(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 509144, 24001]*) (*NotebookOutlinePosition[ 510199, 24037]*) (* CellTagsIndexPosition[ 510155, 24033]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["By: Sandy Emerick", Evaluatable->False, AspectRatioFixed->True, FontSize->14]], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Functions and their Inverses", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["", Evaluatable->False, AspectRatioFixed->True] }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Brief Description:\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[ "This lesson allows the student to discover and use the Ordered Pair Test \ and the Vertical Line Test for functions. The last part of the lesson deals \ with the graphing of the inverse.", Evaluatable->False, AspectRatioFixed->True] }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Input", AspectRatioFixed->True], Cell[TextData["Function and their Inverses"], "Title", Evaluatable->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[CellGroupData[{Cell[TextData["Initialization Cell"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "<False, InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[TextData[ "A function is a set of ordered pairs that assigns to every element in the \ domain exactly one element in the range.\nRecall: \n the domain is the \ set of first coordinates \n the range is the set of second \ coordinates"], "SmallText", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData["Function Tests"], "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[TextData["Ordered Pair Test for a Function"], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData[ "Using the definition of a function, can you determine if set A is a\n\ function?"], "Text", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData["A = {{2,3}, {4,-5}, {-7,0}, {-3,-8}}"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData[StyleBox[ "Observe that 2 is paired with 3\n 4 is paired with \ -5\n -7 is paired with 0\n -3 \ is paired with -8 ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"CalcMath", FontSlant->"Plain", FontColor->GrayLevel[0]]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData[StyleBox[ "These ordered pairs clearly satisify the definition of a function\nsince \ each first coordinate is paired with exactly one second coordinate.", PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"CalcMath", FontWeight->"Plain"]], "Input", PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"CalcMath"], Cell[TextData[" Conclusion: Set A is a function"], "Text", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"CalcMath", FontColor->RGBColor[0, 0, 1]], Cell[TextData["Now, take a look at the set B. 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Remember to use the vertical line test.\n\ Click on the box if you want to see the vertical line \nalong with the \ circle"], "SmallText", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], Cell[TextData["code"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Do[Show[vertfunc[w], circ,\n\ Axes->True,AxesLabel->{\"x\",\"y\"},AspectRatio->1,\n\ PlotRange->{{-1.5,1.5},{-1.5,1.5}}],{w,-.5,1.5,.5}];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Notice that there exists at least one vertical that intersects the circle \ in more than one point. This means that the vertical line fails; and \ therefore, the graph is ", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True], StyleBox["not", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[ " a function\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tTherefore, a circle is ", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True], StyleBox["not", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[" a function\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True], Cell[TextData[ "Let's look at one more graph. Is the following a function? Explain"], "SmallText", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], Cell[TextData["code"], "Subsubsection", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[ "Clear[x] \nwave = Plot[Sin[x],{x,-2 Pi,2 Pi}, \n \ PlotStyle->{{Thickness[0.01],Purple}},\n PlotLabel->sin];\n\n"], "Input",\ AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "I think that we have another function. This curve certainly passes the \ Vertical Line Test. \n In fact, this function is called the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sine Function", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 1]], StyleBox["", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0]] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\nIt's time to try something new. Let's investigate inverses."], "SmallText",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Inverses"], "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontSize->18], Cell[TextData[{ StyleBox["A relation is defined to be ", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], StyleBox["any", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 1, 0]], StyleBox[ " set of ordered pairs. 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To experiment, just type your ordered \ pair in the [ ] that you see above the graph, and click in the\ncell to your \ right. (Be careful not to leave out the comma or the brackets) \nWhat is \ happening with these points in reference to the dotted line?", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["This is called ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["reflection in the line", Evaluatable->False, AspectRatioFixed->True], StyleBox[". \n", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox[ "Do you have any idea what the equation of this line would have to be?", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["You might look at some of the ordered pairs for a clue.", 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Can you support your decision?", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["\n\n", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], StyleBox["Is the yellow graph a function? Explain\n", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["We must conclude that inverses are ", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], StyleBox["not necessarily both functions", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["!!!", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontSize->24, FontColor->RGBColor[1, 0, 0]] }], "Text", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Now, let's try a straight line and its inverse", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]]], "Text", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData["code"], "Subsubsection", Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->True], Cell[TextData[ "Clear[f,x];\nf[x_]:= -3 x + 6\nlinegrph= Plot[f[x], \ {x,-3,10},PlotRange->{{-4,4},{-4,4}},\n \ PlotStyle->{{Thickness[0.009],Green}},\n \ DisplayFunction->Identity];\ng[x_] := -x/3 + 2\nreflect= \ Plot[g[x],{x,-20,20},PlotRange->{{-4,4},{-4,4}},\n \ PlotStyle->{{Thickness[0.009]\n ,Orange}},\n \ DisplayFunction->Identity];\nShow[{reflect,linegrph},\n \ DisplayFunction->$DisplayFunction];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Do these two lines really reflect in our ", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["mystery", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 1]], StyleBox[" line?", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, PageBreakAbove->False, PageBreakBelow->True, AspectRatioFixed->True], Cell[TextData["code"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\n\nShow[{reflect,ident,linegrph},\n \ DisplayFunction->$DisplayFunction];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Notice that both of these lines pass the Vertical Line Test; and therefore, \ are both functions. We say these are function inverses."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Just as expected, these two lines are inverses of each other. A really \ useful idea here is \n If you are graphing with \"pencil and paper\", \ you can find the inverse of your graph by folding your graph paper at a 45 \ degree angle in the first quadrant (the location of the dashed line) . \nBy \ the way, I still do not know the name of the dashing ", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["mystery", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 1]], StyleBox[ " line. Have you figured it out yet? If you know, type the equation here.\ \n_\n", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], StyleBox[" One last curve and its inverse to inspect.", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], Cell[TextData["code"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Clear[f,x,incurve1];\nf[x_]:= 3^x \n curve1 = Plot[f[x], \ {x,-10,10},PlotRange->{{-4,4},{-4,4}},\n \ PlotStyle->{{Thickness[0.009],Green}},\n \ DisplayFunction->Identity];\ng[x_] := Log[3,x]\nincurve1 = \ Plot[g[x],{x,.000001,10},PlotRange->{{-4,4},{-4,4}},\n \ PlotStyle->{{Thickness[0.009],Magenta}},\n \ DisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell[TextData[ "ident= Show[Graphics[{Dashing[{0.02,0.02}],\n \ Line[{{-20,-20},{20,20}}]}],\n DisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Show[{curve1,incurve1,ident},\n DisplayFunction->$DisplayFunction];"], "Input", PageBreakAbove->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "\n\nAre these curves inverses of each other? Explain your reason.\n\nIs the \ green curve a function? Explain\n\n\nIs the magenta curve a function? \ Explain\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox[ "The graph above is the graph of y = 3\:02db and y = log\[Sterling] \tx \ . These plots will be seen often in your advanced classes. 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Since they reflect in our dotted line, we can clearly see that they \ are inverses of each other.", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True] }], "Text", Evaluatable->False, PageBreakBelow->True, AspectRatioFixed->True], Cell[TextData[ "One final note: Since the magenta graph is a function, it has to pass the \ Vertical Line Test. \nWhere would that vertical line be relocated if it were \ reflected along with the magenta line back into our green function?\n\nThe \ answer to that question will give us a New Test to use to check to see if the \ inverse of the original relation will be a function."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[". 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)