906489*^9, 3.814629779407857*^9}, {
  3.814629995800192*^9, 
  3.814630028643722*^9}},ExpressionUUID->"571def01-962f-4037-9479-\
a97b5ac5fa6b"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{{3.814629777954691*^9, 3.814629780000703*^9}},
 CellLabel->"Out[10]=",ExpressionUUID->"50ce8be9-e8a7-443d-817d-00d4d5f60a78"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "+", 
          RowBox[{"2", 
           RowBox[{"f", "[", 
            RowBox[{
             FractionBox["1", 
              RowBox[{"1", "-", "x"}]], ",", "n"}], "]"}]}]}], "\[Equal]", 
         "x"}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.813655029241785*^9, 3.813655032582995*^9}, {
  3.8146298027825537`*^9, 3.814629822677773*^9}},
 CellLabel->"In[11]:=",ExpressionUUID->"4f758a3c-7df1-489a-b05b-8ce513ebdbe1"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "3", "]"}]], "Input",
 CellChangeTimes->{{3.814629827454753*^9, 3.814629840438045*^9}},
 CellLabel->"In[15]:=",ExpressionUUID->"c6ac5a35-c663-49ad-bc41-51d244ee8c91"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "9"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "1", "]"}], "3"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"2", " ", 
          RowBox[{"b", "[", "1", "]"}]}], "3"]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"4", " ", 
           RowBox[{"b", "[", "1", "]"}]}], "9"]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "1", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "12"}], "}"}]], "Output",
 CellChangeTimes->{{3.814629829886037*^9, 3.814629841238141*^9}},
 CellLabel->"Out[15]=",ExpressionUUID->"32b83783-1eed-48af-8526-3aea64a9fa0d"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "3"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "3", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellLabel->"In[16]:=",ExpressionUUID->"5aaf7595-02f8-4586-b64d-4fba9150ae86"],

Cell[BoxData[
 FractionBox[
  RowBox[{"4", "-", 
   RowBox[{"6", " ", "x"}], "+", 
   RowBox[{"3", " ", 
    SuperscriptBox["x", "2"]}], "+", 
   SuperscriptBox["x", "3"]}], 
  RowBox[{"9", " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", "x"}]]], "Output",
 CellChangeTimes->{3.814629865122204*^9},
 CellLabel->"Out[16]=",ExpressionUUID->"e9925b9a-ed73-4ff9-9088-d962852a32c9"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", "//", "Apart"}]], "Input",
 CellChangeTimes->{{3.814629869290065*^9, 3.814629871319487*^9}},
 CellLabel->"In[17]:=",ExpressionUUID->"c55bdaf4-4963-42c8-84f5-c8f5b1722e7b"],

Cell[BoxData[
 RowBox[{
  FractionBox["4", "9"], "+", 
  FractionBox["2", 
   RowBox[{"9", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"-", "1"}], "+", "x"}], ")"}]}]], "-", 
  FractionBox["4", 
   RowBox[{"9", " ", "x"}]], "+", 
  FractionBox["x", "9"]}]], "Output",
 CellChangeTimes->{3.81462987182541*^9},
 CellLabel->"Out[17]=",ExpressionUUID->"e762486b-38f7-4510-a7e3-e54c11737bd1"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"Another", ":", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"2", " ", "f", 
      RowBox[{"(", "x", ")"}]}], " ", "+", " ", 
     RowBox[{"3", " ", "f", 
      RowBox[{"(", 
       RowBox[{"-", "x"}], ")"}]}]}], "\[Equal]", " ", 
    RowBox[{
     SuperscriptBox["x", "2"], "-", "x", "-", "1"}]}]}], " ", "*)"}]], "Input",\

 CellChangeTimes->{{3.8146300386022587`*^9, 
  3.814630088795641*^9}},ExpressionUUID->"59bc333e-1bbb-458f-997d-\
a289b4f5a206"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[18]:=",ExpressionUUID->"cfcaa5f4-5e85-4ff1-9889-17f7a5673caa"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.8146301007554827`*^9},
 CellLabel->"Out[19]=",ExpressionUUID->"12ed0028-42a8-4c4e-aa9c-20e1751dc71e"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"2", " ", 
           RowBox[{"f", "[", 
            RowBox[{"x", ",", "n"}], "]"}]}], "+", 
          RowBox[{"3", 
           RowBox[{"f", "[", 
            RowBox[{
             RowBox[{"-", "x"}], ",", "n"}], "]"}]}]}], "\[Equal]", 
         RowBox[{
          SuperscriptBox["x", "2"], "-", "x", "-", "1"}]}], ",", "x"}], 
       "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.81463011338137*^9, 3.814630133639057*^9}},
 CellLabel->"In[20]:=",ExpressionUUID->"2bf348f2-5a8a-44a9-9cdf-d2a028cc80c7"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.8146301360469007`*^9, 3.814630143758844*^9}},
 CellLabel->"In[23]:=",ExpressionUUID->"3890853f-f8ad-4ca0-a36a-58710eb736cd"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "0", "]"}], "5"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"b", "[", "0", "]"}], "5"]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "2"}], "}"}]], "Output",
 CellChangeTimes->{{3.814630138610314*^9, 3.814630144273979*^9}},
 CellLabel->"Out[23]=",ExpressionUUID->"4a9726ea-b7b8-42b4-8ef0-fec9979574ef"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "2", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.814630165424802*^9, 3.81463017690554*^9}},
 CellLabel->"In[24]:=",ExpressionUUID->"bf840938-7437-4981-be6e-3ebee057848c"],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "5"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "1"}], "+", 
    RowBox[{"5", " ", "x"}], "+", 
    SuperscriptBox["x", "2"]}], ")"}]}]], "Output",
 CellChangeTimes->{3.814630177374551*^9},
 CellLabel->"Out[24]=",ExpressionUUID->"2e7e239e-b338-472a-8433-5f6f39d2b1ab"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", " ", "//", " ", "Apart"}]], "Input",
 CellChangeTimes->{{3.8146301813467083`*^9, 3.8146301832968397`*^9}},
 CellLabel->"In[25]:=",ExpressionUUID->"195e2848-96c7-4fca-b3bb-09c0a9b49e37"],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", 
   FractionBox["1", "5"]}], "+", "x", "+", 
  FractionBox[
   SuperscriptBox["x", "2"], "5"]}]], "Output",
 CellChangeTimes->{3.8146301839967823`*^9},
 CellLabel->"Out[25]=",ExpressionUUID->"4c30276e-1563-4984-9eee-7bf776ffa630"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"ff", "[", "x_", "]"}], ":=", " ", 
  RowBox[{
   RowBox[{"-", 
    FractionBox["1", "5"]}], "+", "x", "+", 
   FractionBox[
    SuperscriptBox["x", "2"], "5"]}]}]], "Input",
 CellChangeTimes->{{3.814630192825633*^9, 3.814630200101433*^9}},
 CellLabel->"In[26]:=",ExpressionUUID->"29a20ac2-00f0-4a90-b39c-f85c88e5b60c"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"ff", "'"}], "[", "1", "]"}]], "Input",
 CellChangeTimes->{{3.8146302019401493`*^9, 3.814630203329705*^9}},
 CellLabel->"In[27]:=",ExpressionUUID->"92f83b6b-a6cf-4307-a356-68489867c182"],

Cell[BoxData[
 FractionBox["7", "5"]], "Output",
 CellChangeTimes->{3.814630203682948*^9},
 CellLabel->"Out[27]=",ExpressionUUID->"bb40301a-f87c-4d8d-a97e-a88170276253"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"Another", ":", " ", 
    RowBox[{
     RowBox[{"f", 
      RowBox[{"(", 
       RowBox[{"x", "+", 
        FractionBox["1", "x"]}], ")"}]}], "\[Equal]", 
     RowBox[{
      SuperscriptBox["x", "2"], "+", 
      FractionBox["1", 
       SuperscriptBox["x", "2"]]}]}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], 
   "\[IndentingNewLine]", 
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.814630340034309*^9, 3.814630411178987*^9}},
 CellLabel->"In[28]:=",ExpressionUUID->"24e4783f-3b7d-4c2a-8c9d-74e2b164fd8e"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.814630411971355*^9},
 CellLabel->"Out[29]=",ExpressionUUID->"b3ee335f-584e-4631-b14c-a654293ecadb"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", "+", 
            FractionBox["1", "x"]}], ",", "n"}], "]"}], "\[Equal]", 
         RowBox[{
          SuperscriptBox["x", "2"], "+", 
          FractionBox["1", 
           SuperscriptBox["x", "2"]]}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.8146304303687277`*^9, 3.814630457296565*^9}},
 CellLabel->"In[30]:=",ExpressionUUID->"71e8ad2d-ed6c-4048-9957-5dcd07b58265"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.81463046160176*^9, 3.8146304962305107`*^9}},
 CellLabel->"In[44]:=",ExpressionUUID->"3813e7aa-fba9-431d-99d2-afedb2bc4400"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
       RowBox[{
        RowBox[{"-", "2"}], " ", 
        RowBox[{"b", "[", "0", "]"}]}]}], ",", 
      RowBox[{
       RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
       RowBox[{"b", "[", "0", "]"}]}], ",", 
      RowBox[{
       RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], "}"}], ",", 
   "1"}], "}"}]], "Output",
 CellChangeTimes->{{3.81463046472703*^9, 3.8146305619775143`*^9}},
 CellLabel->"Out[44]=",ExpressionUUID->"c3758609-e282-4586-845c-cf1cca55a4ac"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "1", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.8146305102367373`*^9, 3.8146305655376883`*^9}},
 CellLabel->"In[45]:=",ExpressionUUID->"616fe1fa-3a85-4768-910e-516d8f0e38c1"],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", "2"}], "+", 
  SuperscriptBox["x", "2"]}]], "Output",
 CellChangeTimes->{{3.814630521546083*^9, 3.814630565865052*^9}},
 CellLabel->"Out[45]=",ExpressionUUID->"93551b9f-313f-4257-8ae3-f36a864563fd"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", "/.", 
  RowBox[{"x", "\[Rule]", " ", "5"}]}]], "Input",
 CellChangeTimes->{{3.81463057490561*^9, 3.814630579384009*^9}},
 CellLabel->"In[46]:=",ExpressionUUID->"240b4d8e-2235-4a1f-a1a7-ad4ebafb7574"],

Cell[BoxData["23"], "Output",
 CellChangeTimes->{3.8146305798293257`*^9},
 CellLabel->"Out[46]=",ExpressionUUID->"215891b6-bc88-45ae-83e2-7dcd80867fca"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{"Another", " ", 
     RowBox[{"(", 
      RowBox[{"difference", " ", "equation"}], ")"}]}], ":", " ", 
    RowBox[{
     RowBox[{
      RowBox[{"f", 
       RowBox[{"(", 
        RowBox[{"x", "+", "1"}], ")"}]}], "-", 
      RowBox[{"f", 
       RowBox[{"(", "x", ")"}]}]}], "\[Equal]", " ", 
     RowBox[{"x", "\[IndentingNewLine]", 
      ButtonBox[
       RowBox[{
        RowBox[{"https", ":"}], "//", 
        RowBox[{
         RowBox[{
          RowBox[{"www", ".", "quora", ".", "com"}], "/", "What"}], "-", "is",
          "-", "f", "-", "x", "-", "when", "-", "f", "-", "x", "-", "1", "-", 
         "f", "-", "x", "-", "x"}]}],
       BaseStyle->"Hyperlink",
       ButtonData->{
         URL["https://www.quora.com/What-is-f-x-when-f-x-1-f-x-x"], None},
       ButtonNote->
        "https://www.quora.com/What-is-f-x-when-f-x-1-f-x-x"]}]}]}], 
   "\[IndentingNewLine]", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{"Total", "@", 
     RowBox[{"Array", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"a", "[", "#", "]"}], " ", 
         RowBox[{"x", "^", "#"}]}], "&"}], ",", 
       RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
   
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.81463059204069*^9, 3.814630604382299*^9}, {
  3.814630924355371*^9, 3.8146309917029667`*^9}},
 CellLabel->"In[60]:=",ExpressionUUID->"7bb25d61-5abd-4eb5-b0b2-dca7d2d090da"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.814630604976173*^9, 3.814630992265297*^9},
 CellLabel->"Out[61]=",ExpressionUUID->"e1fe63b5-32fa-459d-9b04-2b2c68f34bb8"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", "+", "1"}], ",", "n"}], "]"}], "-", 
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}]}], "\[Equal]", "x"}], ",", "x"}], 
       "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.814630619946887*^9, 3.8146306336108723`*^9}},
 CellLabel->"In[75]:=",ExpressionUUID->"528d87d5-04ad-4fd9-a5ea-e93fcac7e8d2"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.814630637827321*^9, 3.814630645514736*^9}, {
  3.814631025849573*^9, 3.8146310270723047`*^9}, {3.814631155250842*^9, 
  3.8146311564793797`*^9}},
 CellLabel->"In[81]:=",ExpressionUUID->"3e16d050-1fe4-4966-bca3-aed849790119"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
       RowBox[{"-", 
        FractionBox["1", "2"]}]}], ",", 
      RowBox[{
       RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
       FractionBox["1", "2"]}]}], "}"}], "}"}], ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{{3.8146306403788147`*^9, 3.814630668435638*^9}, 
   3.814630714261918*^9, {3.8146309955888157`*^9, 3.8146310275333977`*^9}, 
   3.8146310674549923`*^9, {3.814631132364931*^9, 3.814631156953875*^9}, 
   3.814631198466507*^9},
 CellLabel->"Out[81]=",ExpressionUUID->"c7fb006e-ab47-4768-8202-c422366d851c"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "1", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.81463065010885*^9, 3.814630671143035*^9}, {
  3.8146310152093782`*^9, 3.814631037650626*^9}, {3.8146311356264877`*^9, 
  3.814631190745523*^9}},
 CellLabel->"In[82]:=",ExpressionUUID->"48a6947f-e03a-412b-8174-7d58f6ba40c0"],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "2"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "x"}], "+", 
    SuperscriptBox["x", "2"], "+", 
    RowBox[{"2", " ", 
     RowBox[{"a", "[", "0", "]"}]}]}], ")"}]}]], "Output",
 CellChangeTimes->{{3.814630664200156*^9, 3.81463067149594*^9}, 
   3.814630714924802*^9, {3.814630998067272*^9, 3.814631037968223*^9}, 
   3.814631068646147*^9, 3.8146311370588923`*^9, {3.814631168535617*^9, 
   3.814631199090185*^9}},
 CellLabel->"Out[82]=",ExpressionUUID->"57eb1297-c5d4-468e-a934-e3de0e0cde16"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", " ", "//", " ", "Apart"}]], "Input",
 CellChangeTimes->{{3.81463071640807*^9, 3.8146307182786083`*^9}},
 CellLabel->"In[83]:=",ExpressionUUID->"70e187af-f4fe-4be7-bbf9-57219f8e872c"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   FractionBox["1", "2"], " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", "x"}], "+", 
  RowBox[{"a", "[", "0", "]"}]}]], "Output",
 CellChangeTimes->{
  3.814630718713085*^9, {3.814631057280767*^9, 3.814631078631435*^9}, 
   3.81463120268589*^9},
 CellLabel->"Out[83]=",ExpressionUUID->"3224752c-9684-4a74-99a6-55acc1addbee"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Expand", "[", 
  RowBox[{
   RowBox[{
    FractionBox["1", "2"], " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", "x"}], "+", 
   RowBox[{"a", "[", "0", "]"}]}], "]"}]], "Input",
 NumberMarks->False,
 CellLabel->"In[84]:=",ExpressionUUID->"2a8db335-a4a6-4fd9-8689-3b20c0265675"],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", 
   FractionBox["x", "2"]}], "+", 
  FractionBox[
   SuperscriptBox["x", "2"], "2"], "+", 
  RowBox[{"a", "[", "0", "]"}]}]], "Output",
 CellChangeTimes->{3.814631207157961*^9},
 CellLabel->"Out[84]=",ExpressionUUID->"40b3ede0-b42e-4175-b786-8691c2c21361"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"Another", ":", " ", 
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", "x", ")"}], " ", "f", 
     RowBox[{"(", 
      FractionBox["1", "x"], ")"}]}], "\[Equal]", 
    RowBox[{
     RowBox[{"f", 
      RowBox[{"(", "x", ")"}]}], "+", 
     RowBox[{"f", 
      RowBox[{"(", 
       FractionBox["1", "x"], ")"}]}]}]}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.8146314045106173`*^9, 3.814631439623879*^9}},
 CellLabel->"In[85]:=",ExpressionUUID->"584cad65-6afc-4287-a60d-095384b620bf"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.8146318787247458`*^9, 3.8146318804085197`*^9}},
 CellLabel->
  "In[109]:=",ExpressionUUID->"3c8d02b5-56c8-4694-8b14-a67945af62d6"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.814631462621937*^9, 3.814631880876231*^9},
 CellLabel->
  "Out[110]=",ExpressionUUID->"ae8c9439-78e0-4e31-aa42-7a122eb6a570"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], 
          RowBox[{"f", "[", 
           RowBox[{
            FractionBox["1", "x"], ",", "n"}], "]"}]}], "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "+", 
          RowBox[{"f", "[", 
           RowBox[{
            FractionBox["1", "x"], ",", "n"}], "]"}]}]}], ",", "x"}], "]"}]}],
      "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{
        RowBox[{"DeleteDuplicates", "@", "#"}], ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.8146314813780527`*^9, 3.814631514020608*^9}},
 CellLabel->
  "In[111]:=",ExpressionUUID->"8c499ba9-09f6-485d-bbd8-1b88eb1fbf06"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "1", "]"}]], "Input",
 CellChangeTimes->{{3.814631515942017*^9, 3.814631522835994*^9}},
 CellLabel->
  "In[112]:=",ExpressionUUID->"2188de95-1710-4081-8a7c-c032dadf5972"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "2"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "4"}], "}"}]], "Output",
 CellChangeTimes->{{3.814631520469253*^9, 3.814631587068222*^9}, 
   3.814631884397904*^9},
 CellLabel->
  "Out[112]=",ExpressionUUID->"0bc2d9ff-52fb-4b98-ab09-138914ff860e"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "1"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"%", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "2", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.814631546734108*^9, 3.814631590464841*^9}, {
  3.8146318944362593`*^9, 3.814631894497547*^9}},
 CellLabel->
  "In[113]:=",ExpressionUUID->"1642d6e8-4ec0-4526-b7e0-6f39d07a686d"],

Cell[BoxData[
 RowBox[{"1", "-", "x"}]], "Output",
 CellChangeTimes->{{3.814631554679977*^9, 3.814631590795558*^9}, 
   3.8146318949484262`*^9},
 CellLabel->
  "Out[113]=",ExpressionUUID->"302a1a39-a80b-4a2f-ac16-0c533ebeb8c2"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"s2", "=", 
  RowBox[{"sol", "[", "2", "]"}]}]], "Input",
 CellChangeTimes->{{3.814631760529542*^9, 3.814631760601625*^9}, {
   3.814631832328583*^9, 3.8146318335015173`*^9}, 3.8146319219010344`*^9, {
   3.81463198665615*^9, 3.814631987286565*^9}},
 CellLabel->
  "In[119]:=",ExpressionUUID->"096237f9-af46-4bb2-8d77-9fe195fa833f"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "2"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "8"}], "}"}]], "Output",
 CellChangeTimes->{{3.814631761153755*^9, 3.8146317957741213`*^9}, 
   3.814631834443132*^9, 3.8146319226959047`*^9, {3.8146319764969397`*^9, 
   3.814631987630712*^9}},
 CellLabel->
  "Out[119]=",ExpressionUUID->"ffd115a3-926d-494a-aae8-2d5fe4f382f3"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", " ", "2"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"s2", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", "6"}], "}"}]}], "]"}], "//", 
  "Simplify"}]], "Input",
 CellChangeTimes->{{3.814631773059066*^9, 3.814631798797906*^9}, {
   3.814631837664792*^9, 3.8146318404146023`*^9}, 3.81463193151858*^9, {
   3.814631979289124*^9, 3.814632029777975*^9}},
 CellLabel->
  "In[126]:=",ExpressionUUID->"79a95482-8d3d-45a0-80fc-b9d30f524b69"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   RowBox[{"1", "-", "x"}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "2"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "2"]}], ",", 
   RowBox[{"1", "+", "x"}], ",", "2"}], "}"}]], "Output",
 CellChangeTimes->{{3.814631778524825*^9, 3.814631799224771*^9}, 
   3.814631840817371*^9, 3.814631931899025*^9, {3.814631980824004*^9, 
   3.814632030473322*^9}},
 CellLabel->
  "Out[126]=",ExpressionUUID->"9ce51eb7-7d5c-467f-9581-3d25430ace45"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"s3", "=", 
  RowBox[{"sol", "[", "3", "]"}]}]], "Input",
 CellChangeTimes->{{3.814631961853238*^9, 3.81463196257272*^9}, {
  3.8146320423321857`*^9, 3.8146320429856358`*^9}},
 CellLabel->
  "In[132]:=",ExpressionUUID->"4df27f3a-8ead-45c8-8ebe-445fbdabda8b"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "2"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "18"}], "}"}]], "Output",
 CellChangeTimes->{3.814631963066719*^9, 3.814632046476757*^9, 
  3.81463212081423*^9},
 CellLabel->
  "Out[132]=",ExpressionUUID->"e4dc27f4-bb39-45b2-a404-dcc081c8405f"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", " ", "3"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"s3", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"Length", "[", 
       RowBox[{"s3", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "Simplify"}]], "Input",
 CellChangeTimes->{{3.8146319715978107`*^9, 3.814631971654018*^9}, {
  3.8146320497744513`*^9, 3.81463207605991*^9}},
 CellLabel->
  "In[133]:=",ExpressionUUID->"570d149c-7bdf-4caa-aafc-67dbaa76bcc4"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   RowBox[{"1", "-", "x"}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "2"]}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "3"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "3"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "2"]}], ",", 
   RowBox[{"1", "+", "x"}], ",", "2"}], "}"}]], "Output",
 CellChangeTimes->{{3.81463206338908*^9, 3.814632076735107*^9}, 
   3.814632121525592*^9},
 CellLabel->
  "Out[133]=",ExpressionUUID->"e27b3e99-2770-42a9-ab0c-678d98385076"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"s4", "=", 
  RowBox[{"sol", "[", "4", "]"}]}]], "Input",
 CellChangeTimes->{{3.814632105077511*^9, 3.814632116829619*^9}},
 CellLabel->
  "In[131]:=",ExpressionUUID->"f99702d9-c553-43b9-bebb-ff0e4ffd3ce5"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "2"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "4", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "64"}], "}"}]], "Output",
 CellChangeTimes->{{3.8146321063985453`*^9, 3.814632117305982*^9}},
 CellLabel->
  "Out[131]=",ExpressionUUID->"1774f69e-364b-47cb-aebc-85f534c695d2"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", " ", "4"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"s4", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"Length", "[", 
       RowBox[{"s3", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "Simplify"}]], "Input",
 CellChangeTimes->{{3.814632128607085*^9, 3.8146321304796886`*^9}},
 CellLabel->
  "In[134]:=",ExpressionUUID->"917d2874-562d-4157-b967-9d123ecf10ad"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   RowBox[{"1", "-", "x"}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "2"]}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "3"]}], ",", 
   RowBox[{"1", "-", 
    SuperscriptBox["x", "4"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "4"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "3"]}], ",", 
   RowBox[{"1", "+", 
    SuperscriptBox["x", "2"]}]}], "}"}]], "Output",
 CellChangeTimes->{3.814632130942346*^9},
 CellLabel->
  "Out[134]=",ExpressionUUID->"6a6e668f-4113-4bba-8108-6337405d268f"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"and", " ", "so", " ", "on"}], "..."}], " ", "1"}], " ", 
   "\[PlusMinus]", " ", 
   SuperscriptBox["x", "k"]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.814632135415661*^9, 3.814632139415161*^9}, {
  3.814636069454323*^9, 
  3.81463608075784*^9}},ExpressionUUID->"87c31783-a6ce-4140-be75-\
7ff253074d0d"],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"1", "+", 
   SuperscriptBox["x", "k"]}]}]], "Input",
 CellChangeTimes->{{3.815680308894911*^9, 3.815680336553297*^9}, {
  3.81568079917527*^9, 3.8156808072049847`*^9}, {3.818500690998761*^9, 
  3.818500691612986*^9}},
 CellLabel->"In[40]:=",ExpressionUUID->"3628a082-23c8-4488-9d97-f121c0e579e1"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    RowBox[{"Solve", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"f", "[", "3", "]"}], "\[Equal]", "28"}], ",", " ", "k"}], 
     "]"}], "/.", 
    RowBox[{
     RowBox[{"C", "[", "1", "]"}], "\[Rule]", " ", "0"}]}], "//", 
   "FullSimplify"}], "//", "Flatten"}]], "Input",
 CellChangeTimes->{{3.8156805687356853`*^9, 3.815680645907333*^9}},
 CellLabel->"In[41]:=",ExpressionUUID->"e3b2eab5-152d-45a8-911b-d0ec9472eddc"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"k", "\[Rule]", "3"}], "}"}]], "Output",
 CellChangeTimes->{{3.815680793204074*^9, 3.815680808669656*^9}},
 CellLabel->"Out[41]=",ExpressionUUID->"0ef1d87c-c281-4a18-b424-5f4e971a1a87"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{"Another", ":", " ", 
     RowBox[{
      RowBox[{"f", 
       RowBox[{"(", 
        RowBox[{"x", "+", "1"}], ")"}]}], " ", "+", " ", 
      RowBox[{"f", 
       RowBox[{"(", 
        RowBox[{"x", "-", "1"}], ")"}]}]}]}], " ", "=", " ", 
    RowBox[{
     SuperscriptBox["x", "2"], "\[IndentingNewLine]", 
     ButtonBox[
      RowBox[{
       RowBox[{"https", ":"}], "//", 
       RowBox[{
        RowBox[{
         RowBox[{"www", ".", "quora", ".", "com"}], "/", "How"}], "-", "do", 
        "-", "i", "-", "find", "-", "the", "-", "inverse", "-", "of", "-", 
        "f", "-", "x", "-", "Given", "-", "f", "-", "x", "-", "1", "-", "f", 
        "-", "x", "-", "1", "-", "x", "-", "2", "-", "I", "-", "have", "-", 
        "subtituted", "-", "a", "-", "x", "-", "1", "-", "and", "-", "a", "-",
         "x", "-", "1", "-", "and", "-", "got", "-", "f", "-", "x", "-", "f", 
        "-", "x", "-", "2", "-", "x", "-", "1", "-", "2", "-", "and", "-", 
        "f", "-", "x", "-", "2", "-", "f", "-", "x", "-", "x", "-", "1", "-", 
        "2", "-", "Combining", "-", "those", "-", "equations", "-", "I", "-", 
        "got", "-", "f", "-", "x", "-", "2", "-", "f", "-", "x", "-", "2", 
        "-", 
        RowBox[{"4", "x"}], "-", "I", "-", "could", "-", "not", "-", "even", 
        "-", "find", "-", "f", "-", "x", "-", "Please"}]}],
      BaseStyle->"Hyperlink",
      ButtonData->{
        URL[
        "https://www.quora.com/How-do-i-find-the-inverse-of-f-x-Given-f-x-1-f-\
x-1-x-2-I-have-subtituted-a-x-1-and-a-x-1-and-got-f-x-f-x-2-x-1-2-and-f-x-2-f-\
x-x-1-2-Combining-those-equations-I-got-f-x-2-f-x-2-4x-I-could-not-even-find-\
f-x-Please"], None},
      ButtonNote->
       "https://www.quora.com/How-do-i-find-the-inverse-of-f-x-Given-f-x-1-f-\
x-1-x-2-I-have-subtituted-a-x-1-and-a-x-1-and-got-f-x-f-x-2-x-1-2-and-f-x-2-f-\
x-x-1-2-Combining-those-equations-I-got-f-x-2-f-x-2-4x-I-could-not-even-find-\
f-x-Please"]}]}], "\[IndentingNewLine]", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{"Total", "@", 
     RowBox[{"Array", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"a", "[", "#", "]"}], " ", 
         RowBox[{"x", "^", "#"}]}], "&"}], ",", 
       RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
   
