(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 22556, 584] NotebookOptionsPosition[ 21252, 536] NotebookOutlinePosition[ 21589, 551] CellTagsIndexPosition[ 21546, 548] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["2. Image d'une fonction", "Subsubsection", CellChangeTimes->{{3.399729575530548*^9, 3.399729576360353*^9}, { 3.3997296072742453`*^9, 3.399729610464183*^9}}], Cell[TextData[{ StyleBox["D\[EAcute]finition", FontVariations->{"Underline"->True}], " : soit f, une fonction r\[EAcute]elle\n\n", StyleBox["im f = {y \[Element] \[DoubleStruckCapitalR] : y = f(x) et x \ \[Element] dom f }", FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{{3.3997249401964607`*^9, 3.3997250285122423`*^9}, { 3.399730477970831*^9, 3.3997305195686617`*^9}}], Cell["\<\ D\[EAcute]terminer l'image d'une fonction, c'est trouver les r\[EAcute]els y \ qui sont image d'une r\[EAcute]el par f, c'est-\[AGrave]-dire l'ensemble de \ toutes les images. Il n'est pas possible, \[AGrave] ce stade du cours, de trouver l'image d'une \ fonction alg\[EAcute]brique quelconque.\[LineSeparator]Nous nous contentons \ donc des fonctions connues telles que les fonctions \ \[EAcute]l\[EAcute]mentaires (1/x, |x|, ...) et les fonctions du 1er et 2nd \ degr\[EAcute]. Sans oublier les fonctions obtenues par manipulation du graphe \ cart\[EAcute]sien des fonctions \[EAcute]l\[EAcute]mentaires. En fait, il est surtout facile de d\[EAcute]terminer l'image si l'on connait \ le graphe cart\[EAcute]sien de la fonction. \ \>", "Text", CellChangeTimes->{{3.399725060135434*^9, 3.399725199583449*^9}, { 3.399725260880971*^9, 3.3997252869915743`*^9}, {3.399725424121127*^9, 3.3997255499594727`*^9}, {3.399725586881123*^9, 3.3997256679848747`*^9}, { 3.399725716202712*^9, 3.39972571767274*^9}, {3.399725919114477*^9, 3.399725920227268*^9}, {3.399730533546055*^9, 3.3997305882498827`*^9}, { 3.399730619481679*^9, 3.3997307601284323`*^9}, {3.399731302955265*^9, 3.399731304513795*^9}, {3.3997318213301992`*^9, 3.399731856128849*^9}}], Cell[TextData[{ "exemple:\td\[EAcute]terminer l'image d'une fonction du second degr\[EAcute]\ \n\n\tf(x) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}], "+", "2"}], TraditionalForm]]], "\[LineSeparator]Le coefficient a de ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], "\[EAcute]tant positif, la parabole tourne sa concavit\[EAcute] vers le \ haut.\[LineSeparator]" }], "Text", CellChangeTimes->{{3.399730822403512*^9, 3.399730827712171*^9}, { 3.399731310528861*^9, 3.399731394073675*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"gf", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "-", RowBox[{"3", "x"}], "+", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "5"}], ",", "5"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "5"}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Show", "[", "gf", "]"}]}], "Input", CellOpen->False, CellChangeTimes->{{3.399731403254314*^9, 3.399731469963975*^9}}], Cell[BoxData[ FormBox[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJw1lwk0VG8fxwdjKykpbSq0jRKyU/ndaF8VSSqGtCOpRJbGvpf4U7YaoYWK pA31jKWxz5Ut+4x997cMM8b23vc9573n3HPP53ye7/c+z3mec+65ira3z1wR JpFItcT932dlGs3qOvQj0v+uFZh6fyNXLbEH0R8W/DR/uBzbn9MsOWfOQTSN 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FractionBox[ RowBox[{"-", "\:2206"}], RowBox[{"4", "a"}]], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"-", "1"}], "4"], TraditionalForm]]], "\nD\[EGrave]s lors,\n\t\tim f = [", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"-", "1"}], "4"], TraditionalForm]]], ",\[Rule] \n\n" }], "Text", CellChangeTimes->{{3.39973151530968*^9, 3.3997317006334047`*^9}, { 3.399731783859774*^9, 3.399731784393428*^9}, {3.399731994522505*^9, 3.399732117584845*^9}}, FontSlant->"Italic"], Cell[TextData[{ "exemple:\td\[EAcute]terminer l'image d'une fonction \[AGrave] radical\n\n\t\ f(x) =", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ RowBox[{"2", "x"}], "-", "5"}]], TraditionalForm]]], "\[LineSeparator]Le domaine de d\[EAcute]finition est [", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["5", "2"], ",", "\[Rule]"}], TraditionalForm]]], " \nLa valeur minimum de cette racine carr\[EAcute]e est \[EAcute]videmment \ 0 (obtenue pour x = ", Cell[BoxData[ FormBox[ FractionBox["5", "2"], 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