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.8163250702858458`*^9, 3.816325145753942*^9}, {
  3.816326129929203*^9, 3.8163261491121197`*^9}},
 CellLabel->"In[29]:=",ExpressionUUID->"dbcc904e-2525-47de-bb8f-9edd6e5f341c"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.8163251506431847`*^9, 3.8163261503344*^9},
 CellLabel->"Out[30]=",ExpressionUUID->"3637c766-34b9-4b7b-bf6a-329237d3017b"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", "+", "1"}], ",", "n"}], "]"}], "+", 
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", "-", "1"}], ",", "n"}], "]"}]}], "\[Equal]", 
         SuperscriptBox["x", "2"]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.816326086750937*^9, 3.8163260900254383`*^9}, {
  3.816326265364793*^9, 3.816326281226544*^9}},
 CellLabel->"In[38]:=",ExpressionUUID->"f874733b-0217-45a5-814e-53a7d8acf1d6"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", "=", 
  RowBox[{"sol", "[", "2", "]"}]}]], "Input",
 CellChangeTimes->{{3.816326308018092*^9, 3.8163263112811317`*^9}},
 CellLabel->"In[47]:=",ExpressionUUID->"7733d651-30c2-4cd6-9a9e-ff2b1e9c43de"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
       RowBox[{"-", 
        FractionBox["1", "2"]}]}], ",", 
      RowBox[{
       RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
       FractionBox["1", "2"]}]}], "}"}], "}"}], ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{3.8163263119192266`*^9, 3.816326414564521*^9},
 CellLabel->"Out[47]=",ExpressionUUID->"d557611d-b187-43b8-b8fd-d9b2f1060134"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"sol2", "[", 
     RowBox[{"[", "1", "]"}], "]"}], "[", 
    RowBox[{"[", "1", "]"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.816325303682775*^9, 3.816325492061234*^9}, {
  3.816325523690908*^9, 3.8163256109940567`*^9}, {3.8163256955979643`*^9, 
  3.816325705562316*^9}, {3.816326170194887*^9, 3.816326170273652*^9}, {
  3.816326320296322*^9, 3.816326423367837*^9}},
 CellLabel->"In[49]:=",ExpressionUUID->"b4a5a18b-9322-4995-9c5b-d516b7062fec"],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "2"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "1"}], "+", 
    SuperscriptBox["x", "2"]}], ")"}]}]], "Output",
 CellChangeTimes->{
  3.8163254938476553`*^9, {3.81632557605995*^9, 3.816325611604929*^9}, {
   3.816325696674902*^9, 3.816325707153298*^9}, 3.8163260940418587`*^9, {
   3.8163261579106283`*^9, 3.8163261708183107`*^9}, {3.816326323633882*^9, 
   3.816326360523275*^9}, {3.816326402265819*^9, 3.816326423740779*^9}},
 CellLabel->"Out[49]=",ExpressionUUID->"d36bea24-8db8-45bb-97dc-31da798c2fd2"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"%", "//", "Apart"}]], "Input",
 CellChangeTimes->{{3.816326442483822*^9, 3.816326444090461*^9}},
 CellLabel->"In[50]:=",ExpressionUUID->"a11bcf61-0da1-44d0-aa29-d960a8cbb980"],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", 
   FractionBox["1", "2"]}], "+", 
  FractionBox[
   SuperscriptBox["x", "2"], "2"]}]], "Output",
 CellChangeTimes->{3.816326444444765*^9},
 CellLabel->"Out[50]=",ExpressionUUID->"939716cb-2a6f-47c4-bd7c-213da5a373d9"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"Inverse", ":"}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.8163268714815607`*^9, 
  3.8163268747930603`*^9}},ExpressionUUID->"7e906a0b-4d47-4808-ac6c-\
13473aa1fa2e"],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{
   RowBox[{"-", 
    FractionBox["1", "2"]}], "+", 
   FractionBox[
    SuperscriptBox["x", "2"], "2"]}]}]], "Input",
 CellChangeTimes->{{3.816326648333048*^9, 3.816326659922016*^9}},
 CellLabel->"In[54]:=",ExpressionUUID->"a4c993d7-2d8b-4286-b78a-6390a9feb8f7"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{"y", "==", 
    RowBox[{"f", "[", "x", "]"}]}], ",", " ", "x"}], "]"}]], "Input",
 CellChangeTimes->{{3.816326663376626*^9, 3.8163266910008183`*^9}},
 CellLabel->"In[55]:=",ExpressionUUID->"bff2412d-8b38-4082-b100-973791746dc3"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"x", "\[Rule]", 
     RowBox[{"-", 
      SqrtBox[
       RowBox[{"1", "+", 
        RowBox[{"2", " ", "y"}]}]]}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", "\[Rule]", 
     SqrtBox[
      RowBox[{"1", "+", 
       RowBox[{"2", " ", "y"}]}]]}], "}"}]}], "}"}]], "Output",
 CellChangeTimes->{{3.816326669327586*^9, 3.816326692039337*^9}, 
   3.816326878435933*^9},
 CellLabel->"Out[55]=",ExpressionUUID->"b0fc3738-6767-4e56-a542-1d3fbb5cc21a"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"General", " ", "Solution", " ", 
   RowBox[{"example", ":"}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.816326879810212*^9, 
  3.816326890236754*^9}},ExpressionUUID->"79414368-6582-49e7-a872-\
d0f13b68a52a"],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
   RowBox[{
    FractionBox[
     RowBox[{
      SuperscriptBox["x", "2"], "-", "1"}], "2"], "+", 
    RowBox[{"g", "[", "x", "]"}]}]}], "\[IndentingNewLine]", 
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"g", "[", "x", "]"}], " ", "must", " ", "have", " ", "period", 
     " ", "of", " ", "4"}], ":"}], " ", "*)"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"g", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"Sin", "[", 
   RowBox[{
    FractionBox["\[Pi]", "2"], "x"}], "]"}]}]}], "Input",
 CellChangeTimes->{{3.816326891652914*^9, 3.816326958807981*^9}},
 CellLabel->"In[76]:=",ExpressionUUID->"2eb0af19-07bb-4847-893c-5604ababcddc"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     FractionBox[
      RowBox[{
       SuperscriptBox["x", "2"], "-", "1"}], "2"], ",", " ", 
     RowBox[{"g", "[", "x", "]"}], ",", 
     RowBox[{"f", "[", "x", "]"}]}], "}"}], ",", " ", 
   RowBox[{"{", 
    RowBox[{"x", ",", " ", 
     RowBox[{"-", "4"}], ",", " ", "4"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.816326960281057*^9, 3.816327004192423*^9}, {
  3.81632717510938*^9, 3.816327286290769*^9}},
 CellLabel->"In[87]:=",ExpressionUUID->"2a8b8ac3-c4c6-4574-8bca-3720496f504c"],

Cell[BoxData[
 GraphicsBox[{{{}, {}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJw1mHk01H/Ux2XJbiSSGGvWSki0f75UkhCShBLZfrYkWSIklSwlSSohSpSd
QuGOLXvE2MbMfMueXYiR5ZnnnOf563Ne/3zOve/3+55zz5Wxv2bmyMrCwiK0
iYXlf19VtG18Y4NAcgveTd7YIGLXGHIRjBUCyfyzMWV+kYjlF6krzC8SSDcL
vHx+TxAxDSUj+5EJAsl+rsihr4eIaW8Jp7T0EEjjz7X3t+QRMWxwrulFHoGU
FrIvYc2WiJk9aP24z5ZAKi9bu/C4SgLzawv3cK4UIL3u7ad4PhTHyGTe8So7
ftLFa84SVivbMSLpmrC2BB9JUPb+iYCCbdgnt1oOAo2HFDI8HLo3Rhi7Eq0+
2xfGTXqm1/c8ykIIW44LjADERYpVuXRG8oogli5AF0v8tZmEiaUKtZXyYUkG
6t6dDzlIK3p5ZzzXubAwdzt/5dPsJFJ4o7ZWKQcWyPbBz3uKlaRxJ9KWw4EV
swtt29mduYlUKPnTPODfOnpWMZyY5MpCogs7yG/kMdAXXIkvW3YdhG+zKSnZ
LSA5QUzXPPkfuFzxkgnMnkGFKhGuA4PLIGnzccG8dAThdQXSWlp/QdqKQXjx
kYaGdz/TgM/zYJz1qsIqvh1tCc7SUt00B2M/088Jxeej+5+VGgSXZiGW9ZpL
ZHseWpl+f3F+chYEycmN5QG5aMA2I6i0dxZmlbe7P/j+ERXqpFfr5s8CKfW6
evGdDGTG8dr4gu0sSPuGBPZIJCJFMZlWlsAZYPnqqWd1OxZ2a83sfjc/CUr2
ZY4hB0vASTUkpIw2CQ0XFk5P9pRAigKho7V+EtqPmPP/ulkKQqJ7/f6+mgT2
rjB1+/wyWFryrDp1YhIEj72aqZYqh+qy6fPj8RNgEmAjs3GDBJZHp4P3ao0D
/c85WVVCHcTtD/5xXHoceq+S7g9Y1EHzHoGdljzjcN8QKFbJdXBMUrUxlP4b
FsJ9TJ/v/gZyGx5bO+7/Btm48gCkVw/TpKn3Pj1jUE6s+WTo2Qh3T0y1l/mP
wiQjq9/qSStU60+e1bw6CpUv1MLs21uBxWjie67RKIzzSB5XI3yHoPO/W9Jl
R+GY3R5hyejv4Os03BDTMgIDNOK87t02cImgkeykRyCBV67DsqgdDFta87kb
hmCgtu+/LXc6ILK9ZW944RDIqkQO/k3qgAZyc+560hDwL3u4QmkHnKQ1Zs9f
H4KfrW0pjJkOODZdl0kTH4IIu9VMoUudoCZYmVpwbRB6lRiaPepkEDHPjb0o
OgD+2nf2fajrgkKFqC8n2QYg66H/AVNqF5gwnIfUZ36B7CTw4n+6IDJZ5gBP
/S/48+sYMU+qG9bH4vEvvr9gPXZtQtq/G0ZuB6pKdP+EptXbB87s7AHPs08S
K5Rx2MMbFfPlv17IXDCYfLYJh1CncJvDt3th4AU75tlHB4mOVkZabC9YDPmN
Sj6kgxrFJEWhpBdQwGWt0DEa1OaIJmyw9oFg+i6y7nsqpNhnaiw/64PCv3WE
hp0UkGbrVmTPosCjD2nVxxh90OKZLV/5hQJul0NufmrtA/e31HtWLRTY+e1A
f5pvHzzyMPA4NEOBhGcfM2439IK5n3Vvwr5+uKX55Og+9x6YdjknVf25HyzG
PGazsB6Yv/FNN+1bP+xLMkiXFumBKf2nDQ7d/TDJxs4tUNkN1SaH/sUu9sPl
Tl/yGKEbZnoqVu9pUEHX28YtuYgMFrUvvQYyqCCpcFBSJIIMxZ4fysaKqfCv
T+RHlA0ZOAe3Ov6opkKxTptWAAcZHisvDJymU0Fhiy6r+YVOkFMdv9WzlQY8
+YovuP/9AAuRbHHHABq86x+O+R3+HWatH0gbqNKBErQt5fW+7/BG/sS5aW06
CEieyjcdaAUjA1qxvw4dfK9kdpShVnjy9i07Zk4HvWFX0ciVZmhNetCd6E+H
semZVBWvRhBLSj5+o5zpS6x0IV2yEZxEZBxu1dLBRN20Jq61AXJ+hxQ5t9Ch
9Ebh8IpKAyild9AX+unwcNlHpXn4G5Avtjj3MeigwrpS5GZdC2tdgseb1XC4
nK5SJ8VTCyPrlqb7tXGIO2Hd3VlaA41htx/dO4rDyv3y5cPbaiDolfWz3tM4
NPOGHOX7UQUJgu5d+nY4sOTkGZNCqsBwy3OrISccNI1/2vqoVsGae7KAkzsO
SbE6YdQoEpzZr8sm4YdDrwrXo4sxlfCamy5pFoUD5vFfftV4BcgMTOyQisUh
M6+pQ1m/AlJFuk92xOMQoBkjusJaDpExSGniNQ47jgm9eelfBlurnKRGc3EI
C71Rw9pdCo7XL8QNF+IwUU0edt1XCgVBOzWbPuNQfuq5yuHpz6D99hy7XiUO
l0wlivuvFsOS1uYezhYcJucXD96sLYLhX59WFNtwCEpoAwH5Igi30Tfe28Gs
vz+sWWekAHpIIvfme3DYHWxj2q9XAFEfTFzyKTh8ldbq8XmfD3NXPQPNaTgU
N3fUnf2RC50nzdTNBnAgC97/ERaXA8ae28kfh3CYP3+Q9ulcNgQ5VWVMjuCg
8TNlQbw7CybqvvbLTeBgKn+O5ezzTFDdw39sxxQOXq6b+cIs34Osg0770jQO
eQvucmOUtyCtYxbu8geH7wel94onpYPyqGrGv3kcpoM7DxlfSoNhm+mlm4s4
8Nfe17sj+QbM0mIDev7iYCnk89YFUuB6icgu6WUcnidRvn7XeQ06Sbf5TBg4
dCvodGrWvoRakwai8woOwgXvx1/qJcIyefGy4z8czh0WYN3U+AzoSpvbDVaZ
eanzEXM+8xR8DRleoms4tJ/tV2ttjQVAbTrNTJYsvhvrlBoNX1gjdJ3WcXjV
YAIa7x6A8zO5G2NM3k4jTq9nhcG3pTedphs47BrsOphwKRB6lNkc0piswOf1
XNX5OrzYdWonhcmhZCWyeM0l4F/zEGYweTeB5sqx5wTal+ynucHk7rmwFMFl
F8RLcAifZPLsWcU/17x9UfzZfexVTA58qtfkphmK2myHi4KYvB4c8Cpl7z1U
jm49lfrf/12z3TtVIpH1zFxaFrM+Ngv8KKfCY/TR03BgB5NrpB8+eC0Yh3Iq
HpreYPYXniDrpf0oHtlRs2YLmXro8ZVb/uB9juqbsqGPqRdn2Hkdt4cvEB4R
VznK1LNhaVqZgzMJZW6znqIy9Y70iBBKCU9Gcj6bDcuYfmTMxd6WuJOKTN7E
U4KWcHB2VnH6vJ6GwPXig7IFHPRl/azuHn2LNnHGvtNg+q1EqzE2CXqHFr2z
5uPmcBg3u6Q9zniP/pSFS3Ez89LE/2FXyYEstBqvtyY5icPHhr9S4X4fUAma
4JEYx8H9WCyX5GI20mWnfWoexsGIQV0d35eLhCRVzgUP4qBarDxX4p2H4jmt
pLf/wmE5/tek0nI+yj1tb7CJikMZT2BX9c0ipBUpTMtqxyF1j0hnnmwxSvl3
a8W0FYf7JnntSW3FiGjPP0RtxME8YbDZV/kzOi/wWfxVNdMfGcNqFVopKh/c
Rixjzm/PiREQjSxDDWvUV37M+a50Dq1g1/6CfvO1jWz/gENUTnEpPfYrcuHR
/sX2hpmPg8S8uBOVyCZsm0tgDA58NiXZIXOViO/kEX2TCOa8BZt+cE8G1L4a
s8B5F4crA9VmckEklHDmU4GUP7M/pSWjta4qJOq0tXL2Cg6NRbYnCh7WoulH
/FJie3A4Ure7RvFnLRpK+1O1QwGH/G6GTrJWHZr9yKa+IYlDAuMpihqsQ8df
Z93wEsTBATUccjxaj/BV9qiFOTpsNKmpi801osk8xqa7eXTwpq7lPT7VhJRF
kp0LMugwMtWkujm5CQVe83xR/ZoOLYKOu+cNmpEN21O7mCg6vLR4odj6tgVt
zZy4W+tIh/0DmyTvWLah88MUwTJhOqh+uTCnltWGem3/KDfz0EEhLqf2J6MN
ibLmkGtY6MD7Jnt/qlE7qqvgxV0mabBqYkndmG5Hv7dcq79TTQNafp4yqHcg
/T2tI9P/0YCSZLVV1KoDhciJ7nC9TIPuCI41z7AO5KXDjjeb0eD7Fet2yc4O
dG+hnnj8MA0qBTn9gn060U3tRU8WXhqkeF2uPVpCRh5qo7af31EhyYY79xlO
RhZpV3NeJlIhUb/4+RRnF/p3+G60YxQVYqV53F5bdiHfkY8buV5UCG3/tGWN
0YU0tqeXCR6igp0av235kR5UPa0URGjoB7nZr4xD1X3otEvc4ZV2CgS1Hg9f
mOpD1BQuKTOgQPeHZoFcMQpqaxKSi86hQKRjv5zMdQqirAxEZzykwCxlxYhT
uh8F13iP+mAUqKw7mNZ5m4pYl86W+mX2gdWrEgP3gzhaHmFpFLHvhWJ/1CXv
iKPCyMXKAsNeELCot8VjcdQqE9y1V7sXqgR7fMzGcKSkcsahg7cXhI8Hnk06
8xO5ppO5Cop6ILD0ZUDYyk+0+8A7D/JaN5xO72s1thxAL8ePZBQFdsHsvVyn
Oo8B1Ok6fZn7ahckutzdOHx3AGGZuKGuQReM7dmjrpw7gEqt7jRc2t4FEaWh
8Wzsg0gxuerYYeYe09CqaF2aN4hWkvcrxg52wqnlm2MynMPoVsTE6BtV5v5p
JMS2+GkUFS8STZuoLfDIjnuO0TiK7B+v1li/aYHemyz4Om0UcRlqSH51bAHX
5OkvXJvHkLlqesTgVDM8mmnylrAYQw0JL/f3/2uC3id3B47/HUOSS/iMo0Aj
uHYvVMVpjaOkxdrhMf46eGzbE6pWMoliO2dyTk+Vwv3JI2ikZRIJ5/UZapqW
QnBA2tqrgUm0mqMhmlVcAp5PPW5xCkwhvm0aquq3PoNxPZsP3WEKqT12cvy6
UQQCe9Wco4Wm0ayB4dy13znwaD3CaMxjBoVGbDTqicVBvTGR30RoDmEO9tHL
JQVoSkYkuOPlPOqdCxKW4G1HN3122T3e9hf9kSfsN1+loh6qoJlTwzISiq/9
y3J1BF0JODLKafgPRUD4ElvcNKrZpR1Zw7GOZMKmsttUFpACd/nYW30WrKUi
Xl9RnoGI5W4nOSI2YYnvT+6QcltHLCusWd4DrNjKYtrIZgNWbMm04nKdPDuW
rajx/NtlDqxf01ZcPpwD80y60bJzhgvzuiC4RZa6GfP6wmOpkcWHicUPWbIT
ubC0pwe6UncJYrnWqe/n/bkx9Y24n3V6QtiFoyH6DRU8WExONEuzpTBWcNlD
9tAqL2biLK3qFLANW+Y5zlZ2nh+rube8fGJoO2b247AWMVEAc+Y17ufGxDGe
w7xV5a8FsJIcH+9mA3Gs6i3ljE26ALZ/KoLt8XlxTM0/wO5VrgBWf/zWVgk3
cUxAqiRarE4A0zgbOm6UII41umsMivwRwBRwsZuCU+IYxqUcK2BIwJ4Mhq0K
Jklgy9eXd+SaErC/uy792PJeAsvrr39ndIGARW6RTxUulMAk85y+RtsTsFPF
t3dJN0hgq+ffjvAEEDAuSzOG8YIEVpIudZQzg4Blbn94ndeIiHnyz9RnfCRg
2zSPDu+yJGIKfpVmegUErGSSbmZ8lYjFG1xyuVdOwB7snxB/EUDEvOdePmXr
JGA35BUd9TKImIq1q2R6LwH7T0U/07eAiP2qPZilSydgMS0Ww5nlRMwksbfy
zm8C5jp2xGhrJxHjYs08LTNDwGTthf3P0IkYuPmRSQsEjDugL+nebyLm26Vn
e2WFgMnsif5atUDE/u8egv3/PeR/AKBUOgc=
       "]]},
     Annotation[#, "Charting`Private`Tag$28755#1"]& ], 
    TagBox[
     {RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJwtmnlYTd/3x5vvvZUipIgGTZKMIWQfTRSRNGkQkY9QGRKiokhIJUkDTcpQ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       "]]},
     Annotation[#, "Charting`Private`Tag$28755#2"]& ], 
    TagBox[
     {RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJwtWHk0Vd/7vuahXEJCLlEpQxISir1NKaLIUCpRUVGSj6EyVKhkqpRCyVgp
JUqm5DUnKZFMZXavmXvPNVzz737X+v31rmed97xnn+d59rPX2vKnLtm4cpJI
pAAOEul/VQ1JjKysCJfpnjpFXVmh4EtzG8Pn5oXLkvS89WlzFJzzcYcic1q4
jNf4zeafTArW2Gp5ijYqXHabrvUkbZCCd60J66hvFS5bHbn54YkGCsb9jLqE
98Jlcrv2xkomU7DNnR9ZmieFy9wfJf65gyjYvyHs4tlSctl/JbQe31AZ3Ny8
aqTcRahsvCSktYchjSlll8R3yawuU3POu5/vK4U/eVTxCHcKljHs4pKjJtdh
56gd9PYQgbITBiql9UoSmBUbEA6Iv2yXcAwn1Ukcp5O7pOJ7ecucr0/x1RuL
4mfmO7x/3+UpUyz9bTAiIIJDLrhcUdrPXZbS9jHO6v0qHMD1xt97nLNsfDdv
dsAXXuxyo2FTSyZH2eMNYr42Qpw47gs1/pk7qUxjoag9enkeFXdvXf1WYRk+
rzx2ivZnoI0i2Mj2+QIclOkMtrbtQx+Uw937+lngXOkgb7Q7F8k1OheptLPg
xpW/AtXV2SjKX5ff9ycL6GkPNY6uy0KuVSMv+IrY/ZscIKQ8HUk6WfaqxrCA
dGWXmnptGAp6IHrEX4cF6iSLgtBrr2Af65np6uhZqGdF2T2BSuiuzt2grT0D
/2Rrefm/t0Dg4TkTHdUZIO80SLVRbAXJXnxeT2EGpmIg0/pmKxxabMg1IM/A
kNkf3wvabVCmMWFkRpuGYpWnh9cktUNKsrLrkcfTMFa7JLVw7B+cupr++trM
FJQ/vRlc2NADy7yjPwLHpuCzaRO+xeqBxEcaRHDfFISHKZkqy/dC0/tyvdCf
U/DEX+zf4uVeMKZ1f498OQUMqezyijV9sOmwzMQz+ylo/K+4qHp/P1BV4zQg
nwli2kfMDz2lgnLwi4mhLCZQ703sE6mggmfDpzeiqUxQeJiRlzxEBdblFgW3
SCZkdIclqe6kgWDBOjEhZyZMZ0qYLH2ngbphItNRgAn1Uyf/etIHIcAuOW/m
GAFp/4kNG4mPQNnL914brAlY0Xto0a41AtwsUDXfS0DtfqV8fbsRiEnsyUhS
J0DzlOVNr7gRSOmWjzPhISAuUpk/SGwUas5n+MZmM6DN8768Cu8YrAl+ra3G
wYAGQymx7OpxuJ2/tVZklg4Kz2wEdbvGYX7i1VHmGB1WFkN+PJkZh76TLwML
2+iQxm/+9pfiBHwwTK8wyqEDd1n5ze23JsCGJ8nK4SQdxHPeZiTrTkKNvkyP
rh0dPsflGbkdnAQ9v6eXZSzoIGRcG0VynQSFwYRHvdp0aM91kM+5NwnM2scd
HmQ6nBWv0Vnom4QtUvI/SAGTcF9L1y8F00EkQo5e7TYJ57a0Csmb04E1TxGL
tJmESN41AZGH6VD7T/roWuVJCAlxaqO40eFcijhVqX0CMibo73Ui6PBmC/+S
za4JmHL3iTX7SYfYeN4NkhsnQJk7t8e1hQ7XBHiMO8kT4JeWJebZRQfzUY67
Z2nj8C2kYtueCTqMZi+IB8aNw+k+tZMNqxmgqj2p+oI5Bvv1gw/+NmWAm9r1
60WdY6DuWzkhZ8mAZEXhph9fx+DqYOgXG1sGiK7b7j/zdAwo5ePT7qcYMDvr
WW5mMgYj9/aruQUxQJ2+LHZcbQxE3QozBkIZ4D4U4+YlOQZNg/mu+yMY8K8t
e1XC2Cjkawa3VD9mQEXRhN3Io1GIJN3rjWDrupgbnLkcPArSqVh6Tx4Ddr4h
L4ieH4XFyY4vv4sYkJmolrJbfxQ+BYc55FcxIDrAcySKOgL3Dx6129/OgJr/
lvekNoyA2zsPV7UuBpAuxNz7VDQC/Ix5nvk+Bngfz9bqih4B+xjqatMxBhzR
nwjers3uz66terPEgNidwY3GG9jPrcijVRwEfN9G3nREcASGT8bVV7N9aCCr
9u1G1zCktFm3BQgR4C9Ruj6udhiSH2wn71hDQA7ZyvP1h2HwsUmW+CFOwMaV
i2JNt4ch8ptbQdF6Ao7PLrnSvIYhMEDwuKAcAY8nowvnHYfBTbZt1FCBAP6e
dyc2bh+GNRrnaa5bCZgoG3/l0zoE9c+kl29qErClKGg+vHwIzujv+iCoTYBz
rpBlUtYQiJ0JvH1Nh4DfqduI6utD0NUenCWsT8DqxC8mHeeHwO7ZRcGdiIC9
sZZPJg4PwbbPgy8MDQkoDLm4Z93WIQhmlrwXMSWAuLYUoyI6BLG2w6q/2ftS
5b/oXrQ4COL3jbmD9hFwxoOiZUsbhPFZV32yOQHPT7+7fe7XIAS9W+6+bUFA
6zH99sDiQYD4BibtAAEitj9UHmQMwv6T2TfUrAgwP3Ai+EXMIOw5FhZz4iAB
oSbjv4quDIJkrNVmn0MEVOwbO6h1ehBuHbYfuMjOCZLl6M9sy0HI3VlHOWTD
5t96xFJJZxB8cr3apQ8TEGg3XJ+uMAiZpxXkfrJx8dEhC1mhQTjHXTdz3paA
uRODdfGzNNB2OnNugo11TtH2i/XRQNyUFHjMjgA/N2ptdD0NwqNLdXPZOM99
wIy/gAahXE2vJ9mY8OyvCUmlgZHL9WYJewLU/+szXYqkAXkvpUKRjS/591b5
+9GAUBUPkGPj7IAeY8KZBranu/g42HjsenfFBQsahKzwXf7Onqcc1mU4yM5N
5VY1CGbjc+GdZS4baFDoy7kgzcYvo/6hf4I0SBc12ZbCXu/A/b+l9tNUyLlT
40RmY4W4Dv3GbioIm3nGu7L/1yWhvcSijgp+1jv6X7D5SUlq212TRwVZ1XX7
Gtj8daW2FuNkKmS36PzqZfMr87JF9/NdKoiVl4V1svl3fPOncKcPFczLRzwr
2PrEZzfvynGiAlmAO/6+JQEtH37nK+9nz1/vIbCPrefhz415crJUeMzNHPHe
T0As/NJM5KfCH6ktnjQzAn5VNnwQZw5AgEvDSWO2fw7U/8gRqB2AyjsnPxUZ
ERDxq3572IcBkL3Tr/obs3O/+Xv28rMB8BoQ02sxIMC089tb5uUBKE26oJ+g
R0BYb62K5/EBCHxeaHaU7e8K6tc3Q3sHYI86S5iL7X+DierMzvUDoM5nwyW+
g+0HomrLEd4BuC8kYRygxvbDTOXLJno/GIjIu9erELBrpTzja3U/BJpLxKsq
svUUKU3JvdTPPmeck6hSBHiKf5FTdeyHHW++4yQJAt5Kljx/adIPGvb68kiM
ACX54mdPpfpBWtOi2nA1Ww+N/PhblX2gG6reZMzOj7W22fePruuDbM+rUyZ/
GfBBMbLYlKsPeJTI5dMtDDg0d3Zgx2QvOBnr7HnUxICI5/I6gl97Qe6rqWvi
NwYsDz3qLvbrhZ7p5Lj6fAbQggLUZFp6QOrX9+2N0QwIPXTkCH9FD9C/f4lx
CWfAho07Q6be9UBlraN+TwgDHGsn/tTf6gFO1pmPmf4M+Cl6KihoZw9cePNd
6awzAzwPPoj/otQNO6pXP0tXY+fplPlYHEc3+B24QVC2MqAvgRt7tnfB9qZK
kzvyDLAf8B+UvdsFF/x2CWwRZwC66qR9Y6gTOrIDLJ6x2OdZukqz0at/4COz
O+NsKR0+zFQL127qgEjz/zb/1qdDzJu0CoO5dshv4Zm9q0UHD6frvp9+tEPY
xpybaip02FSj8zfNrx3e7VXnMJSkw+O4rJdBtW0wxRP/Vbp/Eq5pPdDXvNAK
iwe/Xkg3nAQj7+Mezz82gzXJ6/f0wDjIKurKrg1vhnWai2dFfo7DQvvaxsjj
zeATbnJXsGAc8gwbtK/yNIPnC96FpPBxUFxjxGnr8BuUO2OdTJXHQTBnS4LA
QiPcX7K/nnd2DF78pUYPh/0EmkBvoWD9CHQESiQnaf6ExZjnhRs+jgBZ1izH
uu8HpEre3CubOAJ+zplNRegHcN8+FPrr7Ajspbqvi5j/DtkzP8rFOUdgaGIy
RdnrGzwMTgnQ1hgGZc75jx7HquA686fRh5uDcMJaJu/v6TywkNB4YTTQD0WC
AX8qfD+ioj1t7XPqbbCzj0P25pEG1BxzSdfSqBA20j/P6VW0o+WW8H35HDVo
f3r7D6sjfchGO7XgatY/ZGopyjX9aRBVAkNpSoWGYlwEGHPfBlF+xd8HmmY0
1OZL6l7uHESTpEXR06doyP35RDE/7xDSrInaERvP7p+s85axH0LWVXmJ27kH
UduD0D7jmSGkISRtdaJtELm3TJXHao+gav+QgN0+w+jeydYb6gVj6O4Hma3S
V8bQ7bE9iFY/hsZ8DS24H4yh4KtpS0/7xpB92qJx++sx5Pnw4jU+8jgSP+H7
3PjvGLL6yuXTdWYcdW6WdJTSH0fk7epno0QnkOabe3eqFsdRzHK45dDFSdQe
LUta8ZhEtyMnVj0PnUSaveaOu0MmUbCkbd3hhEnE8N4ufSp+EnlqyJlB1SRq
cxj96smuWz46bvguREctLzrdKtiVb5AulXaGjmzN9jU0nqKjr1YUoUOiDDRh
8LW5g5uBxNV6JbxlGOjve0fReSEGchZ6seGRIgPN69svCa9j1++qWm16DIQ9
7vfJKTGQ6v49x5xPM1BVSqXsrwMMdN/42GuvPAbiDcvX9ollIEedBNMH9gQi
N+i2/JMg0Mt1Jw5+dCaQwyuWVYssgZgzG47+cSeQYgCrrkaRQFGfMi9IXSdQ
4ljA3yhtAoFG4cPUVwSKZap8e2dHoE3bWntzWQQi3a0+sTeWQOPya4ObEplo
siiqgMHNRDoBqZfLM5goZq/IntxVTBTSvM01J5uJjvIHvDknykQSd/YeiKlg
oqrNs0dBjokMxvylzUeY6NndtQmr9ZjoXn5HfrnuFDJIveClcpGJdlikTOa0
TaHwu0OcGT+YKDBDtT+5bwpRN7ZRPX4zUc1SYUvM2BQSUXz2n2o7EznmNH65
QJpGgWEqWU8G2N9fyxW1des0on2ct1y3wES/u123pvhPo1LJtMxyxSnk66Pi
ck9iBmkJnPGbvTKFJAQnS+zlZxA2I3VaBk+hwuQPkrKqMygvvLT8WegUWqzT
/fXWcAYdUKlQ3xQzhULk96G6CzMo0c6tpSNtCsX8PEPhqZxBsQfjORh1U+jl
1uft1y7NIo/i48qiEtNoX6nLTqNrs8iIi3o+RnoaDR/e/EDg1ix6ECSazC03
jVSvv90XnziLJH85ULq3TKMPf4oLPlXPogqr9EZLXfa6Q1rj6OtZSPl1313K
sWnU+k/Exq2WhViWE68HEqeReaRrm08TC+UYRjT6P59GJbrFTqH/WKgy6K86
X9o0Snl82iOFzkLF46+qpV9Po3OH8sM6JOeQCZ/NQ4mCaTRfeazA6vwcKiLK
dT2appFM1kuZXYLzqJkZ+V8X7wxyvrpnkO/AAjqX8P1AwfkZpHym9s+E/QJK
+DwVssDmgWllW/XHZQGFqIqSkdcMur3pQmq6/wL6PZJ1vspvBr1teHoMpS8g
vEjbXBc6g+Y2LzT4zS+g/DsMi8YkNm9NRYXUzEUULLKn8UDTDDr+xfRV/cdF
FKXscun5nxm0ObMx7mPpIhpJCHvJaJtBhUHD3jebF5G1AVJO6p5BXVultlFI
S4hSqh8rOM5e3/UrqbYOS8jKLdvqIP8sqlTZFVHJs4z+XjNhcRvOIqfttukM
4WX06Oh6fi+TWcTSuFwiJ72M8lX9o7rMZpGqXtZEgNoyIjuF7qixmkWP9skd
1nJYRoUdsxnlx2eRmyufzIvMZSRnFTjvd3UW8Se3vrttsYJ6rOcVzPNnUVra
VE2e3QqS1nR3NCqeRXterunpO7mCDh9ZJuHSWeT1zkIU+ayg1c4Ofy1qZlH7
Z/CbfbaC0Iljoa9aZtGbtlfo3MQKSlwX/CSQNYsUBUqGMvaR8M0arXghQxYi
vqq5p1mS8IjWuuRiUxYqvZ06mmxDwq09ld88zVnInvvOROJxEl57P7Zm7DAL
3VqxnrrvRcKu2y9wGp5loYHpwZXAeBLuPuOV5nCPhdL6xNbZDZGwuC9p+0sq
C3mm3o63GSfhu4xHObajLKTnPCd1iCBh7a11mqsZLNTU2SVjsUjC6z+aXHu8
yEIc7a8VDEU48GuRDS+kxOeQcwNS26bDgQ+vDNsNmswhSomHKU84B07wOhPL
mz2HxKXm5z2jOfAOxT7d8E9zaLVfeE5bLAc2dNBKlfgyhxa3v1j/NokD1/TF
sI7Wz6GO9C6GzUcOrOyz/1vg6Bx6HGn9PKWLAxt4lbUYqc4j8rFds7t3cmI3
X++DVoXziLew+u0LPU7sM+w9FFE+j5bEbU8JY05cNpoV1FI3j8Z+ev3oM+fE
T86Z9iV2zqM6ozfpd505MfNzwpUOrgV0R5lyqDWSE+t92NFjfHgBkeY5X3v3
cWLjjQ0dO5cXkHN/O/P6ICd++uBexAf+RVRWn2MQPcaJWW8vrzcQW0Q3kp2a
X81wYrVrEmtDti6iFZPilX+CXNjr/Pk/pYcX0fK9y/Zmmlz48fiGDol3i2hx
cw+XTBgXdvf1fZ7gsYSOCxdYKd3lwlc4nXVWXVlCJazoBO0YLqygb743OmwJ
BX7fvd06ngvzvuSYqEhaQgteTxxvv+XC9uF8xdO/ltD8Z6tcejMXPj0eGBmq
t4xmrb84VW/mxsSwgtJ96RUU0v6+00CFG8s6Zk2fVF5BZJe044Xq3Bg/lS8z
0VtBil53HLN2c2NptbkwS8cVZBtjbf/AmhsXxzWYnni6gnLqqJYngrhxfVRn
XJkICZ83JutPN3Pjc4fvF0glk/BUHanEs4Mbk3srSU6ZJHzDhqk31M2Nf0rO
PHiXS8LxLm06f0e48Uy2UbpnFQnXXE/TKifx4B5Zvg1XR0hYoURbNVqVB/fe
e3A0j+2jv1on128O48ElvBbvhzo5sNvcTuFbd3mwI3OPpecQB6Z/Wc1FjeHB
uUYCHisEB+Y2Kx7JSODBGgEPd5nws3U4urZ403se/NB2R9QRTU58I+j7kU0d
PDgs51bVe7bOijW7Hm9U58VCdLlUTVO2Tg4iaxT+8eL+ho8nD/zjxqcytydu
7uXF4uD+hzrEje3mrDYq0XgxqzaKFDHNjfWeRu9Up/PizcSLIILMg3m6BR0N
uPlw/aunyMSIBye68aQ7qvLhNu3+bWNveHCV77zWw0A+rNPGGWEfxoulHg0c
4abw4xKrQzcrHfixTqichI8CP85M/RF84hw/tvd2/N2/hR9rUFrNua/y49hD
vywrNfhxjcI5yyuJ/Hi1UIlRyD5+PKGsTsx18uPlWw9VST78mB0spqfPCuAB
fyPSYh37fSPFQ7Whgjj7WMor5hUB3Oe6nyuqbzV20L++r/aLIBZqPJgTNyyC
c50uKugtrsI2MiNleulimCVozFVkJ4T5VWS/hg9IYJvG3dqUeDLe093UM71V
GgvuXlVekkTGXVvrGLO7pHF5RofF8XQyVt1VYLG0VxqrX7nq8jSbjG/NZVev
cZXGZLmCKKlqMj4xdAa7pErjbxc0+tcSZFy0KaE0Tno9xvxK98kHhPEdUjGv
urAMZl1mSWdbC2NqqL/QA1kZ/P7v1xeWDsI4Ojcib2qbDJZ97/Y5ir3Df352
+Fl5QAYv2mXQBK8KY4E/I4k3ImRwQbqcPt9LYWwxWjJ0iZeCPYUmv77MYs+7
0+rNWkvBiv6lNntzhXFE+0Oz0M0U/Mj8xLlbJex57d6fk00o2JuR+JDrtzDu
dcDazBAKVj7mLpveJozpYb3B0bEU3Ful+9qoSxirLcsOKqVR8KH4ttKbw8JY
KWuQdL6cgvk5M/fLTwrjmfHsV6saKRg8/JvLpoSxRmbEkZweCvb7s/ek87ww
Fk6XJDvQKfj/7+dxUuJMxf/u5/8PGENi/A==
       "]]},
     Annotation[#, "Charting`Private`Tag$28755#3"]& ]}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->{True, True},
  AxesLabel->{None, None},
  AxesOrigin->{0, 0},
  DisplayFunction->Identity,
  Frame->{{False, False}, {False, False}},
  FrameLabel->{{None, None}, {None, None}},
  FrameTicks->{{Automatic, 
     Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, 
     Charting`ScaledFrameTicks[{Identity, Identity}]}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImagePadding->All,
  Method->{
   "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "ScalingFunctions" -> None, 
    "CoordinatesToolOptions" -> {"DisplayFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]], 
        (Identity[#]& )[
         Part[#, 2]]}& ), "CopiedValueFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]], 
        (Identity[#]& )[
         Part[#, 2]]}& )}},
  PlotRange->{{-4, 4}, {-1.146033056506329, 7.499999603395332}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{{3.816326965820525*^9, 3.816327004925709*^9}, {
  3.816327164688939*^9, 3.816327286644581*^9}},
 CellLabel->"Out[87]=",ExpressionUUID->"cb915ffc-d5b8-4595-8299-7cb18e79a143"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"f", "[", "x", "]"}]], "Input",
 CellChangeTimes->{{3.816327025754106*^9, 3.8163270263775463`*^9}},
 CellLabel->"In[88]:=",ExpressionUUID->"420cd42e-9ee7-4703-9944-1cd642eeb2d8"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   FractionBox["1", "2"], " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", "1"}], "+", 
     SuperscriptBox["x", "2"]}], ")"}]}], "+", 
  RowBox[{"Sin", "[", 
   FractionBox[
    RowBox[{"\[Pi]", " ", "x"}], "2"], "]"}]}]], "Output",
 CellChangeTimes->{3.816327026738496*^9, 3.816327168322628*^9, 
  3.816337479795702*^9},
 CellLabel->"Out[88]=",ExpressionUUID->"ef778053-dd39-4245-9cec-0a98d4b1be12"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", "+", "1"}], "]"}], "+", 
   RowBox[{"f", "[", 
    RowBox[{"x", "-", "1"}], "]"}]}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.8163270289681664`*^9, 3.816327042157338*^9}},
 CellLabel->"In[89]:=",ExpressionUUID->"e2fb8bdc-9e2b-41e1-a278-493b94ee3c13"],

Cell[BoxData[
 SuperscriptBox["x", "2"]], "Output",
 CellChangeTimes->{{3.816327035139567*^9, 3.816327042437059*^9}, {
   3.8163271549456882`*^9, 3.816327169513623*^9}, 3.816337481042354*^9},
 CellLabel->"Out[89]=",ExpressionUUID->"1708baed-3041-4f68-980f-0e414fc5ee61"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{"Another", ":", " ", 
     RowBox[{"f", 
      RowBox[{"(", 
       RowBox[{"x", " ", "f", 
        RowBox[{"(", "y", ")"}]}], ")"}]}]}], "=", " ", 
    RowBox[{
     RowBox[{"f", 
      RowBox[{"(", 
       RowBox[{"x", " ", "y"}], ")"}]}], " ", "+", " ", "x"}]}], ",", " ", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], " ", "\[Element]", " ", 
    TemplateBox[{},
     "Reals"]}], ",", " ", 
   RowBox[{"independent", " ", "variables"}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.81814008685806*^9, 3.818140157340411*^9}, {
  3.8181681693787127`*^9, 
  3.8181681730238857`*^9}},ExpressionUUID->"16c3961e-3780-4cfe-a6d4-\
4ebfcb30ea78"],

Cell[BoxData[
 RowBox[{
  RowBox[{"$Assumptions", "=", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], "\[Element]", 
    TemplateBox[{},
     "Reals"]}]}], ";"}]], "Input",
 CellChangeTimes->{{3.818140205206332*^9, 3.8181402190930862`*^9}},
 CellLabel->"In[1]:=",ExpressionUUID->"c33f0672-92e8-452d-8350-07fb42faa0c5"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.818140225924383*^9, 3.8181402567654047`*^9}, {
  3.818143660871539*^9, 3.818143661443686*^9}, {3.818168001039897*^9, 
  3.81816800134186*^9}, {3.818168180968347*^9, 3.818168181031329*^9}},
 CellLabel->"In[2]:=",ExpressionUUID->"df20952e-c272-49ef-847a-25613e81da47"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{
  3.818140257211437*^9, 3.81814233516675*^9, 3.818142681114938*^9, {
   3.818143659298221*^9, 3.818143700513908*^9}, {3.8181443339819736`*^9, 
   3.818144342056416*^9}, 3.818145374439254*^9, 3.8181460779070597`*^9, 
   3.818151233932131*^9, 3.818151311979403*^9, 3.818151750245265*^9, 
   3.8181596920042887`*^9, 3.81816733729233*^9, 3.8181675007743807`*^9, 
   3.818167604162397*^9, 3.81816779273059*^9, 3.818168001802363*^9, 
   3.818168191514748*^9, 3.818168233680788*^9},
 CellLabel->"Out[3]=",ExpressionUUID->"35dfbddf-966c-474c-ad97-7042ffc90088"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", 
            RowBox[{"f", "[", 
             RowBox[{"y", ",", " ", "n"}], "]"}]}], ",", " ", "n"}], "]"}], 
         "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", " ", "y"}], ",", "n"}], "]"}], " ", "+", " ", 
          "x"}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818140287733284*^9, 3.818140333229371*^9}, {
  3.818143599643519*^9, 3.818143610636348*^9}, {3.818143685180457*^9, 
  3.8181436933501062`*^9}, {3.818145168396035*^9, 3.818145174851321*^9}, {
  3.818145307984054*^9, 3.818145309711309*^9}, {3.818146084890275*^9, 
  3.818146090735858*^9}, {3.818151344858057*^9, 3.818151351517991*^9}, {
  3.818159325342852*^9, 3.818159354362829*^9}, {3.81815940151412*^9, 
  3.818159410086164*^9}, {3.818159702825521*^9, 3.818159728559155*^9}, {
  3.818167341879003*^9, 3.8181674557640057`*^9}, {3.818167509140544*^9, 
  3.818167511739132*^9}, {3.818167579882263*^9, 3.818167663053783*^9}, {
  3.818168211999712*^9, 3.81816821626082*^9}},
 CellLabel->"In[9]:=",ExpressionUUID->"3c27f791-8498-4a05-90b3-cf90f74c6b7b"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.818140349982601*^9, 3.818140355197728*^9}, {
  3.818142685255755*^9, 3.8181426890763607`*^9}, {3.818143614492283*^9, 
  3.818143615754689*^9}, {3.818143665982217*^9, 3.818143680988772*^9}, {
  3.818143712642333*^9, 3.818143714729566*^9}, {3.818143752810752*^9, 
  3.8181437530014353`*^9}, {3.8181598115127277`*^9, 3.818159823309122*^9}, {
  3.818167498365052*^9, 3.818167525108748*^9}, {3.818167804204111*^9, 
  3.818167824917338*^9}, {3.8181678624102573`*^9, 3.8181678624934807`*^9}, {
  3.818168006083107*^9, 3.818168006186816*^9}, {3.8181681943001957`*^9, 
  3.818168194361513*^9}, {3.8181682691465273`*^9, 3.8181682691730328`*^9}},
 CellLabel->"In[10]:=",ExpressionUUID->"3654ee2c-edee-476e-b5aa-258837e3f189"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
      RowBox[{"y", "+", 
       FractionBox["1", 
        RowBox[{"a", "[", "1", "]"}]], "-", 
       RowBox[{"y", " ", 
        RowBox[{"a", "[", "1", "]"}]}]}]}], "}"}], "}"}], ",", "1"}], 
  "}"}]], "Output",
 CellChangeTimes->{{3.8181403533645277`*^9, 3.8181403555683928`*^9}, 
   3.818140862401368*^9, 3.8181423499478827`*^9, {3.8181426829244747`*^9, 
   3.8181426896259193`*^9}, 3.8181427401464777`*^9, 3.81814361618382*^9, {
   3.818143695715423*^9, 3.818143715038835*^9}, 3.818143753423613*^9, 
   3.818144159844943*^9, 3.818144344980282*^9, 3.8181453112077103`*^9, 
   3.818145377184569*^9, 3.818146095905328*^9, 3.8181512354842443`*^9, 
   3.818151316017922*^9, 3.8181513527588787`*^9, {3.818151735029253*^9, 
   3.818151754202927*^9}, 3.818159412347094*^9, 3.81815969642979*^9, 
   3.818159730177293*^9, {3.818159812099925*^9, 3.8181598236927967`*^9}, {
   3.818167349230927*^9, 3.8181673659531107`*^9}, 3.818167457404048*^9, {
   3.818167495930997*^9, 3.818167525484861*^9}, {3.8181676436807327`*^9, 
   3.8181676649747458`*^9}, {3.8181677944171667`*^9, 
   3.8181678054393883`*^9}, {3.818167856657599*^9, 3.818167863525917*^9}, 
   3.818167894773676*^9, 3.8181679856608257`*^9, {3.8181681899437027`*^9, 
   3.818168219639114*^9}, {3.818168262643518*^9, 3.818168269716029*^9}, 
   3.818168310941964*^9},
 CellLabel->"Out[10]=",ExpressionUUID->"ad7094dd-dfcd-4d0e-9bb7-8eae058faf21"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", ",", " ", "1"}], "]"}], "/.", 
  RowBox[{
   RowBox[{
    RowBox[{"sol", "[", "1", "]"}], "[", 
    RowBox[{"[", "1", "]"}], "]"}], "[", 
   RowBox[{"[", "1", "]"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.8181404002472277`*^9, 3.818140456506225*^9}, {
  3.818140496342524*^9, 3.818140515406921*^9}, {3.818140631210018*^9, 
  3.818140645383547*^9}, {3.8181425076346188`*^9, 3.818142516017757*^9}, {
  3.8181427202705*^9, 3.818142773600771*^9}, {3.818143622701044*^9, 
  3.818143623954439*^9}, {3.818143736521529*^9, 3.818143758851022*^9}, {
  3.818159502029887*^9, 3.818159504762895*^9}, {3.818167810773844*^9, 
  3.818167840476286*^9}, {3.8181678746254053`*^9, 3.818167924249042*^9}, {
  3.8181679910000467`*^9, 3.8181679921504593`*^9}, {3.8181682768967733`*^9, 
  3.8181682802875566`*^9}},
 CellLabel->"In[11]:=",ExpressionUUID->"5a46d4f1-ac46-4891-a5f3-c5f622732f57"],

Cell[BoxData[
 RowBox[{
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  FractionBox[
   RowBox[{"1", "+", 
    RowBox[{"y", " ", 
     RowBox[{"a", "[", "1", "]"}]}], "-", 
    RowBox[{"y", " ", 
     SuperscriptBox[
      RowBox[{"a", "[", "1", "]"}], "2"]}]}], 
   RowBox[{"a", "[", "1", "]"}]]}]], "Output",
 CellChangeTimes->{{3.818140419233189*^9, 3.81814045690171*^9}, {
   3.818140497583053*^9, 3.818140515688933*^9}, {3.818140635985394*^9, 
   3.8181406457431803`*^9}, 3.818140863773431*^9, 3.8181423882266273`*^9, 
   3.818142512638859*^9, {3.818142706690268*^9, 3.81814277390701*^9}, 
   3.81814362453546*^9, {3.8181437319324503`*^9, 3.818143759467441*^9}, 
   3.8181441627946177`*^9, 3.818144345943077*^9, 3.818145315329323*^9, 
   3.818145379791527*^9, 3.818146098043954*^9, 3.818151237016879*^9, 
   3.81815135712315*^9, 3.818151762032042*^9, 3.818159416509089*^9, 
   3.818159505136392*^9, 3.818159697599337*^9, 3.818159734669541*^9, 
   3.818159780399888*^9, 3.818159820239729*^9, {3.8181676465210648`*^9, 
   3.818167675352768*^9}, {3.818167796602661*^9, 3.818167811290407*^9}, {
   3.818167875884679*^9, 3.818167883415781*^9}, {3.818167919023307*^9, 
   3.8181679258635473`*^9}, {3.8181679889760427`*^9, 3.818167993288959*^9}, 
   3.818168196949212*^9, 3.818168280816876*^9, 3.818168311885614*^9},
 CellLabel->"Out[11]=",ExpressionUUID->"05951f65-ca24-4ac6-bdee-79d63134afa6"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
   "Now", " ", "we", " ", "need", " ", "to", " ", "determine", " ", "the", 
    " ", "value", " ", "of", " ", "the", " ", "coefficient", " ", "that", " ",
     "makes", " ", "y", " ", "disappear", " ", "in", " ", "all", " ", "cases",
     " ", 
    RowBox[{"(", 
     RowBox[{
     "so", " ", "that", " ", "it", " ", "can", " ", "be", " ", "an", " ", 
      "independent", " ", "variable"}], ")"}]}], " ", "*)"}], 
  "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{"Solve", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"x", " ", 
        RowBox[{"a", "[", "1", "]"}]}], "+", 
       FractionBox[
        RowBox[{"1", "+", 
         RowBox[{"y", " ", 
          RowBox[{"a", "[", "1", "]"}]}], "-", 
         RowBox[{"y", " ", 
          SuperscriptBox[
           RowBox[{"a", "[", "1", "]"}], "2"]}]}], 
        RowBox[{"a", "[", "1", "]"}]]}], "\[Equal]", 
      RowBox[{
       RowBox[{"x", " ", 
        RowBox[{"a", "[", "1", "]"}]}], "+", "1"}]}], ",", " ", 
     RowBox[{"a", "[", "1", "]"}]}], "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.818168350827271*^9, 3.818168501478005*^9}, {
  3.818168579207617*^9, 3.818168595311913*^9}},
 CellLabel->"In[19]:=",ExpressionUUID->"e040c7b8-50f6-45f5-9918-2d05744c3687"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
     RowBox[{"-", 
      FractionBox["1", "y"]}]}], "}"}]}], "}"}]], "Output",
 CellChangeTimes->{{3.818168454673211*^9, 3.8181685019347467`*^9}, 
   3.818168596095661*^9},
 CellLabel->"Out[19]=",ExpressionUUID->"5f9112af-e532-4858-af0a-81c987e62743"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"So", ",", " ", 
    RowBox[{
     RowBox[{"a", "[", "1", "]"}], " ", "has", " ", "to", " ", "be", " ", "1",
      " ", 
     RowBox[{"(", 
      RowBox[{"the", " ", "only", " ", "value", " ", 
       RowBox[{"that", "'"}], "s", " ", "truly", " ", "independent", " ", 
       "of", " ", "y"}], ")"}]}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"x", " ", 
        RowBox[{"a", "[", "1", "]"}]}], "+", 
       FractionBox[
        RowBox[{"1", "+", 
         RowBox[{"y", " ", 
          RowBox[{"a", "[", "1", "]"}]}], "-", 
         RowBox[{"y", " ", 
          SuperscriptBox[
           RowBox[{"a", "[", "1", "]"}], "2"]}]}], 
        RowBox[{"a", "[", "1", "]"}]]}], "/.", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", " ", "1"}]}], "//", 
     "Simplify"}]}], "\[IndentingNewLine]", 
   RowBox[{"f", "[", "x", "]"}]}]}]], "Input",
 CellChangeTimes->{{3.818168521193161*^9, 3.8181685724100924`*^9}, {
  3.818168604236229*^9, 3.8181686191367693`*^9}},
 CellLabel->"In[22]:=",ExpressionUUID->"3bfe0e89-5a61-4fe8-bcbd-cb6a444c2e02"],

Cell[BoxData[
 RowBox[{"1", "+", "x"}]], "Output",
 CellChangeTimes->{
  3.818168572739149*^9, {3.818168620086589*^9, 3.8181686218835363`*^9}},
 CellLabel->"Out[23]=",ExpressionUUID->"9092bcf3-d7a8-4cf4-b910-edbd0f52d57f"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{"A", " ", "related", " ", "one", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"with", " ", "y"}], " ", "=", " ", 
       RowBox[{"f", 
        RowBox[{"(", "x", ")"}]}]}], ")"}]}], ":", " ", 
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", " ", "f", 
       RowBox[{"(", 
        RowBox[{"f", 
         RowBox[{"(", "x", ")"}]}], ")"}]}], ")"}]}]}], " ", "=", " ", 
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", 
      RowBox[{"x", " ", "f", 
       RowBox[{"(", "x", ")"}]}], ")"}]}], " ", "+", " ", "x"}]}], " ", 
  "*)"}]], "Input",
 CellChangeTimes->{{3.8181686836064167`*^9, 3.8181687142719927`*^9}, {
  3.8181688352134943`*^9, 3.818168843534184*^9}},
 CellLabel->"In[32]:=",ExpressionUUID->"609d1368-5c71-4c22-867f-7c93aa710c4e"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[38]:=",ExpressionUUID->"f37a35e4-4031-4d11-8cf2-7921c30df61e"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.8181687242728233`*^9, 3.818168845214616*^9, 
  3.818248852821267*^9, 3.818324121200102*^9},
 CellLabel->"Out[39]=",ExpressionUUID->"8452fa1c-a8fe-44ad-b368-cbb4b3f6866c"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", 
            RowBox[{"f", "[", 
             RowBox[{
              RowBox[{"f", "[", 
               RowBox[{"x", ",", "n"}], "]"}], ",", " ", "n"}], "]"}]}], ",", 
           " ", "n"}], "]"}], "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", " ", 
             RowBox[{"f", "[", 
              RowBox[{"x", ",", "n"}], "]"}]}], ",", "n"}], "]"}], " ", "+", 
          " ", "x"}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818168738706807*^9, 3.818168745488062*^9}},
 CellLabel->"In[40]:=",ExpressionUUID->"8ad4b9ec-e566-4d5d-9c42-05e2a8441aa6"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]], "Input",
 CellLabel->"In[41]:=",ExpressionUUID->"923cffb2-cf7c-4e8a-a93b-0381d84c59df"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}]}], "}"}], "}"}], ",", 
   "1"}], "}"}]], "Output",
 CellChangeTimes->{3.818168755604456*^9, 3.818168846543312*^9, 
  3.818168894283401*^9, 3.8182488630622597`*^9, 3.8183241237127533`*^9},
 CellLabel->"Out[41]=",ExpressionUUID->"d912b88c-3bab-4fc6-95a5-8372a7102885"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", ",", " ", "1"}], "]"}], "/.", 
  RowBox[{
   RowBox[{"sol", "[", "1", "]"}], "[", 
   RowBox[{"[", "1", "]"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.81816876661067*^9, 3.818168767504759*^9}},
 CellLabel->"In[42]:=",ExpressionUUID->"0e157153-400d-481f-9f22-32649f41b36c"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"1", "+", "x"}], "}"}]], "Output",
 CellChangeTimes->{3.81816876904758*^9, 3.818168847151311*^9, 
  3.818168895059722*^9, 3.81824886576577*^9, 3.818324124676523*^9},
 CellLabel->"Out[42]=",ExpressionUUID->"0247a037-6774-4f11-a7b1-d2c6f3164d62"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"gives", " ", "the", " ", "solution", " ", "directly", 
    RowBox[{"(", "!", ")"}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
   RowBox[{"x", "+", "1"}]}]}]], "Input",
 CellChangeTimes->{{3.8181687766742773`*^9, 3.818168798470777*^9}, {
  3.8181689521270227`*^9, 3.818168953371207*^9}},
 CellLabel->"In[43]:=",ExpressionUUID->"f12e9b9e-298c-409f-8c7e-89819eb48923"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", " ", 
     RowBox[{"f", "[", 
      RowBox[{"f", "[", "x", "]"}], "]"}]}], "]"}], "\[Equal]", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", " ", 
      RowBox[{"f", "[", "x", "]"}]}], "]"}], " ", "+", " ", "x"}]}], "//", 
  "Simplify"}]], "Input",
 CellChangeTimes->{{3.818168809973505*^9, 3.818168825292602*^9}},
 CellLabel->"In[44]:=",ExpressionUUID->"3fb9f7e1-10d5-4670-b232-e03dc66952d2"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.818168813310916*^9, 3.818168849717059*^9}, 
   3.818168896744481*^9, 3.8183241269374743`*^9},
 CellLabel->"Out[44]=",ExpressionUUID->"137ef1d1-a8e3-45ee-aa0e-eeaff49b8deb"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "2", "]"}], "//", "Simplify"}]], "Input",
 CellChangeTimes->{{3.818248876594552*^9, 3.818248876660611*^9}},
 CellLabel->"In[45]:=",ExpressionUUID->"91c64797-e3a0-4dd2-b251-ee9e49c7249d"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}]}], "}"}]}], "}"}], ",",
    "2"}], "}"}]], "Output",
 CellChangeTimes->{3.818248877772533*^9, 3.818324128742465*^9},
 CellLabel->"Out[45]=",ExpressionUUID->"caf09821-e0ce-4a8b-af35-f8f97f7cd6f8"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{"same", " ", 
    RowBox[{"solution", "."}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{"sol", "[", "3", "]"}]}]], "Input",
 CellChangeTimes->{{3.8182488873412447`*^9, 3.818248904470332*^9}, {
  3.818248944696342*^9, 3.818248947296399*^9}, {3.818324142596221*^9, 
  3.818324187807136*^9}},
 CellLabel->"In[49]:=",ExpressionUUID->"bc487467-6292-48de-a315-267b6cf4e5ec"],

Cell[BoxData["$Aborted"], "Output",
 CellChangeTimes->{3.81832421661486*^9},
 CellLabel->"Out[49]=",ExpressionUUID->"42945a86-dbd0-44e6-ab21-fb57a282bdaf"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{"Let", "'"}], "s", " ", "explore", " ", "rational", " ", 
    RowBox[{"functions", ":"}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], 
   "\[IndentingNewLine]", 
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.818324225068468*^9, 3.8183242254641542`*^9}},
 CellLabel->"In[51]:=",ExpressionUUID->"8e7f7c9d-9d8c-4d0e-98f1-6503889fd86c"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.818248948755651*^9, 3.8183239001768713`*^9, 
  3.81832422625014*^9},
 CellLabel->"Out[52]=",ExpressionUUID->"7edee4ca-5912-4c76-9f4c-e38741de52aa"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", 
            RowBox[{"f", "[", 
             RowBox[{
              RowBox[{"f", "[", 
               RowBox[{"x", ",", "n"}], "]"}], ",", " ", "n"}], "]"}]}], ",", 
           " ", "n"}], "]"}], "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{
            RowBox[{"x", " ", 
             RowBox[{"f", "[", 
              RowBox[{"x", ",", "n"}], "]"}]}], ",", "n"}], "]"}], " ", "+", 
          " ", "x"}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818323953013014*^9, 3.818324002690906*^9}},
 CellLabel->"In[53]:=",ExpressionUUID->"881a6343-0adc-4da9-9b38-18141e2d37a5"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", " ", "=", " ", 
  RowBox[{
   RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.818249038729312*^9, 3.818249041182695*^9}},
 CellLabel->"In[54]:=",ExpressionUUID->"55ae1bfa-bd55-4bae-a391-583a5dccccae"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "0", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "0", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "0", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          SuperscriptBox[
           RowBox[{"b", "[", "0", "]"}], "2"], 
          RowBox[{"b", "[", "1", "]"}]]}]}]}], "}"}]}], "}"}], ",", "9"}], 
  "}"}]], "Output",
 CellChangeTimes->{
  3.818248979614541*^9, 3.818249041948392*^9, 3.818323910142065*^9, {
   3.818323980802349*^9, 3.81832400454135*^9}, 3.8183242360531588`*^9},
 CellLabel->"Out[54]=",ExpressionUUID->"1359adfc-380c-4f0f-a785-98b511c8f80d"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "1"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"sol1", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"sol1", "[", 
       RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "FullSimplify"}]], "Input",
 CellChangeTimes->{{3.8182490202322807`*^9, 3.818249100395512*^9}},
 CellLabel->"In[55]:=",ExpressionUUID->"bde15c33-edca-4611-abaf-993ac17cdb34"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "ComplexInfinity", ",", "ComplexInfinity", ",", "0", ",", "0", ",", "0", 
   ",", "0", ",", 
   RowBox[{"1", "+", "x"}], ",", 
   RowBox[{"-", "1"}], ",", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"b", "[", "0", "]"}], 
     RowBox[{"b", "[", "1", "]"}]]}]}], "}"}]], "Output",
 CellChangeTimes->{{3.818249078715963*^9, 3.818249100713414*^9}, 
   3.8183239128270683`*^9, 3.8183240070646343`*^9, 3.818324238722933*^9},
 CellLabel->"Out[55]=",ExpressionUUID->"68a85739-2dc3-4369-aedc-c1e1a7b30f45"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"1", "+", "x"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", " ", 
     RowBox[{"f", "[", 
      RowBox[{"f", "[", "x", "]"}], "]"}]}], "]"}], "\[Equal]", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", " ", 
      RowBox[{"f", "[", "x", "]"}]}], "]"}], " ", "+", " ", "x"}]}], "//", 
  "Simplify"}]}], "Input",
 CellChangeTimes->{{3.818249118006629*^9, 3.8182491246440563`*^9}, {
  3.8183242527576723`*^9, 3.818324253105504*^9}},
 CellLabel->"In[59]:=",ExpressionUUID->"5653ca6a-b8e4-49bc-bc08-be02dd629392"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{3.8183242536951647`*^9},
 CellLabel->"Out[60]=",ExpressionUUID->"d8253ec7-0421-43ca-88d6-f7c58cc446ac"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"-", "1"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", " ", 
     RowBox[{"f", "[", 
      RowBox[{"f", "[", "x", "]"}], "]"}]}], "]"}], "\[Equal]", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", " ", 
      RowBox[{"f", "[", "x", "]"}]}], "]"}], " ", "+", " ", "x"}]}], "//", 
  "Simplify"}]}], "Input",
 CellChangeTimes->{{3.818249118006629*^9, 3.818249178752565*^9}, {
  3.818324024830134*^9, 3.818324061528077*^9}, {3.8183242593461*^9, 
  3.8183242599431543`*^9}},
 CellLabel->"In[61]:=",ExpressionUUID->"d7d9774e-7978-4221-ae2b-1fcee4c0ab4f"],

Cell[BoxData[
 RowBox[{"x", "\[Equal]", "0"}]], "Output",
 CellChangeTimes->{3.818324260445039*^9},
 CellLabel->"Out[62]=",ExpressionUUID->"da1e951f-0779-4535-b5b9-c31c4f183b3e"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", " ", "=", " ", 
  RowBox[{
   RowBox[{"sol", "[", "2", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.818249192817402*^9, 3.818249194624784*^9}, {
  3.818307192774982*^9, 3.818307198586802*^9}, {3.818324097265915*^9, 
  3.818324099106659*^9}},
 CellLabel->"In[63]:=",ExpressionUUID->"6424e545-1924-4d81-90fc-69be8d3c5178"],

Cell[BoxData["$Aborted"], "Output",
 CellChangeTimes->{
  3.818307201293109*^9, {3.818324095147367*^9, 3.818324102619742*^9}, 
   3.81832428455519*^9},
 CellLabel->"Out[63]=",ExpressionUUID->"4b3ebc5a-1302-4933-9de6-edd0ec5a3b02"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", 
  RowBox[{
   RowBox[{
    RowBox[{" ", 
     RowBox[{"Another", ":", " ", 
      RowBox[{"f", 
       RowBox[{"(", 
        RowBox[{"x", " ", "y"}], ")"}]}]}], ")"}], "=", " ", 
    RowBox[{"f", 
     RowBox[{"(", "x", ")"}], " ", "f", 
     RowBox[{"(", "y", ")"}]}]}], ",", " ", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], " ", "\[Element]", " ", 
    TemplateBox[{},
     "Reals"]}], ",", " ", 
   RowBox[{"independent", " ", "variables"}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.81814008685806*^9, 3.818140157340411*^9}, {
  3.8181681693787127`*^9, 3.8181681730238857`*^9}, {3.8197165424668493`*^9, 
  3.8197165736509447`*^9}},ExpressionUUID->"009ce6a1-02fe-4b53-98b1-\
58af54d707fb"],

Cell[BoxData[
 RowBox[{
  RowBox[{"$Assumptions", "=", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], "\[Element]", 
    TemplateBox[{},
     "Reals"]}]}], ";"}]], "Input",
 CellChangeTimes->{{3.818140205206332*^9, 3.8181402190930862`*^9}},
 CellLabel->"In[1]:=",ExpressionUUID->"af026db1-7e50-4aae-807b-1ce6f8b44ca4"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.818140225924383*^9, 3.8181402567654047`*^9}, {
  3.818143660871539*^9, 3.818143661443686*^9}, {3.818168001039897*^9, 
  3.81816800134186*^9}, {3.818168180968347*^9, 3.818168181031329*^9}},
 CellLabel->"In[2]:=",ExpressionUUID->"d6752bf8-8b3b-4ebb-96ac-65127c588d72"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{
  3.818140257211437*^9, 3.81814233516675*^9, 3.818142681114938*^9, {
   3.818143659298221*^9, 3.818143700513908*^9}, {3.8181443339819736`*^9, 
   3.818144342056416*^9}, 3.818145374439254*^9, 3.8181460779070597`*^9, 
   3.818151233932131*^9, 3.818151311979403*^9, 3.818151750245265*^9, 
   3.8181596920042887`*^9, 3.81816733729233*^9, 3.8181675007743807`*^9, 
   3.818167604162397*^9, 3.81816779273059*^9, 3.818168001802363*^9, 
   3.818168191514748*^9, 3.818168233680788*^9, 3.819716584739538*^9},
 CellLabel->"Out[3]=",ExpressionUUID->"48fa33fb-b49a-4f7e-924f-94986588040a"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", "y"}], ",", " ", "n"}], "]"}], "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], " ", 
          RowBox[{"f", "[", 
           RowBox[{"y", ",", "n"}], "]"}]}]}], ",", 
        RowBox[{"{", 
         RowBox[{"x", ",", "y"}], "}"}]}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818140287733284*^9, 3.818140333229371*^9}, {
  3.818143599643519*^9, 3.818143610636348*^9}, {3.818143685180457*^9, 
  3.8181436933501062`*^9}, {3.818145168396035*^9, 3.818145174851321*^9}, {
  3.818145307984054*^9, 3.818145309711309*^9}, {3.818146084890275*^9, 
  3.818146090735858*^9}, {3.818151344858057*^9, 3.818151351517991*^9}, {
  3.818159325342852*^9, 3.818159354362829*^9}, {3.81815940151412*^9, 
  3.818159410086164*^9}, {3.818159702825521*^9, 3.818159728559155*^9}, {
  3.818167341879003*^9, 3.8181674557640057`*^9}, {3.818167509140544*^9, 
  3.818167511739132*^9}, {3.818167579882263*^9, 3.818167663053783*^9}, {
  3.818168211999712*^9, 3.81816821626082*^9}, {3.819716591801076*^9, 
  3.819716614333769*^9}},
 CellLabel->"In[4]:=",ExpressionUUID->"66b90039-4add-41be-9dc8-e3801d299919"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", "=", 
  RowBox[{
   RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.818140349982601*^9, 3.818140355197728*^9}, {
  3.818142685255755*^9, 3.8181426890763607`*^9}, {3.818143614492283*^9, 
  3.818143615754689*^9}, {3.818143665982217*^9, 3.818143680988772*^9}, {
  3.818143712642333*^9, 3.818143714729566*^9}, {3.818143752810752*^9, 
  3.8181437530014353`*^9}, {3.8181598115127277`*^9, 3.818159823309122*^9}, {
  3.818167498365052*^9, 3.818167525108748*^9}, {3.818167804204111*^9, 
  3.818167824917338*^9}, {3.8181678624102573`*^9, 3.8181678624934807`*^9}, {
  3.818168006083107*^9, 3.818168006186816*^9}, {3.8181681943001957`*^9, 
  3.818168194361513*^9}, {3.8181682691465273`*^9, 3.8181682691730328`*^9}, {
  3.819716643348901*^9, 3.819716645422996*^9}},
 CellLabel->"In[6]:=",ExpressionUUID->"94e5eb8a-6f30-420f-9172-71a4e1afeade"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "3"}], "}"}]], "Output",
 CellChangeTimes->{{3.819716617478744*^9, 3.819716645787863*^9}},
 CellLabel->"Out[6]=",ExpressionUUID->"9177df49-3d82-4876-bca4-79f219607d8d"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "1"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"sol1", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"sol1", "[", 
       RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "FullSimplify"}]], "Input",
 CellLabel->"In[7]:=",ExpressionUUID->"b8e8d113-f858-4728-94b9-e25b2cf80238"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", "x", ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{3.819716652741992*^9},
 CellLabel->"Out[7]=",ExpressionUUID->"7d3db860-8108-4ee3-b2bd-7a4d322aa694"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", "0"}]], "Input",
 CellChangeTimes->{{3.819716678706465*^9, 3.819716683560316*^9}},
 CellLabel->"In[8]:=",ExpressionUUID->"f07fe4ad-8c93-4c7e-9e7d-5dfa19decc43"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", " ", "y"}], "]"}], "\[Equal]", " ", 
  RowBox[{
   RowBox[{"f", "[", "x", "]"}], 
   RowBox[{"f", "[", "y", "]"}]}]}]], "Input",
 CellChangeTimes->{{3.8197166867860537`*^9, 3.819716692632689*^9}},
 CellLabel->"In[9]:=",ExpressionUUID->"9e5b8a11-56cc-4a00-a935-0e3132fe5ca1"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{3.819716693041692*^9},
 CellLabel->"Out[9]=",ExpressionUUID->"b6590e3a-3593-490b-847f-aca9c42ccd0d"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", "x"}]], "Input",
 CellChangeTimes->{{3.819716697058649*^9, 3.819716699593566*^9}},
 CellLabel->"In[10]:=",ExpressionUUID->"70047a44-b22c-41de-9c25-cfea2d8d4924"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", " ", "y"}], "]"}], "\[Equal]", " ", 
  RowBox[{
   RowBox[{"f", "[", "x", "]"}], 
   RowBox[{"f", "[", "y", "]"}]}]}]], "Input",
 CellLabel->"In[11]:=",ExpressionUUID->"099bd213-44ba-4af0-94fb-31bc5f977a96"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{3.819716704313181*^9},
 CellLabel->"Out[11]=",ExpressionUUID->"1d52eeaa-fbec-4e6f-8e5c-9489bfd46970"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", "1"}]], "Input",
 CellChangeTimes->{{3.819716707290917*^9, 3.819716712738429*^9}},
 CellLabel->"In[12]:=",ExpressionUUID->"f923e5a3-1ab1-4d6b-86ae-d3b6f6e5d9a7"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", " ", "y"}], "]"}], "\[Equal]", " ", 
  RowBox[{
   RowBox[{"f", "[", "x", "]"}], 
   RowBox[{"f", "[", "y", "]"}]}]}]], "Input",
 CellLabel->"In[13]:=",ExpressionUUID->"7dfcb28d-db1e-4025-bd6f-c36e40d7f337"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{3.819716717170175*^9},
 CellLabel->"Out[13]=",ExpressionUUID->"83427341-5d4a-441f-a2c0-f74c9afcd1e3"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", "=", 
  RowBox[{
   RowBox[{"sol", "[", "2", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.8197167320514402`*^9, 3.819716733475559*^9}},
 CellLabel->"In[14]:=",ExpressionUUID->"7955e0a2-4504-47e6-98d8-b797338a8bd5"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "4"}], "}"}]], "Output",
 CellChangeTimes->{3.819716733975112*^9},
 CellLabel->"Out[14]=",ExpressionUUID->"0612841b-ab52-4a8a-8b20-ab35e2bd20ec"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "2"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"sol2", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"sol2", "[", 
       RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "FullSimplify"}]], "Input",
 CellChangeTimes->{{3.819716743580082*^9, 3.8197167524375563`*^9}},
 CellLabel->"In[16]:=",ExpressionUUID->"8b1f4623-eb62-409c-8c36-2cbb1545f8b6"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   SuperscriptBox["x", "2"], ",", "x", ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{{3.819716749327526*^9, 3.81971675286996*^9}},
 CellLabel->"Out[16]=",ExpressionUUID->"c9c6bb9a-111d-4fb6-b041-c35705bee489"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  SuperscriptBox["x", "2"]}]], "Input",
 CellChangeTimes->{{3.819716763262199*^9, 3.8197167692709227`*^9}},
 CellLabel->"In[17]:=",ExpressionUUID->"83efcd74-2934-4b7e-8324-8e196ac24f6a"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", " ", "y"}], "]"}], "\[Equal]", 
  RowBox[{
   RowBox[{"f", "[", "x", "]"}], 
   RowBox[{"f", "[", "y", "]"}]}]}]], "Input",
 CellChangeTimes->{{3.8197167713123407`*^9, 3.819716782008157*^9}},
 CellLabel->"In[18]:=",ExpressionUUID->"1ddf41c7-8adb-441d-8a10-b126e8aad916"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{3.819716782394905*^9},
 CellLabel->"Out[18]=",ExpressionUUID->"79a7d73e-082e-48b9-afc7-cce869073430"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol3", "=", 
  RowBox[{
   RowBox[{"sol", "[", "3", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.8197168212076387`*^9, 3.8197168241060658`*^9}},
 CellLabel->"In[19]:=",ExpressionUUID->"ce65cb7a-9bc6-41ab-bf62-d91b94a145dd"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "1"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "5"}], "}"}]], "Output",
 CellChangeTimes->{3.8197168246625843`*^9},
 CellLabel->"Out[19]=",ExpressionUUID->"5d5e8d4b-9277-4aa2-9a7c-dd51dea21ac4"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      RowBox[{"x", ",", "3"}], "]"}], "/.", 
     RowBox[{
      RowBox[{"sol3", "[", 
       RowBox[{"[", "1", "]"}], "]"}], "[", 
      RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"n", ",", " ", "1", ",", " ", 
      RowBox[{"sol3", "[", 
       RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
  "FullSimplify"}]], "Input",
 CellChangeTimes->{{3.819716832347541*^9, 3.8197168361480722`*^9}},
 CellLabel->"In[20]:=",ExpressionUUID->"4def0740-684f-4d64-ad32-1dbd26a48a54"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", 
   SuperscriptBox["x", "3"], ",", 
   SuperscriptBox["x", "2"], ",", "x", ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{3.819716836748328*^9},
 CellLabel->"Out[20]=",ExpressionUUID->"acaac3a2-e698-496c-9e99-a62ad0164dbd"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"So", ",", " ", 
   RowBox[{"all", " ", "\"\<pure\>\"", " ", "polynomials", " ", "work"}]}], 
  " ", "*)"}]], "Input",
 CellChangeTimes->{{3.819716956824527*^9, 3.8197169706577063`*^9}, 
   3.819717014641987*^9, 
   3.819718813130185*^9},ExpressionUUID->"6e94d565-1c62-470e-b8cf-\
b4753ed0c294"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{"Let", "'"}], "s", " ", "explore", " ", "rational", " ", 
    RowBox[{"functions", ":"}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
    RowBox[{
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"a", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
     RowBox[{"Total", "@", 
      RowBox[{"Array", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"b", "[", "#", "]"}], " ", 
          RowBox[{"x", "^", "#"}]}], "&"}], ",", 
        RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], 
   "\[IndentingNewLine]", 
   RowBox[{"f", "[", 
    RowBox[{"x", ",", " ", "2"}], "]"}]}]}]], "Input",
 CellLabel->"In[5]:=",ExpressionUUID->"bc74323b-d76d-445d-838c-63340ebd2712"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.819718846613448*^9, 3.819719108799942*^9},
 CellLabel->"Out[6]=",ExpressionUUID->"dd59ad75-4280-4709-9b4b-15992f2c4bb5"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", "y"}], ",", " ", "n"}], "]"}], "\[Equal]", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], " ", 
          RowBox[{"f", "[", 
           RowBox[{"y", ",", "n"}], "]"}]}]}], ",", 
        RowBox[{"{", 
         RowBox[{"x", ",", "y"}], "}"}]}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818140287733284*^9, 3.818140333229371*^9}, {
  3.818143599643519*^9, 3.818143610636348*^9}, {3.818143685180457*^9, 
  3.8181436933501062`*^9}, {3.818145168396035*^9, 3.818145174851321*^9}, {
  3.818145307984054*^9, 3.818145309711309*^9}, {3.818146084890275*^9, 
  3.818146090735858*^9}, {3.818151344858057*^9, 3.818151351517991*^9}, {
  3.818159325342852*^9, 3.818159354362829*^9}, {3.81815940151412*^9, 
  3.818159410086164*^9}, {3.818159702825521*^9, 3.818159728559155*^9}, {
  3.818167341879003*^9, 3.8181674557640057`*^9}, {3.818167509140544*^9, 
  3.818167511739132*^9}, {3.818167579882263*^9, 3.818167663053783*^9}, {
  3.818168211999712*^9, 3.81816821626082*^9}, {3.819716591801076*^9, 
  3.819716614333769*^9}},
 CellLabel->"In[7]:=",ExpressionUUID->"a3bc9fdf-2f06-44d5-a02c-12df11ccdd4e"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", "=", 
  RowBox[{
   RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.818140349982601*^9, 3.818140355197728*^9}, {
  3.818142685255755*^9, 3.8181426890763607`*^9}, {3.818143614492283*^9, 
  3.818143615754689*^9}, {3.818143665982217*^9, 3.818143680988772*^9}, {
  3.818143712642333*^9, 3.818143714729566*^9}, {3.818143752810752*^9, 
  3.8181437530014353`*^9}, {3.8181598115127277`*^9, 3.818159823309122*^9}, {
  3.818167498365052*^9, 3.818167525108748*^9}, {3.818167804204111*^9, 
  3.818167824917338*^9}, {3.8181678624102573`*^9, 3.8181678624934807`*^9}, {
  3.818168006083107*^9, 3.818168006186816*^9}, {3.8181681943001957`*^9, 
  3.818168194361513*^9}, {3.8181682691465273`*^9, 3.8181682691730328`*^9}, {
  3.819716643348901*^9, 3.819716645422996*^9}, {3.819719135548263*^9, 
  3.819719146656966*^9}},
 CellLabel->"In[14]:=",ExpressionUUID->"1eb4d59c-b158-46ab-89ce-6c67277b3c69"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "1", "]"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "1", "]"}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "0", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "1", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"b", "[", "1", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"a", "[", "1", "]"}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "11"}], "}"}]], "Output",
 CellChangeTimes->{{3.819716617478744*^9, 3.819716645787863*^9}, {
  3.819718891800782*^9, 3.819718900725925*^9}, {3.819719110401062*^9, 
  3.819719148992489*^9}},
 CellLabel->"Out[14]=",ExpressionUUID->"f2a7bb48-61c9-492e-a482-55e746c4440b"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "1"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol1", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol1", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.819719100574862*^9, 3.819719115895104*^9}, {
  3.8197191511707277`*^9, 3.819719153416442*^9}},
 CellLabel->"In[15]:=",ExpressionUUID->"23664627-6bc4-4f0c-854c-1d5b40933c93"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", "1", ",", "ComplexInfinity", ",", "x", ",", 
   FractionBox["1", "x"]}], "}"}]], "Output",
 CellChangeTimes->{
  3.819718903251059*^9, {3.819719105187028*^9, 3.819719153735024*^9}},
 CellLabel->"Out[15]=",ExpressionUUID->"83c4e837-9d9b-4fb0-9706-dbee844de21a"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol2", "=", 
   RowBox[{
    RowBox[{"sol", "[", "2", "]"}], "//", "Simplify"}]}], ";"}]], "Input",
 CellChangeTimes->{{3.819719163152821*^9, 3.819719188635125*^9}},
 CellLabel->"In[17]:=",ExpressionUUID->"0c78b518-b46f-4414-aa5f-ea69b307ab4f"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "2"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol2", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol2", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.819719194178013*^9, 3.819719198899843*^9}},
 CellLabel->"In[18]:=",ExpressionUUID->"40344889-5436-462d-a399-fd5b3a413923"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"0", ",", "1", ",", "ComplexInfinity", ",", "x", ",", 
   SuperscriptBox["x", "2"], ",", 
   FractionBox["1", "x"], ",", 
   FractionBox["1", 
    SuperscriptBox["x", "2"]]}], "}"}]], "Output",
 CellChangeTimes->{3.819719199685376*^9},
 CellLabel->"Out[18]=",ExpressionUUID->"8e983ada-92b2-4e76-ab77-d2c1e8153977"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{"Another", ":", " ", 
     RowBox[{
      RowBox[{"f", 
       RowBox[{"(", "x", ")"}]}], " ", "+", " ", 
      RowBox[{"f", 
       RowBox[{"(", 
        FractionBox[
         RowBox[{"x", "-", "1"}], "x"], ")"}]}]}]}], " ", "=", " ", 
    RowBox[{"1", "+", "x"}]}], ",", " ", 
   RowBox[{"x", " ", "\[NotEqual]", " ", "0"}], ",", " ", "1"}], " ", 
  "*)"}]], "Input",
 CellChangeTimes->{{3.821687854116405*^9, 
  3.8216879038306026`*^9}},ExpressionUUID->"9d994d20-3739-46f4-8744-\
8b280ea55a4b"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[1]:=",ExpressionUUID->"09700ef3-64a5-4f2f-8dd8-342b651b7694"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.8216879234279737`*^9},
 CellLabel->"Out[2]=",ExpressionUUID->"b6b484c1-721c-4f42-aba5-39a9c6483444"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", " ", "n"}], "]"}], "+", 
          RowBox[{"f", "[", 
           RowBox[{
            FractionBox[
             RowBox[{"x", "-", "1"}], "x"], ",", "n"}], "]"}]}], "\[Equal]", 
         RowBox[{"1", "+", "x"}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.821687939504148*^9, 3.821687972363283*^9}},
 CellLabel->"In[3]:=",ExpressionUUID->"50eb3c06-3b30-4d71-af35-0c2e53947a51"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", "=", 
  RowBox[{"sol", "[", "1", "]"}]}]], "Input",
 CellChangeTimes->{{3.821687984933279*^9, 3.8216879859691343`*^9}, {
  3.8216880512446413`*^9, 3.8216880521407623`*^9}},
 CellLabel->"In[7]:=",ExpressionUUID->"13845a6e-625f-4c21-bc86-15ee1a866242"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "2"}], "}"}]], "Output",
 CellChangeTimes->{3.82168798637558*^9, 3.82168805311043*^9},
 CellLabel->"Out[7]=",ExpressionUUID->"0da23133-1c49-4514-a15d-59dc943e3b04"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "1"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol1", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol1", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellLabel->"In[8]:=",ExpressionUUID->"01a564b2-9f37-4e3c-8315-ff0acb3b144f"],

Cell[BoxData[
 RowBox[{"{", "ComplexInfinity", "}"}]], "Output",
 CellChangeTimes->{3.821688060904935*^9},
 CellLabel->"Out[8]=",ExpressionUUID->"699ea491-1e48-495c-9ac5-a9ca41853518"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", "=", 
  RowBox[{"sol", "[", "2", "]"}]}]], "Input",
 CellChangeTimes->{{3.8216879967149878`*^9, 3.8216879985799923`*^9}, {
  3.8216880651119137`*^9, 3.821688067101589*^9}},
 CellLabel->"In[9]:=",ExpressionUUID->"ec6ffebe-0e17-4797-a7ba-f195f6f6d100"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "18"}], "}"}]], "Output",
 CellChangeTimes->{3.821687998981958*^9, 3.821688067557928*^9},
 CellLabel->"Out[9]=",ExpressionUUID->"a325413e-7c6a-4239-a0e1-038343131b2e"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "2"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol2", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol2", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.821688076264772*^9, 3.821688080523036*^9}},
 CellLabel->"In[10]:=",ExpressionUUID->"ba3370c9-b18e-474e-b38e-36d403b0d84a"],

Cell[BoxData[
 RowBox[{"{", "ComplexInfinity", "}"}]], "Output",
 CellChangeTimes->{3.821688081783235*^9},
 CellLabel->"Out[10]=",ExpressionUUID->"159b82a4-5d09-4831-bf9d-c51cceddc747"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol3", "=", 
  RowBox[{"sol", "[", "3", "]"}]}]], "Input",
 CellChangeTimes->{{3.821688003941123*^9, 3.82168801668366*^9}, {
  3.821688083987816*^9, 3.821688090003964*^9}},
 CellLabel->"In[12]:=",ExpressionUUID->"8f07d548-f7d2-4273-af2c-ad56864b5523"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "2", "]"}], "2"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "2", "]"}], "2"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"b", "[", "2", "]"}], "2"]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "2", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "2", "]"}], "2"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "2", "]"}], "2"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "3", "]"}], "\[Rule]", 
        FractionBox[
         RowBox[{"b", "[", "2", "]"}], "2"]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         RowBox[{"b", "[", "2", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "3", "]"}], "\[Rule]", "0"}]}], "}"}]}], "}"}], ",",
    "12"}], "}"}]], "Output",
 CellChangeTimes->{
  3.8216880172825203`*^9, {3.821688085889225*^9, 3.821688090684661*^9}},
 CellLabel->"Out[12]=",ExpressionUUID->"60529652-50a2-403d-a4af-c3c8e8afd724"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "3"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol3", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol3", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.8216880975767927`*^9, 3.821688102401791*^9}},
 CellLabel->"In[13]:=",ExpressionUUID->"3cbdb7fd-2e01-46db-8848-9b17427060c9"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"ComplexInfinity", ",", 
   FractionBox[
    RowBox[{"1", "+", 
     SuperscriptBox["x", "2"], "-", 
     SuperscriptBox["x", "3"]}], 
    RowBox[{
     RowBox[{"2", " ", "x"}], "-", 
     RowBox[{"2", " ", 
      SuperscriptBox["x", "2"]}]}]]}], "}"}]], "Output",
 CellChangeTimes->{3.821688103413803*^9},
 CellLabel->"Out[13]=",ExpressionUUID->"50b00552-d113-4a42-80d6-1073eb2a44be"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  FractionBox[
   RowBox[{"1", "+", 
    SuperscriptBox["x", "2"], "-", 
    SuperscriptBox["x", "3"]}], 
   RowBox[{
    RowBox[{"2", " ", "x"}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["x", "2"]}]}]]}]], "Input",
 CellLabel->"In[19]:=",ExpressionUUID->"d09521fa-4ea8-4b06-894e-83faa241d05a"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", "x", "]"}], "+", 
   RowBox[{"f", "[", 
    FractionBox[
     RowBox[{"x", "-", "1"}], "x"], "]"}]}], "//", "FullSimplify"}]], "Input",\

 CellChangeTimes->{{3.821688145926312*^9, 3.821688186566733*^9}},
 CellLabel->"In[20]:=",ExpressionUUID->"d7b5e1d8-1217-47af-86a8-46ad08197629"],

Cell[BoxData[
 RowBox[{"1", "+", "x"}]], "Output",
 CellChangeTimes->{{3.8216881546311502`*^9, 3.821688158772398*^9}, 
   3.821688209539764*^9},
 CellLabel->"Out[20]=",ExpressionUUID->"a9a8149d-0dd9-412c-8c2c-864ebee2c7e9"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x", "]"}], "//", "Expand"}]], "Input",
 CellChangeTimes->{{3.821688210705925*^9, 3.821688215960594*^9}},
 CellLabel->"In[21]:=",ExpressionUUID->"3e0d5300-fc21-4323-b4ac-0ef7436de1de"],

Cell[BoxData[
 RowBox[{
  FractionBox["1", 
   RowBox[{
    RowBox[{"2", " ", "x"}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["x", "2"]}]}]], "+", 
  FractionBox[
   SuperscriptBox["x", "2"], 
   RowBox[{
    RowBox[{"2", " ", "x"}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["x", "2"]}]}]], "-", 
  FractionBox[
   SuperscriptBox["x", "3"], 
   RowBox[{
    RowBox[{"2", " ", "x"}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["x", "2"]}]}]]}]], "Output",
 CellChangeTimes->{3.8216882163367577`*^9},
 CellLabel->"Out[21]=",ExpressionUUID->"e7d2f455-2614-4b5c-b385-65fa751592ec"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{
   SubscriptBox["f", "RazimanTV"], "[", "x_", "]"}], ":=", " ", 
  RowBox[{
   FractionBox["x", "2"], "-", 
   FractionBox["1", 
    RowBox[{"2", 
     RowBox[{"(", 
      RowBox[{"x", "-", "1"}], ")"}]}]], "+", 
   FractionBox["1", 
    RowBox[{"2", "x"}]]}]}]], "Input",
 CellChangeTimes->{{3.821688218774795*^9, 3.821688248000114*^9}},
 CellLabel->"In[22]:=",ExpressionUUID->"9c491baa-037b-47a7-abe4-92f18b3b7088"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    SubscriptBox["f", "RazimanTV"], "[", "x", "]"}], "+", 
   RowBox[{
    SubscriptBox["f", "RazimanTV"], "[", 
    FractionBox[
     RowBox[{"x", "-", "1"}], "x"], "]"}]}], "//", "FullSimplify"}]], "Input",\

 CellChangeTimes->{{3.8216882576646833`*^9, 3.8216882676370707`*^9}},
 CellLabel->"In[23]:=",ExpressionUUID->"85fc9405-657f-45d0-a8db-f4ee013b6431"],

Cell[BoxData[
 RowBox[{"1", "+", "x"}]], "Output",
 CellChangeTimes->{3.821688268233007*^9},
 CellLabel->"Out[23]=",ExpressionUUID->"342378a1-62d9-47c6-ab8a-82dd8f783192"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"f", "[", "x", "]"}]], "Input",
 CellChangeTimes->{{3.821688313892152*^9, 3.821688317627591*^9}, 
   3.8216890748907127`*^9},
 CellLabel->"In[36]:=",ExpressionUUID->"fca9b7d0-8667-42f1-bf62-ce431ba342c2"],

Cell[BoxData[
 FractionBox[
  RowBox[{"1", "+", 
   SuperscriptBox["x", "2"], "-", 
   SuperscriptBox["x", "3"]}], 
  RowBox[{
   RowBox[{"2", " ", "x"}], "-", 
   RowBox[{"2", " ", 
    SuperscriptBox["x", "2"]}]}]]], "Output",
 CellChangeTimes->{3.821688317927129*^9, 3.821689075742227*^9},
 CellLabel->"Out[36]=",ExpressionUUID->"533dc832-c76b-4c7f-833a-6e6324243a69"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   SubscriptBox["f", "RazimanTV"], "[", "x", "]"}], "//", 
  "Together"}]], "Input",
 CellChangeTimes->{{3.821688349897644*^9, 3.821688408291264*^9}, {
  3.821688446223259*^9, 3.821688468948935*^9}},
 CellLabel->"In[37]:=",ExpressionUUID->"763ab6bd-7e90-4ab6-a713-f52eb11ddca8"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"-", "1"}], "-", 
   SuperscriptBox["x", "2"], "+", 
   SuperscriptBox["x", "3"]}], 
  RowBox[{"2", " ", 
   RowBox[{"(", 
    RowBox[{
     RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", "x"}]]], "Output",
 CellChangeTimes->{{3.8216883618550863`*^9, 3.821688374948608*^9}, 
   3.821688408824586*^9, {3.821688446654861*^9, 3.821688469520266*^9}, 
   3.821689077280223*^9},
 CellLabel->"Out[37]=",ExpressionUUID->"1e75edc3-8818-4cfe-a0f9-6b7bd43ed55a"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    SubscriptBox["f", "RazimanTV"], "[", "x", "]"}], "\[Equal]", 
   RowBox[{"f", "[", "x", "]"}]}], "//", "Reduce"}]], "Input",
 CellChangeTimes->{{3.821688271602127*^9, 3.82168828741805*^9}},
 CellLabel->"In[38]:=",ExpressionUUID->"49d1b79f-e357-46a8-a215-9a6e1dff3ade"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.82168828197086*^9, 3.821688287772771*^9}, {
  3.8216890614387608`*^9, 3.8216890790605087`*^9}},
 CellLabel->"Out[38]=",ExpressionUUID->"d339b426-21ab-478c-a336-397e86297950"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{"Another", ":", " ", 
     RowBox[{"f", 
      RowBox[{"(", 
       SqrtBox[
        RowBox[{
         SuperscriptBox["x", "2"], " ", "+", " ", 
         SuperscriptBox["y", "2"]}]], ")"}]}]}], " ", "=", " ", 
    RowBox[{
     RowBox[{"f", 
      RowBox[{"(", "x", ")"}]}], " ", "+", " ", 
     RowBox[{"f", 
      RowBox[{"(", "y", ")"}]}]}]}], ",", " ", 
   RowBox[{
    RowBox[{"f", 
     RowBox[{"(", "1", ")"}]}], " ", "=", " ", "1"}], " ", ",", " ", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], " ", "\[Element]", " ", 
    TemplateBox[{},
     "Reals"]}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.821689081673437*^9, 
  3.821689161259256*^9}},ExpressionUUID->"14709f19-df4a-40a2-88b7-\
a94fd5da0cf7"],

Cell[BoxData[
 RowBox[{
  RowBox[{"$Assumptions", "=", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"x", ",", " ", "y"}], "}"}], "\[Element]", 
    TemplateBox[{},
     "Reals"]}]}], ";"}]], "Input",
 CellChangeTimes->{{3.818140205206332*^9, 3.8181402190930862`*^9}, 
   3.821689198189568*^9},
 CellLabel->"In[39]:=",ExpressionUUID->"6bdeeb2b-f695-4a87-81ca-51a192f17b0a"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.818140225924383*^9, 3.8181402567654047`*^9}, {
   3.818143660871539*^9, 3.818143661443686*^9}, {3.818168001039897*^9, 
   3.81816800134186*^9}, {3.818168180968347*^9, 3.818168181031329*^9}, 
   3.821719607850214*^9},
 CellLabel->"In[13]:=",ExpressionUUID->"abfb7ca4-dc48-45f9-b5d1-bd1fbe4bb6b7"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.821689201387569*^9, 3.821719624480822*^9},
 CellLabel->"Out[14]=",ExpressionUUID->"edced59b-2ef3-491f-9ef2-9b102410f077"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           SqrtBox[
            RowBox[{
             SuperscriptBox["x", "2"], "+", 
             SuperscriptBox["y", "2"]}]], ",", " ", "n"}], "]"}], "\[Equal]", 
         
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"x", ",", "n"}], "]"}], "+", 
          RowBox[{"f", "[", 
           RowBox[{"y", ",", "n"}], "]"}]}]}], ",", 
        RowBox[{"{", 
         RowBox[{"x", ",", "y"}], "}"}]}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.8216892337114983`*^9, 3.8216892933391*^9}},
 CellLabel->"In[15]:=",ExpressionUUID->"da5ed8a9-0923-40f2-a011-b030da2a242b"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", "=", 
  RowBox[{"sol", "[", "1", "]"}]}]], "Input",
 CellChangeTimes->{{3.821689271122007*^9, 3.8216892744248466`*^9}},
 CellLabel->"In[16]:=",ExpressionUUID->"e320a826-24a8-42ed-b217-07b501f1feef"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], "}"}], ",", 
   "1"}], "}"}]], "Output",
 CellChangeTimes->{{3.821689274784301*^9, 3.8216892972881737`*^9}, {
  3.82171962247607*^9, 3.821719626299408*^9}},
 CellLabel->"Out[16]=",ExpressionUUID->"b81a656b-fd96-4c7f-b744-be25dc8a66ba"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", "=", 
  RowBox[{"sol", "[", "2", "]"}]}]], "Input",
 CellChangeTimes->{{3.8216892779717493`*^9, 3.821689282001639*^9}},
 CellLabel->"In[17]:=",ExpressionUUID->"4fd52582-5bc9-4bfd-a633-619a42962906"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], "}"}], ",", 
   "1"}], "}"}]], "Output",
 CellChangeTimes->{{3.8216892823562717`*^9, 3.821689299048053*^9}, 
   3.821719628729157*^9},
 CellLabel->"Out[17]=",ExpressionUUID->"f90e8e47-4710-4174-b2f9-05fcd96b37ae"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "2"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol2", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol2", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.821689371317877*^9, 3.8216893754064407`*^9}},
 CellLabel->"In[18]:=",ExpressionUUID->"b651f813-2072-4746-bed9-ebf67f8627e0"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}], "}"}]], "Output",
 CellChangeTimes->{3.8216893757953453`*^9, 3.821719630312377*^9},
 CellLabel->"Out[18]=",ExpressionUUID->"0f552eec-e6e2-4397-ab70-645052ac15e6"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"a", " ", 
   SuperscriptBox["x", "2"]}]}]], "Input",
 CellLabel->"In[1]:=",ExpressionUUID->"2c8907c2-1ff9-45ef-ad15-88ea075a815c"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"f", "[", 
    SqrtBox[
     RowBox[{
      SuperscriptBox["x", "2"], "+", 
      SuperscriptBox["y", "2"]}]], "]"}], "\[Equal]", 
   RowBox[{
    RowBox[{"f", "[", "x", "]"}], "+", 
    RowBox[{"f", "[", "y", "]"}]}]}], "//", "FullSimplify"}]], "Input",
 CellLabel->"In[2]:=",ExpressionUUID->"6da0b098-a2c4-4bc5-96dd-effe07454c1f"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{
  3.821690908386408*^9, 3.82169102030403*^9, {3.8216911183767757`*^9, 
   3.821691156436167*^9}, 3.8216912349649267`*^9},
 CellLabel->"Out[2]=",ExpressionUUID->"b0274d25-86e5-42f0-b2f3-48cd1cc85b34"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{
     RowBox[{"But", " ", "f", 
      RowBox[{"(", "1", ")"}]}], " ", "=", " ", "1"}], ",", " ", 
    RowBox[{"so", ":"}]}], " ", "*)"}], "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"f", "[", 
      SqrtBox[
       RowBox[{
        SuperscriptBox["x", "2"], "+", 
        SuperscriptBox["y", "2"]}]], "]"}], "\[Equal]", 
     RowBox[{
      RowBox[{"f", "[", "x", "]"}], "+", 
      RowBox[{"f", "[", "y", "]"}]}]}], "&&", 
    RowBox[{
     RowBox[{"f", "[", "1", "]"}], "\[Equal]", "1"}]}], "//", 
   "Reduce"}]}]], "Input",
 CellChangeTimes->{{3.82169087955284*^9, 3.821690925275201*^9}, {
  3.8216910256351023`*^9, 3.821691032249464*^9}, {3.821691122784709*^9, 
  3.82169112309041*^9}},
 CellLabel->"In[3]:=",ExpressionUUID->"a6b491af-6117-41c0-a178-cb6f88908635"],

Cell[BoxData[
 RowBox[{"a", "\[Equal]", "1"}]], "Output",
 CellChangeTimes->{3.821691123630673*^9, 3.8216911579982357`*^9, 
  3.82169123616409*^9},
 CellLabel->"Out[3]=",ExpressionUUID->"c7fdda57-1ec0-45c2-89a5-c7b3080b09e0"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{
   RowBox[{"a", " ", 
    SuperscriptBox["x", "2"]}], "/.", 
   RowBox[{"a", "\[Rule]", "1"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", "x", "]"}]}], "Input",
 CellChangeTimes->{{3.8216911862979507`*^9, 3.821691198866687*^9}},
 CellLabel->"In[4]:=",ExpressionUUID->"fc4bc0ad-2afc-4e5d-8799-5578aadd2397"],

Cell[BoxData[
 SuperscriptBox["x", "2"]], "Output",
 CellChangeTimes->{3.821691199174707*^9, 3.821691238789707*^9},
 CellLabel->"Out[5]=",ExpressionUUID->"1f29c81a-fb7c-4980-af5f-424f3016b2e8"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{"Another", ":", " ", 
    RowBox[{
     RowBox[{"2", " ", "f", 
      RowBox[{"(", "x", ")"}]}], " ", "-", " ", 
     RowBox[{"x", " ", "f", 
      RowBox[{"(", 
       FractionBox[
        RowBox[{
         RowBox[{"2", " ", "x"}], " ", "+", " ", "3"}], 
        RowBox[{"x", "-", "2"}]], ")"}]}]}]}], " ", "=", " ", "3"}], " ", 
  "*)"}]], "Input",
 CellChangeTimes->{{3.821689081673437*^9, 3.821689161259256*^9}, {
  3.8217194066013403`*^9, 
  3.821719468277017*^9}},ExpressionUUID->"a0c08030-c94c-4bcc-aac0-\
d08000514a4f"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}], "/", 
   RowBox[{"Total", "@", 
    RowBox[{"Array", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "#", "]"}], " ", 
        RowBox[{"x", "^", "#"}]}], "&"}], ",", 
      RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellChangeTimes->{{3.818140225924383*^9, 3.8181402567654047`*^9}, {
   3.818143660871539*^9, 3.818143661443686*^9}, {3.818168001039897*^9, 
   3.81816800134186*^9}, {3.818168180968347*^9, 3.818168181031329*^9}, 
   3.821719637767322*^9},
 CellLabel->"In[19]:=",ExpressionUUID->"e9708de2-c47e-4663-9911-0ae162dbe73e"],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{"a", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"a", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"a", "[", "2", "]"}]}]}], 
  RowBox[{
   RowBox[{"b", "[", "0", "]"}], "+", 
   RowBox[{"x", " ", 
    RowBox[{"b", "[", "1", "]"}]}], "+", 
   RowBox[{
    SuperscriptBox["x", "2"], " ", 
    RowBox[{"b", "[", "2", "]"}]}]}]]], "Output",
 CellChangeTimes->{3.821689201387569*^9, 3.821719481414136*^9, 
  3.821719571144347*^9, 3.8217196386697807`*^9},
 CellLabel->"Out[20]=",ExpressionUUID->"e91bd8c2-21d5-4622-acd4-2ab61f4025da"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"2", 
           RowBox[{"f", "[", 
            RowBox[{"x", ",", " ", "n"}], "]"}]}], "-", 
          RowBox[{"x", " ", 
           RowBox[{"f", "[", 
            RowBox[{
             FractionBox[
              RowBox[{
               RowBox[{"2", "x"}], "+", "3"}], 
              RowBox[{"x", "-", "2"}]], ",", "n"}], "]"}]}]}], "\[Equal]", 
         "3"}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.8216892337114983`*^9, 3.8216892933391*^9}, {
  3.821719485881229*^9, 3.821719545457506*^9}},
 CellLabel->"In[21]:=",ExpressionUUID->"2c0847fc-441d-4a34-8bd9-552ead695d80"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol1", "=", 
  RowBox[{"sol", "[", "1", "]"}]}]], "Input",
 CellChangeTimes->{{3.821689271122007*^9, 3.8216892744248466`*^9}},
 CellLabel->"In[22]:=",ExpressionUUID->"09f349ef-385c-4cbf-a0f2-1ecd7224a391"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
      RowBox[{
       RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}]}], "}"}], "}"}], ",", 
   "1"}], "}"}]], "Output",
 CellChangeTimes->{3.821719551843914*^9, 3.8217196412208233`*^9},
 CellLabel->"Out[22]=",ExpressionUUID->"0e023ad0-28d4-4a58-aba5-f2ecfebe4487"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "1"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol1", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol1", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.821719665375942*^9, 3.821719669628168*^9}},
 CellLabel->"In[23]:=",ExpressionUUID->"c8f49b10-fa87-437d-8bb5-2ddf390ddabe"],

Cell[BoxData[
 RowBox[{"{", "ComplexInfinity", "}"}]], "Output",
 CellChangeTimes->{3.821719670187768*^9},
 CellLabel->"Out[23]=",ExpressionUUID->"ef0bd746-7ea9-4336-a212-83192b41e54b"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol2", "=", 
  RowBox[{"sol", "[", "2", "]"}]}]], "Input",
 CellChangeTimes->{{3.821719681937316*^9, 3.8217196841915617`*^9}},
 CellLabel->"In[24]:=",ExpressionUUID->"a5e269b5-84a7-4406-8f2a-882fae466bce"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"b", "[", "2", "]"}], "\[Rule]", "0"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "2", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"3", " ", 
           RowBox[{"b", "[", "2", "]"}]}], "2"]}]}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"6", " ", 
         RowBox[{"b", "[", "2", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "0", "]"}], "\[Rule]", 
        RowBox[{"4", " ", 
         RowBox[{"b", "[", "2", "]"}]}]}], ",", 
       RowBox[{
        RowBox[{"b", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", 
         FractionBox[
          RowBox[{"b", "[", "2", "]"}], "2"]}]}]}], "}"}]}], "}"}], ",", 
   "2"}], "}"}]], "Output",
 CellChangeTimes->{3.821719684789361*^9},
 CellLabel->"Out[24]=",ExpressionUUID->"ea55b337-76b1-448f-9bf8-11f9adc6acdd"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "2"}], "]"}], "/.", 
      RowBox[{
       RowBox[{"sol2", "[", 
        RowBox[{"[", "1", "]"}], "]"}], "[", 
       RowBox[{"[", "n", "]"}], "]"}]}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"n", ",", " ", "1", ",", " ", 
       RowBox[{"sol2", "[", 
        RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "]"}], "//", 
   "FullSimplify"}], "//", "DeleteDuplicates"}]], "Input",
 CellChangeTimes->{{3.821719692993984*^9, 3.821719697665454*^9}},
 CellLabel->"In[25]:=",ExpressionUUID->"3f3344d2-b382-4c5d-91f1-01fed4e7e2ab"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"ComplexInfinity", ",", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"3", " ", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "4"}], "+", 
        SuperscriptBox["x", "2"]}], ")"}]}], 
     RowBox[{"8", "+", 
      RowBox[{"x", " ", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", 
         RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}]}], "}"}]], "Output",
 CellChangeTimes->{3.8217196984555063`*^9},
 CellLabel->"Out[25]=",ExpressionUUID->"0ac3e8b1-7eb6-46cc-9dd8-bd753d470ec2"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  FractionBox[
   RowBox[{"3", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"-", "4"}], "+", 
      SuperscriptBox["x", "2"]}], ")"}]}], 
   RowBox[{"8", "+", 
    RowBox[{"x", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "1"}], "+", 
       RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}]], "Input",
 CellChangeTimes->{{3.8217197058924437`*^9, 3.821719712204706*^9}},
 CellLabel->"In[26]:=",ExpressionUUID->"20484ec3-ea22-4842-9d73-650cf496352c"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x", "]"}], "//", "Expand"}]], "Input",
 CellChangeTimes->{{3.821719714340198*^9, 3.821719718794381*^9}},
 CellLabel->"In[27]:=",ExpressionUUID->"f09f73c7-a145-4bf9-8705-02893244cb1d"],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", 
   FractionBox["12", 
    RowBox[{"8", "+", 
     RowBox[{"x", " ", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "1"}], "+", 
        RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}], "+", 
  FractionBox[
   RowBox[{"3", " ", 
    SuperscriptBox["x", "2"]}], 
   RowBox[{"8", "+", 
    RowBox[{"x", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "1"}], "+", 
       RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}]], "Output",
 CellChangeTimes->{3.821719719155734*^9},
 CellLabel->"Out[27]=",ExpressionUUID->"92914d6d-1454-446b-a8a5-25505be5386d"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{
    RowBox[{"-", 
     FractionBox["12", 
      RowBox[{"8", "+", 
       RowBox[{"x", " ", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"-", "1"}], "+", 
          RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}], "+", 
    FractionBox[
     RowBox[{"3", " ", 
      SuperscriptBox["x", "2"]}], 
     RowBox[{"8", "+", 
      RowBox[{"x", " ", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", 
         RowBox[{"2", " ", "x"}]}], ")"}]}]}]]}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"-", "8"}], ",", "8"}], "}"}]}], "]"}]], "Input",
 NumberMarks->False,
 CellLabel->"In[28]:=",ExpressionUUID->"15495087-6e7d-456d-a2e7-f3ec1039714e"],

Cell[BoxData[
 GraphicsBox[{{{}, {}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
      1.], LineBox[CompressedData["
1:eJwtmHk01N//x0nZmxCyjyWyLyVS6XUrUVmyJSGkpNAmpU1Joki0ECVLUqQQ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       "]]},
     Annotation[#, "Charting`Private`Tag$10911#1"]& ]}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->{True, True},
  AxesLabel->{None, None},
  AxesOrigin->{0, 0},
  DisplayFunction->Identity,
  Frame->{{False, False}, {False, False}},
  FrameLabel->{{None, None}, {None, None}},
  FrameTicks->{{Automatic, 
     Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic, 
     Charting`ScaledFrameTicks[{Identity, Identity}]}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImagePadding->All,
  Method->{
   "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "ScalingFunctions" -> None, 
    "CoordinatesToolOptions" -> {"DisplayFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]], 
        (Identity[#]& )[
         Part[#, 2]]}& ), "CopiedValueFunction" -> ({
        (Identity[#]& )[
         Part[#, 1]], 
        (Identity[#]& )[
         Part[#, 2]]}& )}},
  PlotRange->{{-8, 8}, {-1.5118541480048886`, 1.4062499887595654`}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.821719749162456*^9},
 CellLabel->"Out[28]=",ExpressionUUID->"befa1589-6c69-4962-be43-48f0e1497a00"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{"Another", " ", "one", " ", "f", 
    RowBox[{"(", 
     RowBox[{"f", 
      RowBox[{"(", "x", ")"}]}], ")"}]}], " ", "=", " ", 
   RowBox[{"x", " ", "+", " ", "1"}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.8181686836064167`*^9, 3.8181687142719927`*^9}, {
  3.8181688352134943`*^9, 3.818168843534184*^9}, {3.830135303426394*^9, 
  3.83013532872329*^9}},ExpressionUUID->"36bc8a91-497e-4f38-a1a7-\
4f4a9182b6ba"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[1]:=",ExpressionUUID->"2d764d9d-5802-4b43-94f3-a1fa1b980c65"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.8181687242728233`*^9, 3.818168845214616*^9, 
  3.818248852821267*^9, 3.818324121200102*^9, 3.830135335633893*^9},
 CellLabel->"Out[2]=",ExpressionUUID->"7dbbd60b-6ff9-473e-8983-531d354d1115"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"f", "[", 
            RowBox[{"x", ",", "n"}], "]"}], ",", "n"}], "]"}], "\[Equal]", 
         RowBox[{"x", "+", "1"}]}], ",", "x"}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.818168738706807*^9, 3.818168745488062*^9}, {
  3.830135340166987*^9, 3.830135362885034*^9}},
 CellLabel->"In[3]:=",ExpressionUUID->"81d884ef-c4dd-40ce-939a-440969e55313"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "1", "]"}], "//", "Simplify"}]], "Input",
 CellLabel->"In[4]:=",ExpressionUUID->"f1d7d73f-51f6-4b57-a3d6-9e7344a3331f"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}], ",", 
      RowBox[{
       RowBox[{"a", "[", "0", "]"}], "\[Rule]", 
       FractionBox["1", "2"]}]}], "}"}], "}"}], ",", "1"}], "}"}]], "Output",
 CellChangeTimes->{3.818168755604456*^9, 3.818168846543312*^9, 
  3.818168894283401*^9, 3.8182488630622597`*^9, 3.8183241237127533`*^9, 
  3.830135368885817*^9},
 CellLabel->"Out[4]=",ExpressionUUID->"6dd7ed31-509d-4a30-a709-cb4aaf499562"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x", ",", " ", "1"}], "]"}], "/.", 
  RowBox[{
   RowBox[{"sol", "[", "1", "]"}], "[", 
   RowBox[{"[", "1", "]"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.81816876661067*^9, 3.818168767504759*^9}},
 CellLabel->"In[5]:=",ExpressionUUID->"88f97da0-448a-4177-b0d4-b541d14aa18a"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   FractionBox["1", "2"], "+", "x"}], "}"}]], "Output",
 CellChangeTimes->{3.81816876904758*^9, 3.818168847151311*^9, 
  3.818168895059722*^9, 3.81824886576577*^9, 3.818324124676523*^9, 
  3.830135380777945*^9},
 CellLabel->"Out[5]=",ExpressionUUID->"f1a48b3d-ec46-4766-bca2-c9a71a163e10"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"x", "+", 
   FractionBox["1", "2"]}]}]], "Input",
 CellChangeTimes->{{3.830135397823656*^9, 3.8301354032392693`*^9}},
 CellLabel->"In[87]:=",ExpressionUUID->"d0051a68-adb3-493b-acf8-41a8714932c0"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"f", "[", "x", "]"}], "]"}], "\[Equal]", 
  RowBox[{"x", "+", "1"}]}]], "Input",
 CellChangeTimes->{{3.8301354050498257`*^9, 3.830135411088541*^9}},
 CellLabel->"In[88]:=",ExpressionUUID->"4dda54db-0dfa-4441-822d-30e77729c41c"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.830135407851488*^9, 3.830135411495545*^9}, 
   3.830136312321085*^9, 3.830136535181312*^9},
 CellLabel->"Out[88]=",ExpressionUUID->"cdf1c950-6d99-4149-b62a-b6ba92c2e948"]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{
    RowBox[{"Let", "'"}], "s", " ", "try", " ", "this", " ", "one", " ", "f", 
    
    RowBox[{"(", 
     RowBox[{"f", 
      RowBox[{"(", "x", ")"}], "f", 
      RowBox[{"(", "y", ")"}]}], ")"}]}], " ", "=", " ", 
   RowBox[{"f", 
    RowBox[{"(", 
     RowBox[{"x", " ", "y"}], ")"}]}]}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.8701555497374687`*^9, 3.870155571305573*^9}, 
   3.870156159135041*^9},ExpressionUUID->"f3caff71-34e2-49dd-ad8e-\
df312c28893f"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "n_"}], "]"}], ":=", 
  RowBox[{"Total", "@", 
   RowBox[{"Array", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"a", "[", "#", "]"}], " ", 
       RowBox[{"x", "^", "#"}]}], "&"}], ",", 
     RowBox[{"n", "+", "1"}], ",", "0"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"f", "[", 
  RowBox[{"x", ",", " ", "2"}], "]"}]}], "Input",
 CellLabel->"In[1]:=",ExpressionUUID->"3e60e770-aec1-4f55-9b99-702c6629ac2c"],

Cell[BoxData[
 RowBox[{
  RowBox[{"a", "[", "0", "]"}], "+", 
  RowBox[{"x", " ", 
   RowBox[{"a", "[", "1", "]"}]}], "+", 
  RowBox[{
   SuperscriptBox["x", "2"], " ", 
   RowBox[{"a", "[", "2", "]"}]}]}]], "Output",
 CellChangeTimes->{3.870155634127933*^9},
 CellLabel->"Out[2]=",ExpressionUUID->"cc386be5-ad78-4b1c-9d54-59692e8e9e5a"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"sol", "[", "n_", "]"}], ":=", 
  RowBox[{"With", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{"sol", "=", 
      RowBox[{"SolveAlways", "[", 
       RowBox[{
        RowBox[{
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{
            RowBox[{"f", "[", 
             RowBox[{"x", ",", " ", "n"}], "]"}], " ", 
            RowBox[{"f", "[", 
             RowBox[{"y", ",", "n"}], "]"}]}], ",", "n"}], "]"}], "\[Equal]", 
         
         RowBox[{"f", "[", 
          RowBox[{
           RowBox[{"x", " ", "y"}], ",", "n"}], "]"}]}], ",", 
        RowBox[{"{", 
         RowBox[{"x", ",", "y"}], "}"}]}], "]"}]}], "}"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"{", 
       RowBox[{"#", ",", 
        RowBox[{"Length", "@", "#"}]}], "}"}], "&"}], "@", "sol"}]}], 
   "]"}]}]], "Input",
 CellChangeTimes->{{3.870155671492216*^9, 3.870155702506338*^9}, {
  3.8701557675828323`*^9, 3.870155767886963*^9}},
 CellLabel->"In[7]:=",ExpressionUUID->"02d9a0b5-3f7f-4ee5-b9d8-95d6c9f96e7f"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "1", "]"}]], "Input",
 CellChangeTimes->{{3.870155705153057*^9, 3.870155726470509*^9}, {
  3.870155770449601*^9, 3.870155770557686*^9}},
 CellLabel->"In[8]:=",ExpressionUUID->"79972568-346f-4ea0-aa6b-65b445933581"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], "}"}]}], "}"}], ",", 
   "3"}], "}"}]], "Output",
 CellChangeTimes->{
  3.870155706788568*^9, {3.870155761384091*^9, 3.870155770940366*^9}},
 CellLabel->"Out[8]=",ExpressionUUID->"f4f9d0e9-8e77-4574-9885-b644197c925a"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"sol", "[", "1", "]"}]], "Input",
 CellChangeTimes->{{3.8701557926438293`*^9, 3.870155849108245*^9}},
 CellLabel->"In[37]:=",ExpressionUUID->"8c062cb1-b5a1-4192-ab2a-bebd2bb72fa9"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", 
        RowBox[{"-", "1"}]}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{
        RowBox[{"a", "[", "0", "]"}], "\[Rule]", "0"}], ",", 
       RowBox[{
        RowBox[{"a", "[", "1", "]"}], "\[Rule]", "1"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"a", "[", "1", "]"}], "\[Rule]", "0"}], "}"}]}], "}"}], ",", 
   "3"}], "}"}]], "Output",
 CellChangeTimes->{{3.870155804161964*^9, 3.870155849375763*^9}, 
   3.870156068262355*^9},
 CellLabel->"Out[37]=",ExpressionUUID->"eb2002c5-3d03-4914-bd50-6cd1626c4acd"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"fs", "=", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "1"}], "]"}], "/.", 
   RowBox[{
    RowBox[{"sol", "[", "1", "]"}], "[", 
    RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input",
 CellChangeTimes->{{3.870155850911108*^9, 3.870155858398325*^9}, {
  3.8701559688301992`*^9, 3.8701559873364153`*^9}},
 CellLabel->"In[26]:=",ExpressionUUID->"2f0463d5-a649-4dce-96bb-a35587ff351c"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"-", "x"}], ",", "x", ",", 
   RowBox[{"a", "[", "0", "]"}]}], "}"}]], "Output",
 CellChangeTimes->{
  3.870155858731686*^9, {3.8701559776504803`*^9, 3.870155987778438*^9}},
 CellLabel->"Out[26]=",ExpressionUUID->"9b816bbd-fa5b-42e4-9ee6-33b093145703"]
}, Open  ]],

Cell[BoxData[{
 RowBox[{
  RowBox[{"f1", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"fs", "[", 
   RowBox[{"[", "1", "]"}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"f2", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"fs", "[", 
   RowBox[{"[", "2", "]"}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"f3", "[", "x_", "]"}], ":=", " ", 
  RowBox[{"fs", "[", 
   RowBox[{"[", "3", "]"}], "]"}]}]}], "Input",
 CellChangeTimes->{{3.8701558803325768`*^9, 3.87015589294797*^9}, {
  3.870155954171337*^9, 3.870155955337023*^9}, {3.870155999825872*^9, 
  3.870156032878455*^9}},ExpressionUUID->"c3952f97-c9c3-4dd7-8ee8-\
c8e5e3b03344"],

Cell[CellGroupData[{

Cell[BoxData[{
 RowBox[{
  RowBox[{"f1", "[", 
   RowBox[{
    RowBox[{"f1", "[", "x", "]"}], 
    RowBox[{"f1", "[", "y", "]"}]}], "]"}], "\[Equal]", 
  RowBox[{"f1", "[", 
   RowBox[{"x", " ", "y"}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"f2", "[", 
   RowBox[{
    RowBox[{"f2", "[", "x", "]"}], 
    RowBox[{"f2", "[", "y", "]"}]}], "]"}], "\[Equal]", 
  RowBox[{"f2", "[", 
   RowBox[{"x", " ", "y"}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"f3", "[", 
   RowBox[{
    RowBox[{"f3", "[", "x", "]"}], 
    RowBox[{"f3", "[", "y", "]"}]}], "]"}], "\[Equal]", 
  RowBox[{"f3", "[", 
   RowBox[{"x", " ", "y"}], "]"}]}]}], "Input",
 CellChangeTimes->{{3.870155897105281*^9, 3.870155964055229*^9}, {
  3.870156036453414*^9, 3.870156062827673*^9}},
 CellLabel->"In[34]:=",ExpressionUUID->"1187969e-8114-4c54-bc30-0f619a623854"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.8701559333785133`*^9, 3.870155964525078*^9}, 
   3.870156003416745*^9, {3.870156046434147*^9, 3.870156063308625*^9}},
 CellLabel->"Out[34]=",ExpressionUUID->"13cf0588-5bcd-44e0-9762-54bd3e501fb9"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.8701559333785133`*^9, 3.870155964525078*^9}, 
   3.870156003416745*^9, {3.870156046434147*^9, 3.870156063310013*^9}},
 CellLabel->"Out[35]=",ExpressionUUID->"f66e0ecd-701b-4688-b342-15a7b040050e"],

Cell[BoxData["True"], "Output",
 CellChangeTimes->{{3.8701559333785133`*^9, 3.870155964525078*^9}, 
   3.870156003416745*^9, {3.870156046434147*^9, 3.870156063311227*^9}},
 CellLabel->"Out[36]=",ExpressionUUID->"c0f3abec-aa9f-4acf-afa4-265f9e06a503"]
}, Open  ]]
},
WindowSize->{843, 465},
WindowMargins->{{Automatic, 6}, {169, Automatic}},
FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 8, 2019)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[558, 20, 695, 15, 46, "Text",ExpressionUUID->"6cd68e5d-db9f-40e0-a667-b077a0e31723"],
Cell[CellGroupData[{
Cell[1278, 39, 1061, 28, 52, "Input",ExpressionUUID->"c32c2ce9-46ef-4340-82c6-89c9a026c5b6"],
Cell[2342, 69, 816, 25, 56, "Output",ExpressionUUID->"532a617b-30b5-4f4f-b55e-57a6c06fe878"]
}, Open  ]],
Cell[3173, 97, 1236, 35, 75, "Input",ExpressionUUID->"4e9afd22-5845-4aa8-950e-18bc179db96c"],
Cell[CellGroupData[{
Cell[4434, 136, 596, 10, 30, "Input",ExpressionUUID->"5675651d-e8c0-4435-b1ea-b03c061dcc69"],
Cell[5033, 148, 3285, 93, 157, "Output",ExpressionUUID->"cfebd1c7-961b-490f-9ff5-8798d9cf3a54"]
}, Open  ]],
Cell[CellGroupData[{
Cell[8355, 246, 721, 16, 30, "Input",ExpressionUUID->"bb63cb6d-3335-41bf-8d2a-b6e21db96c17"],
Cell[9079, 264, 657, 18, 55, "Output",ExpressionUUID->"f585aeb1-c19a-4ad9-94d2-66d13bd7b38b"]
}, Open  ]],
Cell[CellGroupData[{
Cell[9773, 287, 214, 4, 30, "Input",ExpressionUUID->"feb70026-6ae8-4b28-8577-06a902d4cf0c"],
Cell[9990, 293, 424, 13, 52, "Output",ExpressionUUID->"531da635-f869-4063-9328-00bd36e631ca"]
}, Open  ]],
Cell[CellGroupData[{
Cell[10451, 311, 1468, 40, 73, "Input",ExpressionUUID->"e8f44cff-0d38-4287-8574-428bd4fc3930"],
Cell[11922, 353, 827, 21, 56, "Output",ExpressionUUID->"9bb3c44d-8c53-47d0-ba38-fda7b42a4240"]
}, Open  ]],
Cell[12764, 377, 1021, 28, 76, "Input",ExpressionUUID->"09ac34d6-f9cd-42dc-83a1-d4ddd87a8717"],
Cell[CellGroupData[{
Cell[13810, 409, 597, 9, 30, "Input",ExpressionUUID->"a67b81f1-bfd4-436d-9ee1-40d53b5641c5"],
Cell[14410, 420, 899, 25, 34, "Output",ExpressionUUID->"87b8f662-9db2-4964-8257-8cd2d3a00b14"]
}, Open  ]],
Cell[CellGroupData[{
Cell[15346, 450, 1540, 48, 73, "Input",ExpressionUUID->"7047b3c7-5077-4696-85e7-4b6be5dd8186"],
Cell[16889, 500, 717, 23, 58, "Output",ExpressionUUID->"82372910-5879-4821-b148-39b30ac896f4"]
}, Open  ]],
Cell[17621, 526, 759, 24, 76, "Input",ExpressionUUID->"759a3f5d-ba90-4456-bdd9-b7be4bdefb2a"],
Cell[CellGroupData[{
Cell[18405, 554, 203, 3, 30, "Input",ExpressionUUID->"83867158-21bb-4231-9580-935830752bec"],
Cell[18611, 559, 156, 2, 34, "Output",ExpressionUUID->"733d9d73-af7d-40f0-ab82-c7e5db6bc96f"]
}, Open  ]],
Cell[CellGroupData[{
Cell[18804, 566, 2027, 60, 99, "Input",ExpressionUUID->"571def01-962f-4037-9479-a97b5ac5fa6b"],
Cell[20834, 628, 585, 17, 56, "Output",ExpressionUUID->"50ce8be9-e8a7-443d-817d-00d4d5f60a78"]
}, Open  ]],
Cell[21434, 648, 947, 28, 75, "Input",ExpressionUUID->"4f758a3c-7df1-489a-b05b-8ce513ebdbe1"],
Cell[CellGroupData[{
Cell[22406, 680, 201, 3, 30, "Input",ExpressionUUID->"c6ac5a35-c663-49ad-bc41-51d244ee8c91"],
Cell[22610, 685, 1999, 58, 92, "Output",ExpressionUUID->"32b83783-1eed-48af-8526-3aea64a9fa0d"]
}, Open  ]],
Cell[CellGroupData[{
Cell[24646, 748, 329, 9, 30, "Input",ExpressionUUID->"5aaf7595-02f8-4586-b64d-4fba9150ae86"],
Cell[24978, 759, 414, 12, 55, "Output",ExpressionUUID->"e9925b9a-ed73-4ff9-9088-d962852a32c9"]
}, Open  ]],
Cell[CellGroupData[{
Cell[25429, 776, 199, 3, 30, "Input",ExpressionUUID->"c55bdaf4-4963-42c8-84f5-c8f5b1722e7b"],
Cell[25631, 781, 395, 12, 52, "Output",ExpressionUUID->"e762486b-38f7-4510-a7e3-e54c11737bd1"]
}, Open  ]],
Cell[26041, 796, 497, 15, 30, "Input",ExpressionUUID->"59bc333e-1bbb-458f-997d-a289b4f5a206"],
Cell[CellGroupData[{
Cell[26563, 815, 767, 23, 52, "Input",ExpressionUUID->"cfcaa5f4-5e85-4ff1-9889-17f7a5673caa"],
Cell[27333, 840, 563, 17, 56, "Output",ExpressionUUID->"12ed0028-42a8-4c4e-aa9c-20e1751dc71e"]
}, Open  ]],
Cell[27911, 860, 960, 29, 56, "Input",ExpressionUUID->"2bf348f2-5a8a-44a9-9cdf-d2a028cc80c7"],
Cell[CellGroupData[{
Cell[28896, 893, 203, 3, 30, "Input",ExpressionUUID->"3890853f-f8ad-4ca0-a36a-58710eb736cd"],
Cell[29102, 898, 1145, 33, 52, "Output",ExpressionUUID->"4a9726ea-b7b8-42b4-8ef0-fec9979574ef"]
}, Open  ]],
Cell[CellGroupData[{
Cell[30284, 936, 394, 10, 30, "Input",ExpressionUUID->"bf840938-7437-4981-be6e-3ebee057848c"],
Cell[30681, 948, 320, 9, 51, "Output",ExpressionUUID->"2e7e239e-b338-472a-8433-5f6f39d2b1ab"]
}, Open  ]],
Cell[CellGroupData[{
Cell[31038, 962, 213, 3, 30, "Input",ExpressionUUID->"195e2848-96c7-4fca-b3bb-09c0a9b49e37"],
Cell[31254, 967, 269, 7, 54, "Output",ExpressionUUID->"4c30276e-1563-4984-9eee-7bf776ffa630"]
}, Open  ]],
Cell[31538, 977, 353, 9, 51, "Input",ExpressionUUID->"29a20ac2-00f0-4a90-b39c-f85c88e5b60c"],
Cell[CellGroupData[{
Cell[31916, 990, 220, 4, 30, "Input",ExpressionUUID->"92f83b6b-a6cf-4307-a356-68489867c182"],
Cell[32139, 996, 169, 3, 51, "Output",ExpressionUUID->"bb40301a-f87c-4d8d-a97e-a88170276253"]
}, Open  ]],
Cell[CellGroupData[{
Cell[32345, 1004, 1245, 38, 79, "Input",ExpressionUUID->"24e4783f-3b7d-4c2a-8c9d-74e2b164fd8e"],
Cell[33593, 1044, 561, 17, 56, "Output",ExpressionUUID->"b3ee335f-584e-4631-b14c-a654293ecadb"]
}, Open  ]],
Cell[34169, 1064, 880, 26, 77, "Input",ExpressionUUID->"71e8ad2d-ed6c-4048-9957-5dcd07b58265"],
Cell[CellGroupData[{
Cell[35074, 1094, 202, 3, 30, "Input",ExpressionUUID->"3813e7aa-fba9-431d-99d2-afedb2bc4400"],
Cell[35279, 1099, 777, 22, 34, "Output",ExpressionUUID->"c3758609-e282-4586-845c-cf1cca55a4ac"]
}, Open  ]],
Cell[CellGroupData[{
Cell[36093, 1126, 399, 10, 30, "Input",ExpressionUUID->"616fe1fa-3a85-4768-910e-516d8f0e38c1"],
Cell[36495, 1138, 237, 5, 34, "Output",ExpressionUUID->"93551b9f-313f-4257-8ae3-f36a864563fd"]
}, Open  ]],
Cell[CellGroupData[{
Cell[36769, 1148, 228, 4, 31, "Input",ExpressionUUID->"240b4d8e-2235-4a1f-a1a7-ad4ebafb7574"],
Cell[37000, 1154, 152, 2, 35, "Output",ExpressionUUID->"215891b6-bc88-45ae-83e2-7dcd80867fca"]
}, Open  ]],
Cell[CellGroupData[{
Cell[37189, 1161, 1626, 46, 124, "Input",ExpressionUUID->"7bb25d61-5abd-4eb5-b0b2-dca7d2d090da"],
Cell[38818, 1209, 360, 9, 37, "Output",ExpressionUUID->"e1fe63b5-32fa-459d-9b04-2b2c68f34bb8"]
}, Open  ]],
Cell[39193, 1221, 834, 25, 55, "Input",ExpressionUUID->"528d87d5-04ad-4fd9-a5ea-e93fcac7e8d2"],
Cell[CellGroupData[{
Cell[40052, 1250, 303, 5, 31, "Input",ExpressionUUID->"3e16d050-1fe4-4966-bca3-aed849790119"],
Cell[40358, 1257, 679, 17, 53, "Output",ExpressionUUID->"c7fb006e-ab47-4768-8202-c422366d851c"]
}, Open  ]],
Cell[CellGroupData[{
Cell[41074, 1279, 496, 12, 31, "Input",ExpressionUUID->"48a6947f-e03a-412b-8174-7d58f6ba40c0"],
Cell[41573, 1293, 546, 13, 53, "Output",ExpressionUUID->"57eb1297-c5d4-468e-a934-e3de0e0cde16"]
}, Open  ]],
Cell[CellGroupData[{
Cell[42156, 1311, 210, 3, 31, "Input",ExpressionUUID->"70e187af-f4fe-4be7-bbf9-57219f8e872c"],
Cell[42369, 1316, 398, 11, 50, "Output",ExpressionUUID->"3224752c-9684-4a74-99a6-55acc1addbee"]
}, Open  ]],
Cell[CellGroupData[{
Cell[42804, 1332, 341, 10, 47, "Input",ExpressionUUID->"2a8db335-a4a6-4fd9-8689-3b20c0265675"],
Cell[43148, 1344, 295, 8, 53, "Output",ExpressionUUID->"40b3ede0-b42e-4175-b786-8691c2c21361"]
}, Open  ]],
Cell[43458, 1355, 532, 15, 35, "Input",ExpressionUUID->"584cad65-6afc-4287-a60d-095384b620bf"],
Cell[CellGroupData[{
Cell[44015, 1374, 572, 16, 52, "Input",ExpressionUUID->"3c8d02b5-56c8-4694-8b14-a67945af62d6"],
Cell[44590, 1392, 364, 10, 34, "Output",ExpressionUUID->"ae8c9439-78e0-4e31-aa42-7a122eb6a570"]
}, Open  ]],
Cell[44969, 1405, 1026, 32, 75, "Input",ExpressionUUID->"8c499ba9-09f6-485d-bbd8-1b88eb1fbf06"],
Cell[CellGroupData[{
Cell[46020, 1441, 205, 4, 30, "Input",ExpressionUUID->"2188de95-1710-4081-8a7c-c032dadf5972"],
Cell[46228, 1447, 1101, 34, 34, "Output",ExpressionUUID->"0bc2d9ff-52fb-4b98-ab09-138914ff860e"]
}, Open  ]],
Cell[CellGroupData[{
Cell[47366, 1486, 450, 12, 30, "Input",ExpressionUUID->"1642d6e8-4ec0-4526-b7e0-6f39d07a686d"],
Cell[47819, 1500, 227, 5, 34, "Output",ExpressionUUID->"302a1a39-a80b-4a2f-ac16-0c533ebeb8c2"]
}, Open  ]],
Cell[CellGroupData[{
Cell[48083, 1510, 354, 7, 30, "Input",ExpressionUUID->"096237f9-af46-4bb2-8d77-9fe195fa833f"],
Cell[48440, 1519, 2069, 60, 56, "Output",ExpressionUUID->"ffd115a3-926d-494a-aae8-2d5fe4f382f3"]
}, Open  ]],
Cell[CellGroupData[{
Cell[50546, 1584, 672, 18, 30, "Input",ExpressionUUID->"79a95482-8d3d-45a0-80fc-b9d30f524b69"],
Cell[51221, 1604, 508, 13, 37, "Output",ExpressionUUID->"9ce51eb7-7d5c-467f-9581-3d25430ace45"]
}, Open  ]],
Cell[CellGroupData[{
Cell[51766, 1622, 281, 6, 30, "Input",ExpressionUUID->"4df27f3a-8ead-45c8-8ebe-445fbdabda8b"],
Cell[52050, 1630, 3190, 92, 98, "Output",ExpressionUUID->"e4dc27f4-bb39-45b2-a404-dcc081c8405f"]
}, Open  ]],
Cell[CellGroupData[{
Cell[55277, 1727, 701, 20, 30, "Input",ExpressionUUID->"570d149c-7bdf-4caa-aafc-67dbaa76bcc4"],
Cell[55981, 1749, 555, 16, 37, "Output",ExpressionUUID->"e27b3e99-2770-42a9-ab0c-678d98385076"]
}, Open  ]],
Cell[CellGroupData[{
Cell[56573, 1770, 229, 5, 30, "Input",ExpressionUUID->"f99702d9-c553-43b9-bebb-ff0e4ffd3ce5"],
Cell[56805, 1777, 4684, 132, 224, "Output",ExpressionUUID->"1774f69e-364b-47cb-aebc-85f534c695d2"]
}, Open  ]],
Cell[CellGroupData[{
Cell[61526, 1914, 651, 19, 31, "Input",ExpressionUUID->"917d2874-562d-4157-b967-9d123ecf10ad"],
Cell[62180, 1935, 582, 18, 40, "Output",ExpressionUUID->"6a6e668f-4113-4bba-8108-6337405d268f"]
}, Open  ]],
Cell[62777, 1956, 394, 11, 31, "Input",ExpressionUUID->"87c31783-a6ce-4140-be75-7ff253074d0d"],
Cell[63174, 1969, 376, 8, 33, "Input",ExpressionUUID->"3628a082-23c8-4488-9d97-f121c0e579e1"],
Cell[CellGroupData[{
Cell[63575, 1981, 473, 13, 31, "Input",ExpressionUUID->"e3b2eab5-152d-45a8-911b-d0ec9472eddc"],
Cell[64051, 1996, 224, 4, 35, "Output",ExpressionUUID->"0ef1d87c-c281-4a18-b424-5f4e971a1a87"]
}, Open  ]],
Cell[CellGroupData[{
Cell[64312, 2005, 2706, 61, 168, "Input",ExpressionUUID->"dbcc904e-2525-47de-bb8f-9edd6e5f341c"],
Cell[67021, 2068, 360, 9, 34, "Output",ExpressionUUID->"3637c766-34b9-4b7b-bf6a-329237d3017b"]
}, Open  ]],
Cell[67396, 2080, 895, 26, 33, "Input",ExpressionUUID->"f874733b-0217-45a5-814e-53a7d8acf1d6"],
Cell[CellGroupData[{
Cell[68316, 2110, 229, 4, 30, "Input",ExpressionUUID->"7733d651-30c2-4cd6-9a9e-ff2b1e9c43de"],
Cell[68548, 2116, 577, 16, 50, "Output",ExpressionUUID->"d557611d-b187-43b8-b8fd-d9b2f1060134"]
}, Open  ]],
Cell[CellGroupData[{
Cell[69162, 2137, 595, 13, 30, "Input",ExpressionUUID->"b4a5a18b-9322-4995-9c5b-d516b7062fec"],
Cell[69760, 2152, 559, 12, 50, "Output",ExpressionUUID->"d36bea24-8db8-45bb-97dc-31da798c2fd2"]
}, Open  ]],
Cell[CellGroupData[{
Cell[70356, 2169, 199, 3, 30, "Input",ExpressionUUID->"a11bcf61-0da1-44d0-aa29-d960a8cbb980"],
Cell[70558, 2174, 257, 7, 53, "Output",ExpressionUUID->"939716cb-2a6f-47c4-bd7c-213da5a373d9"]
}, Open  ]],
Cell[70830, 2184, 215, 5, 30, "Input",ExpressionUUID->"7e906a0b-4d47-4808-ac6c-13473aa1fa2e"],
Cell[71048, 2191, 342, 9, 50, "Input",ExpressionUUID->"a4c993d7-2d8b-4286-b78a-6390a9feb8f7"],
Cell[CellGroupData[{
Cell[71415, 2204, 288, 6, 30, "Input",ExpressionUUID->"bff2412d-8b38-4082-b100-973791746dc3"],
Cell[71706, 2212, 518, 16, 45, "Output",ExpressionUUID->"b0fc3738-6767-4e56-a542-1d3fbb5cc21a"]
}, Open  ]],
Cell[72239, 2231, 258, 6, 30, "Input",ExpressionUUID->"79414368-6582-49e7-a872-d0f13b68a52a"],
Cell[72500, 2239, 733, 20, 107, "Input",ExpressionUUID->"2eb0af19-07bb-4847-893c-5604ababcddc"],
Cell[CellGroupData[{
Cell[73258, 2263, 574, 15, 50, "Input",ExpressionUUID->"2a8b8ac3-c4c6-4574-8bca-3720496f504c"],
Cell[73835, 2280, 30639, 526, 247, "Output",ExpressionUUID->"cb915ffc-d5b8-4595-8299-7cb18e79a143"]
}, Open  ]],
Cell[CellGroupData[{
Cell[104511, 2811, 201, 3, 30, "Input",ExpressionUUID->"420cd42e-9ee7-4703-9944-1cd642eeb2d8"],
Cell[104715, 2816, 441, 13, 50, "Output",ExpressionUUID->"ef778053-dd39-4245-9cec-0a98d4b1be12"]
}, Open  ]],
Cell[CellGroupData[{
Cell[105193, 2834, 335, 8, 30, "Input",ExpressionUUID->"e2fb8bdc-9e2b-41e1-a278-493b94ee3c13"],
Cell[105531, 2844, 270, 4, 34, "Output",ExpressionUUID->"1708baed-3041-4f68-980f-0e414fc5ee61"]
}, Open  ]],
Cell[105816, 2851, 740, 22, 30, "Input",ExpressionUUID->"16c3961e-3780-4cfe-a6d4-4ebfcb30ea78"],
Cell[106559, 2875, 344, 9, 30, "Input",ExpressionUUID->"c33f0672-92e8-452d-8350-07fb42faa0c5"],
Cell[CellGroupData[{
Cell[106928, 2888, 708, 17, 52, "Input",ExpressionUUID->"df20952e-c272-49ef-847a-25613e81da47"],
Cell[107639, 2907, 791, 16, 34, "Output",ExpressionUUID->"35dfbddf-966c-474c-ad97-7042ffc90088"]
}, Open  ]],
Cell[108445, 2926, 1549, 37, 30, "Input",ExpressionUUID->"3c27f791-8498-4a05-90b3-cf90f74c6b7b"],
Cell[CellGroupData[{
Cell[110019, 2967, 824, 12, 30, "Input",ExpressionUUID->"3654ee2c-edee-476e-b5aa-258837e3f189"],
Cell[110846, 2981, 1531, 29, 52, "Output",ExpressionUUID->"ad7094dd-dfcd-4d0e-9bb7-8eae058faf21"]
}, Open  ]],
Cell[CellGroupData[{
Cell[112414, 3015, 932, 18, 30, "Input",ExpressionUUID->"5a46d4f1-ac46-4891-a5f3-c5f622732f57"],
Cell[113349, 3035, 1409, 26, 55, "Output",ExpressionUUID->"05951f65-ca24-4ac6-bdee-79d63134afa6"]
}, Open  ]],
Cell[CellGroupData[{
Cell[114795, 3066, 1308, 34, 95, "Input",ExpressionUUID->"e040c7b8-50f6-45f5-9918-2d05744c3687"],
Cell[116106, 3102, 464, 13, 53, "Output",ExpressionUUID->"5f9112af-e532-4858-af0a-81c987e62743"]
}, Open  ]],
Cell[CellGroupData[{
Cell[116607, 3120, 1233, 33, 98, "Input",ExpressionUUID->"3bfe0e89-5a61-4fe8-bcbd-cb6a444c2e02"],
Cell[117843, 3155, 222, 4, 34, "Output",ExpressionUUID->"9092bcf3-d7a8-4cf4-b910-edbd0f52d57f"]
}, Open  ]],
Cell[118080, 3162, 825, 24, 30, "Input",ExpressionUUID->"609d1368-5c71-4c22-867f-7c93aa710c4e"],
Cell[CellGroupData[{
Cell[118930, 3190, 498, 14, 52, "Input",ExpressionUUID->"f37a35e4-4031-4d11-8cf2-7921c30df61e"],
Cell[119431, 3206, 409, 10, 37, "Output",ExpressionUUID->"8452fa1c-a8fe-44ad-b368-cbb4b3f6866c"]
}, Open  ]],
Cell[119855, 3219, 1075, 32, 55, "Input",ExpressionUUID->"8ad4b9ec-e566-4d5d-9c42-05e2a8441aa6"],
Cell[CellGroupData[{
Cell[120955, 3255, 166, 3, 30, "Input",ExpressionUUID->"923cffb2-cf7c-4e8a-a93b-0381d84c59df"],
Cell[121124, 3260, 500, 13, 34, "Output",ExpressionUUID->"d912b88c-3bab-4fc6-95a5-8372a7102885"]
}, Open  ]],
Cell[CellGroupData[{
Cell[121661, 3278, 334, 8, 30, "Input",ExpressionUUID->"0e157153-400d-481f-9f22-32649f41b36c"],
Cell[121998, 3288, 283, 5, 34, "Output",ExpressionUUID->"0247a037-6774-4f11-a7b1-d2c6f3164d62"]
}, Open  ]],
Cell[122296, 3296, 477, 10, 52, "Input",ExpressionUUID->"f12e9b9e-298c-409f-8c7e-89819eb48923"],
Cell[CellGroupData[{
Cell[122798, 3310, 474, 13, 30, "Input",ExpressionUUID->"3fb9f7e1-10d5-4670-b232-e03dc66952d2"],
Cell[123275, 3325, 226, 3, 34, "Output",ExpressionUUID->"137ef1d1-a8e3-45ee-aa0e-eeaff49b8deb"]
}, Open  ]],
Cell[CellGroupData[{
Cell[123538, 3333, 232, 4, 30, "Input",ExpressionUUID->"91c64797-e3a0-4dd2-b251-ee9e49c7249d"],
Cell[123773, 3339, 803, 23, 34, "Output",ExpressionUUID->"caf09821-e0ce-4a8b-af35-f8f97f7cd6f8"]
}, Open  ]],
Cell[CellGroupData[{
Cell[124613, 3367, 431, 9, 52, "Input",ExpressionUUID->"bc487467-6292-48de-a315-267b6cf4e5ec"],
Cell[125047, 3378, 155, 2, 34, "Output",ExpressionUUID->"42945a86-dbd0-44e6-ab21-fb57a282bdaf"]
}, Open  ]],
Cell[CellGroupData[{
Cell[125239, 3385, 1081, 31, 73, "Input",ExpressionUUID->"8e7f7c9d-9d8c-4d0e-98f1-6503889fd86c"],
Cell[126323, 3418, 609, 18, 62, "Output",ExpressionUUID->"7edee4ca-5912-4c76-9f4c-e38741de52aa"]
}, Open  ]],
Cell[126947, 3439, 1075, 32, 52, "Input",ExpressionUUID->"881a6343-0adc-4da9-9b38-18141e2d37a5"],
Cell[CellGroupData[{
Cell[128047, 3475, 269, 5, 30, "Input",ExpressionUUID->"55ae1bfa-bd55-4bae-a391-583a5dccccae"],
Cell[128319, 3482, 3041, 89, 126, "Output",ExpressionUUID->"1359adfc-380c-4f0f-a785-98b511c8f80d"]
}, Open  ]],
Cell[CellGroupData[{
Cell[131397, 3576, 611, 17, 30, "Input",ExpressionUUID->"bde15c33-edca-4611-abaf-993ac17cdb34"],
Cell[132011, 3595, 544, 13, 53, "Output",ExpressionUUID->"68a85739-2dc3-4369-aedc-c1e1a7b30f45"]
}, Open  ]],
Cell[CellGroupData[{
Cell[132592, 3613, 637, 17, 52, "Input",ExpressionUUID->"5653ca6a-b8e4-49bc-bc08-be02dd629392"],
Cell[133232, 3632, 154, 2, 34, "Output",ExpressionUUID->"d8253ec7-0421-43ca-88d6-f7c58cc446ac"]
}, Open  ]],
Cell[CellGroupData[{
Cell[133423, 3639, 677, 18, 52, "Input",ExpressionUUID->"d7d9774e-7978-4221-ae2b-1fcee4c0ab4f"],
Cell[134103, 3659, 178, 3, 34, "Output",ExpressionUUID->"da1e951f-0779-4535-b5b9-c31c4f183b3e"]
}, Open  ]],
Cell[CellGroupData[{
Cell[134318, 3667, 367, 7, 30, "Input",ExpressionUUID->"6424e545-1924-4d81-90fc-69be8d3c5178"],
Cell[134688, 3676, 230, 4, 34, "Output",ExpressionUUID->"4b3ebc5a-1302-4933-9de6-edd0ec5a3b02"]
}, Open  ]],
Cell[134933, 3683, 756, 21, 30, "Input",ExpressionUUID->"009ce6a1-02fe-4b53-98b1-58af54d707fb"],
Cell[135692, 3706, 344, 9, 30, "Input",ExpressionUUID->"af026db1-7e50-4aae-807b-1ce6f8b44ca4"],
Cell[CellGroupData[{
Cell[136061, 3719, 708, 17, 52, "Input",ExpressionUUID->"d6752bf8-8b3b-4ebb-96ac-65127c588d72"],
Cell[136772, 3738, 813, 16, 34, "Output",ExpressionUUID->"48fa33fb-b49a-4f7e-924f-94986588040a"]
}, Open  ]],
Cell[137600, 3757, 1584, 37, 30, "Input",ExpressionUUID->"66b90039-4add-41be-9dc8-e3801d299919"],
Cell[CellGroupData[{
Cell[139209, 3798, 899, 14, 30, "Input",ExpressionUUID->"94e5eb8a-6f30-420f-9172-71a4e1afeade"],
Cell[140111, 3814, 847, 25, 34, "Output",ExpressionUUID->"9177df49-3d82-4876-bca4-79f219607d8d"]
}, Open  ]],
Cell[CellGroupData[{
Cell[140995, 3844, 542, 16, 30, "Input",ExpressionUUID->"b8e8d113-f858-4728-94b9-e25b2cf80238"],
Cell[141540, 3862, 203, 4, 34, "Output",ExpressionUUID->"7d3db860-8108-4ee3-b2bd-7a4d322aa694"]
}, Open  ]],
Cell[141758, 3869, 228, 4, 30, "Input",ExpressionUUID->"f07fe4ad-8c93-4c7e-9e7d-5dfa19decc43"],
Cell[CellGroupData[{
Cell[142011, 3877, 335, 8, 30, "Input",ExpressionUUID->"9e5b8a11-56cc-4a00-a935-0e3132fe5ca1"],
Cell[142349, 3887, 151, 2, 34, "Output",ExpressionUUID->"b6590e3a-3593-490b-847f-aca9c42ccd0d"]
}, Open  ]],
Cell[142515, 3892, 229, 4, 30, "Input",ExpressionUUID->"70047a44-b22c-41de-9c25-cfea2d8d4924"],
Cell[CellGroupData[{
Cell[142769, 3900, 268, 7, 31, "Input",ExpressionUUID->"099bd213-44ba-4af0-94fb-31bc5f977a96"],
Cell[143040, 3909, 152, 2, 35, "Output",ExpressionUUID->"1d52eeaa-fbec-4e6f-8e5c-9489bfd46970"]
}, Open  ]],
Cell[143207, 3914, 229, 4, 31, "Input",ExpressionUUID->"f923e5a3-1ab1-4d6b-86ae-d3b6f6e5d9a7"],
Cell[CellGroupData[{
Cell[143461, 3922, 268, 7, 30, "Input",ExpressionUUID->"7dfcb28d-db1e-4025-bd6f-c36e40d7f337"],
Cell[143732, 3931, 152, 2, 34, "Output",ExpressionUUID->"83427341-5d4a-441f-a2c0-f74c9afcd1e3"]
}, Open  ]],
Cell[CellGroupData[{
Cell[143921, 3938, 261, 5, 30, "Input",ExpressionUUID->"7955e0a2-4504-47e6-98d8-b797338a8bd5"],
Cell[144185, 3945, 1335, 39, 56, "Output",ExpressionUUID->"0612841b-ab52-4a8a-8b20-ab35e2bd20ec"]
}, Open  ]],
Cell[CellGroupData[{
Cell[145557, 3989, 611, 17, 30, "Input",ExpressionUUID->"8b1f4623-eb62-409c-8c36-2cbb1545f8b6"],
Cell[146171, 4008, 262, 5, 37, "Output",ExpressionUUID->"c9c6bb9a-111d-4fb6-b041-c35705bee489"]
}, Open  ]],
Cell[146448, 4016, 255, 5, 30, "Input",ExpressionUUID->"83efcd74-2934-4b7e-8324-8e196ac24f6a"],
Cell[CellGroupData[{
Cell[146728, 4025, 331, 8, 30, "Input",ExpressionUUID->"1ddf41c7-8adb-441d-8a10-b126e8aad916"],
Cell[147062, 4035, 152, 2, 34, "Output",ExpressionUUID->"79a7d73e-082e-48b9-afc7-cce869073430"]
}, Open  ]],
Cell[CellGroupData[{
Cell[147251, 4042, 263, 5, 30, "Input",ExpressionUUID->"ce65cb7a-9bc6-41ab-bf62-d91b94a145dd"],
Cell[147517, 4049, 2004, 57, 77, "Output",ExpressionUUID->"5d5e8d4b-9277-4aa2-9a7c-dd51dea21ac4"]
}, Open  ]],
Cell[CellGroupData[{
Cell[149558, 4111, 611, 17, 30, "Input",ExpressionUUID->"4def0740-684f-4d64-ad32-1dbd26a48a54"],
Cell[150172, 4130, 274, 6, 37, "Output",ExpressionUUID->"acaac3a2-e698-496c-9e99-a62ad0164dbd"]
}, Open  ]],
Cell[150461, 4139, 340, 8, 30, "Input",ExpressionUUID->"6e94d565-1c62-470e-b8cf-b4753ed0c294"],
Cell[CellGroupData[{
Cell[150826, 4151, 1012, 30, 73, "Input",ExpressionUUID->"bc74323b-d76d-445d-838c-63340ebd2712"],
Cell[151841, 4183, 582, 17, 56, "Output",ExpressionUUID->"dd59ad75-4280-4709-9b4b-15992f2c4bb5"]
}, Open  ]],
Cell[152438, 4203, 1584, 37, 30, "Input",ExpressionUUID->"a3bc9fdf-2f06-44d5-a02c-12df11ccdd4e"],
Cell[CellGroupData[{
Cell[154047, 4244, 949, 15, 30, "Input",ExpressionUUID->"1eb4d59c-b158-46ab-89ce-6c67277b3c69"],
Cell[154999, 4261, 3503, 101, 119, "Output",ExpressionUUID->"f2a7bb48-61c9-492e-a482-55e746c4440b"]
}, Open  ]],
Cell[CellGroupData[{
Cell[158539, 4367, 713, 19, 30, "Input",ExpressionUUID->"23664627-6bc4-4f0c-854c-1d5b40933c93"],
Cell[159255, 4388, 309, 6, 50, "Output",ExpressionUUID->"83c4e837-9d9b-4fb0-9706-dbee844de21a"]
}, Open  ]],
Cell[159579, 4397, 279, 6, 30, "Input",ExpressionUUID->"0c78b518-b46f-4414-aa5f-ea69b307ab4f"],
Cell[CellGroupData[{
Cell[159883, 4407, 662, 18, 30, "Input",ExpressionUUID->"40344889-5436-462d-a399-fd5b3a413923"],
Cell[160548, 4427, 353, 8, 51, "Output",ExpressionUUID->"8e983ada-92b2-4e76-ab77-d2c1e8153977"]
}, Open  ]],
Cell[160916, 4438, 567, 17, 35, "Input",ExpressionUUID->"9d994d20-3739-46f4-8744-8b280ea55a4b"],
Cell[CellGroupData[{
Cell[161508, 4459, 766, 23, 52, "Input",ExpressionUUID->"09700ef3-64a5-4f2f-8dd8-342b651b7694"],
Cell[162277, 4484, 562, 17, 56, "Output",ExpressionUUID->"b6b484c1-721c-4f42-aba5-39a9c6483444"]
}, Open  ]],
Cell[162854, 4504, 846, 25, 47, "Input",ExpressionUUID->"50eb3c06-3b30-4d71-af35-0c2e53947a51"],
Cell[CellGroupData[{
Cell[163725, 4533, 281, 5, 30, "Input",ExpressionUUID->"13845a6e-625f-4c21-bc86-15ee1a866242"],
Cell[164009, 4540, 644, 19, 34, "Output",ExpressionUUID->"0da23133-1c49-4514-a15d-59dc943e3b04"]
}, Open  ]],
Cell[CellGroupData[{
Cell[164690, 4564, 595, 17, 30, "Input",ExpressionUUID->"01a564b2-9f37-4e3c-8315-ff0acb3b144f"],
Cell[165288, 4583, 184, 3, 34, "Output",ExpressionUUID->"699ea491-1e48-495c-9ac5-a9ca41853518"]
}, Open  ]],
Cell[CellGroupData[{
Cell[165509, 4591, 281, 5, 30, "Input",ExpressionUUID->"ec6ffebe-0e17-4797-a7ba-f195f6f6d100"],
Cell[165793, 4598, 5235, 151, 140, "Output",ExpressionUUID->"a325413e-7c6a-4239-a0e1-038343131b2e"]
}, Open  ]],
Cell[CellGroupData[{
Cell[171065, 4754, 662, 18, 30, "Input",ExpressionUUID->"ba3370c9-b18e-474e-b38e-36d403b0d84a"],
Cell[171730, 4774, 185, 3, 34, "Output",ExpressionUUID->"159b82a4-5d09-4831-bf9d-c51cceddc747"]
}, Open  ]],
Cell[CellGroupData[{
Cell[171952, 4782, 275, 5, 30, "Input",ExpressionUUID->"8f07d548-f7d2-4273-af2c-ad56864b5523"],
Cell[172230, 4789, 5601, 160, 202, "Output",ExpressionUUID->"60529652-50a2-403d-a4af-c3c8e8afd724"]
}, Open  ]],
Cell[CellGroupData[{
Cell[177868, 4954, 664, 18, 30, "Input",ExpressionUUID->"3cbdb7fd-2e01-46db-8848-9b17427060c9"],
Cell[178535, 4974, 424, 12, 54, "Output",ExpressionUUID->"50b00552-d113-4a42-80d6-1073eb2a44be"]
}, Open  ]],
Cell[178974, 4989, 371, 11, 51, "Input",ExpressionUUID->"d09521fa-4ea8-4b06-894e-83faa241d05a"],
Cell[CellGroupData[{
Cell[179370, 5004, 338, 9, 47, "Input",ExpressionUUID->"d7b5e1d8-1217-47af-86a8-46ad08197629"],
Cell[179711, 5015, 223, 4, 34, "Output",ExpressionUUID->"a9a8149d-0dd9-412c-8c2c-864ebee2c7e9"]
}, Open  ]],
Cell[CellGroupData[{
Cell[179971, 5024, 228, 4, 30, "Input",ExpressionUUID->"3e0d5300-fc21-4323-b4ac-0ef7436de1de"],
Cell[180202, 5030, 599, 20, 54, "Output",ExpressionUUID->"e7d2f455-2614-4b5c-b385-65fa751592ec"]
}, Open  ]],
Cell[180816, 5053, 452, 13, 49, "Input",ExpressionUUID->"9c491baa-037b-47a7-abe4-92f18b3b7088"],
Cell[CellGroupData[{
Cell[181293, 5070, 406, 11, 47, "Input",ExpressionUUID->"85fc9405-657f-45d0-a8db-f4ee013b6431"],
Cell[181702, 5083, 171, 3, 34, "Output",ExpressionUUID->"342378a1-62d9-47c6-ab8a-82dd8f783192"]
}, Open  ]],
Cell[CellGroupData[{
Cell[181910, 5091, 227, 4, 31, "Input",ExpressionUUID->"fca9b7d0-8667-42f1-bf62-ce431ba342c2"],
Cell[182140, 5097, 371, 10, 54, "Output",ExpressionUUID->"533dc832-c76b-4c7f-833a-6e6324243a69"]
}, Open  ]],
Cell[CellGroupData[{
Cell[182548, 5112, 313, 7, 30, "Input",ExpressionUUID->"763ab6bd-7e90-4ab6-a713-f52eb11ddca8"],
Cell[182864, 5121, 498, 13, 55, "Output",ExpressionUUID->"1e75edc3-8818-4cfe-a0f9-6b7bd43ed55a"]
}, Open  ]],
Cell[CellGroupData[{
Cell[183399, 5139, 319, 7, 30, "Input",ExpressionUUID->"49d1b79f-e357-46a8-a215-9a6e1dff3ade"],
Cell[183721, 5148, 228, 3, 34, "Output",ExpressionUUID->"d339b426-21ab-478c-a336-397e86297950"]
}, Open  ]],
Cell[183964, 5154, 812, 26, 45, "Input",ExpressionUUID->"14709f19-df4a-40a2-88b7-a94fd5da0cf7"],
Cell[184779, 5182, 371, 10, 30, "Input",ExpressionUUID->"6bdeeb2b-f695-4a87-81ca-51a192f17b0a"],
Cell[CellGroupData[{
Cell[185175, 5196, 737, 18, 52, "Input",ExpressionUUID->"abfb7ca4-dc48-45f9-b5d1-bd1fbe4bb6b7"],
Cell[185915, 5216, 360, 9, 34, "Output",ExpressionUUID->"edced59b-2ef3-491f-9ef2-9b102410f077"]
}, Open  ]],
Cell[186290, 5228, 1011, 31, 71, "Input",ExpressionUUID->"da5ed8a9-0923-40f2-a011-b030da2a242b"],
Cell[CellGroupData[{
Cell[187326, 5263, 229, 4, 30, "Input",ExpressionUUID->"e320a826-24a8-42ed-b217-07b501f1feef"],
Cell[187558, 5269, 479, 13, 34, "Output",ExpressionUUID->"b81a656b-fd96-4c7f-b744-be25dc8a66ba"]
}, Open  ]],
Cell[CellGroupData[{
Cell[188074, 5287, 229, 4, 30, "Input",ExpressionUUID->"4fd52582-5bc9-4bfd-a633-619a42962906"],
Cell[188306, 5293, 457, 13, 34, "Output",ExpressionUUID->"f90e8e47-4710-4174-b2f9-05fcd96b37ae"]
}, Open  ]],
Cell[CellGroupData[{
Cell[188800, 5311, 664, 18, 30, "Input",ExpressionUUID->"b651f813-2072-4746-bed9-ebf67f8627e0"],
Cell[189467, 5331, 272, 6, 37, "Output",ExpressionUUID->"0f552eec-e6e2-4397-ab70-645052ac15e6"]
}, Open  ]],
Cell[189754, 5340, 210, 5, 30, "Input",ExpressionUUID->"2c8907c2-1ff9-45ef-ad15-88ea075a815c"],
Cell[CellGroupData[{
Cell[189989, 5349, 378, 11, 42, "Input",ExpressionUUID->"6da0b098-a2c4-4bc5-96dd-effe07454c1f"],
Cell[190370, 5362, 251, 4, 34, "Output",ExpressionUUID->"b0274d25-86e5-42f0-b2f3-48cd1cc85b34"]
}, Open  ]],
Cell[CellGroupData[{
Cell[190658, 5371, 867, 25, 63, "Input",ExpressionUUID->"a6b491af-6117-41c0-a178-cb6f88908635"],
Cell[191528, 5398, 225, 4, 34, "Output",ExpressionUUID->"c7fdda57-1ec0-45c2-89a5-c7b3080b09e0"]
}, Open  ]],
Cell[CellGroupData[{
Cell[191790, 5407, 391, 9, 52, "Input",ExpressionUUID->"fc4bc0ad-2afc-4e5d-8799-5578aadd2397"],
Cell[192184, 5418, 193, 3, 34, "Output",ExpressionUUID->"1f29c81a-fb7c-4980-af5f-424f3016b2e8"]
}, Open  ]],
Cell[192392, 5424, 584, 17, 35, "Input",ExpressionUUID->"a0c08030-c94c-4bcc-aac0-d08000514a4f"],
Cell[CellGroupData[{
Cell[193001, 5445, 1006, 27, 52, "Input",ExpressionUUID->"e9708de2-c47e-4663-9911-0ae162dbe73e"],
Cell[194010, 5474, 632, 18, 56, "Output",ExpressionUUID->"e91bd8c2-21d5-4622-acd4-2ab61f4025da"]
}, Open  ]],
Cell[194657, 5495, 1005, 30, 47, "Input",ExpressionUUID->"2c0847fc-441d-4a34-8bd9-552ead695d80"],
Cell[CellGroupData[{
Cell[195687, 5529, 229, 4, 30, "Input",ExpressionUUID->"09f349ef-385c-4cbf-a0f2-1ecd7224a391"],
Cell[195919, 5535, 429, 12, 34, "Output",ExpressionUUID->"0e023ad0-28d4-4a58-aba5-f2ecfebe4487"]
}, Open  ]],
Cell[CellGroupData[{
Cell[196385, 5552, 662, 18, 30, "Input",ExpressionUUID->"c8f49b10-fa87-437d-8bb5-2ddf390ddabe"],
Cell[197050, 5572, 185, 3, 34, "Output",ExpressionUUID->"ef0bd746-7ea9-4336-a212-83192b41e54b"]
}, Open  ]],
Cell[CellGroupData[{
Cell[197272, 5580, 229, 4, 30, "Input",ExpressionUUID->"a5e269b5-84a7-4406-8f2a-882fae466bce"],
Cell[197504, 5586, 1273, 38, 79, "Output",ExpressionUUID->"ea55b337-76b1-448f-9bf8-11f9adc6acdd"]
}, Open  ]],
Cell[CellGroupData[{
Cell[198814, 5629, 662, 18, 30, "Input",ExpressionUUID->"3f3344d2-b382-4c5d-91f1-01fed4e7e2ab"],
Cell[199479, 5649, 552, 17, 56, "Output",ExpressionUUID->"0ac3e8b1-7eb6-46cc-9dd8-bd753d470ec2"]
}, Open  ]],
Cell[200046, 5669, 530, 16, 54, "Input",ExpressionUUID->"20484ec3-ea22-4842-9d73-650cf496352c"],
Cell[CellGroupData[{
Cell[200601, 5689, 228, 4, 30, "Input",ExpressionUUID->"f09f73c7-a145-4bf9-8705-02893244cb1d"],
Cell[200832, 5695, 595, 20, 55, "Output",ExpressionUUID->"92914d6d-1454-446b-a8a5-25505be5386d"]
}, Open  ]],
Cell[CellGroupData[{
Cell[201464, 5720, 742, 25, 52, "Input",ExpressionUUID->"15495087-6e7d-456d-a2e7-f3ec1039714e"],
Cell[202209, 5747, 8099, 152, 242, "Output",ExpressionUUID->"befa1589-6c69-4962-be43-48f0e1497a00"]
}, Open  ]],
Cell[210323, 5902, 474, 11, 30, "Input",ExpressionUUID->"36bc8a91-497e-4f38-a1a7-4f4a9182b6ba"],
Cell[CellGroupData[{
Cell[210822, 5917, 497, 14, 52, "Input",ExpressionUUID->"2d764d9d-5802-4b43-94f3-a1fa1b980c65"],
Cell[211322, 5933, 430, 10, 34, "Output",ExpressionUUID->"7dbbd60b-6ff9-473e-8983-531d354d1115"]
}, Open  ]],
Cell[211767, 5946, 795, 23, 30, "Input",ExpressionUUID->"81d884ef-c4dd-40ce-939a-440969e55313"],
Cell[CellGroupData[{
Cell[212587, 5973, 165, 3, 30, "Input",ExpressionUUID->"f1d7d73f-51f6-4b57-a3d6-9e7344a3331f"],
Cell[212755, 5978, 546, 14, 50, "Output",ExpressionUUID->"6dd7ed31-509d-4a30-a709-cb4aaf499562"]
}, Open  ]],
Cell[CellGroupData[{
Cell[213338, 5997, 333, 8, 30, "Input",ExpressionUUID->"88f97da0-448a-4177-b0d4-b541d14aa18a"],
Cell[213674, 6007, 329, 7, 50, "Output",ExpressionUUID->"f1a48b3d-ec46-4766-bca2-c9a71a163e10"]
}, Open  ]],
Cell[214018, 6017, 276, 6, 47, "Input",ExpressionUUID->"d0051a68-adb3-493b-acf8-41a8714932c0"],
Cell[CellGroupData[{
Cell[214319, 6027, 283, 6, 30, "Input",ExpressionUUID->"4dda54db-0dfa-4441-822d-30e77729c41c"],
Cell[214605, 6035, 224, 3, 34, "Output",ExpressionUUID->"cdf1c950-6d99-4149-b62a-b6ba92c2e948"]
}, Open  ]],
Cell[214844, 6041, 523, 15, 30, "Input",ExpressionUUID->"f3caff71-34e2-49dd-ad8e-df312c28893f"],
Cell[CellGroupData[{
Cell[215392, 6060, 497, 14, 52, "Input",ExpressionUUID->"3e60e770-aec1-4f55-9b99-702c6629ac2c"],
Cell[215892, 6076, 337, 9, 34, "Output",ExpressionUUID->"cc386be5-ad78-4b1c-9d54-59692e8e9e5a"]
}, Open  ]],
Cell[216244, 6088, 1045, 31, 52, "Input",ExpressionUUID->"02d9a0b5-3f7f-4ee5-b9d8-95d6c9f96e7f"],
Cell[CellGroupData[{
Cell[217314, 6123, 249, 4, 30, "Input",ExpressionUUID->"79972568-346f-4ea0-aa6b-65b445933581"],
Cell[217566, 6129, 799, 24, 34, "Output",ExpressionUUID->"f4f9d0e9-8e77-4574-9885-b644197c925a"]
}, Open  ]],
Cell[CellGroupData[{
Cell[218402, 6158, 203, 3, 30, "Input",ExpressionUUID->"8c062cb1-b5a1-4192-ab2a-bebd2bb72fa9"],
Cell[218608, 6163, 801, 24, 34, "Output",ExpressionUUID->"eb2002c5-3d03-4914-bd50-6cd1626c4acd"]
}, Open  ]],
Cell[CellGroupData[{
Cell[219446, 6192, 412, 10, 30, "Input",ExpressionUUID->"2f0463d5-a649-4dce-96bb-a35587ff351c"],
Cell[219861, 6204, 303, 7, 34, "Output",ExpressionUUID->"9b816bbd-fa5b-42e4-9ee6-33b093145703"]
}, Open  ]],
Cell[220179, 6214, 640, 16, 73, "Input",ExpressionUUID->"c3952f97-c9c3-4dd7-8ee8-c8e5e3b03344"],
Cell[CellGroupData[{
Cell[220844, 6234, 855, 24, 73, "Input",ExpressionUUID->"1187969e-8114-4c54-bc30-0f619a623854"],
Cell[221702, 6260, 250, 3, 34, "Output",ExpressionUUID->"13cf0588-5bcd-44e0-9762-54bd3e501fb9"],
Cell[221955, 6265, 250, 3, 34, "Output",ExpressionUUID->"f66e0ecd-701b-4688-b342-15a7b040050e"],
Cell[222208, 6270, 250, 3, 34, "Output",ExpressionUUID->"c0f3abec-aa9f-4acf-afa4-265f9e06a503"]
}, Open  ]]
}
]
*)