MathPlotAI Logo MathPlotAI

Finding the Inverse Function and Domain Example

Finding the Inverse Function and Domain Example
Loading the math board and drawing, please wait…
2014-08-06 01:37
FunctionCoordinate GeometryGraphing
In collections UCSS Secondary Math III Unit 4A Lesson 1
This example demonstrates how to find the inverse of the function f(x) = x(x-1) and determine its domain. The core mathematical concepts involved include the definition of an inverse function, methods for finding inverses, and determining the domain of a function. The construction utilizes GeoGebra commands and objects such as LaTeX, If, Simplify, TableText, and elements like f, text0, text1, etc., to solve and display the inverse function. It is suitable for high school mathematics teaching and learning about inverse functions.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
  3. LaTeX((("\text{Since the inverse function results from switching the input and output variables}\cr\text{of the original function, identify which variable represents the input and which}\cr\text{represents the output. For the function }\textcolor{#000099}{f(x) = " * f) * "}\text{, }\textcolor{#000099}{x}\text{ is the input (independent)}\cr\text{variable, and }\textcolor{#000099}{f(x)}\text{ is the output (dependent) variable. Therefore, swap }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}.")) → text2
  4. LaTeX("\text{Determine the domain of }\textcolor{#000099}{f(x)}.") → text16
  5. LaTeX("\text{Determine whether the function }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text18
  6. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}.") → text23
  7. LaTeX("\text{Match domains to ranges to find the inverse(s) of }\textcolor{#000099}{f(x)}.") → text25
  8. If(a ≥ 0, "", "-") → signtext
  9. If(a ≥ 0, "non-negative", "negative") → numberwords
  10. If(a ≥ 0, LaTeX["[0, + \infty)"], LaTeX["(-\infty,0]"]) → domaing
  11. LaTeX("\text{Switch the domain and function variables, and then rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x).}") → text1
  12. LaTeX("\text{Next, rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x)}\text{, since this new relation represents the inverse of the}\cr\text{original function.}") → text4
  13. Simplify(ac / ((2 * ab))) → gcon
  14. If(a ≟ 1, LaTeX["f^{-1}(x)"], If[a ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(a * "f^{-1}(x)")]]) → inversefront
  15. If(b ≟ 1, LaTeX["f^{-1}(x)"], If[b ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(b * "f^{-1}(x)")]]) → inversemid
  16. If(c < 0, LaTeX[("+" * abs(c))], LaTeX[("-" * c)]) → inverseend
  17. LaTeX(((((inversefront * "(") * inversemid) * inverseend) * ")")) → inversefunc
  18. LaTeX((((("\text{The resulting inverse relation, }\textcolor{#006633}{x = " * inversefunc) * "}\text{, is the possible inverse function of}\cr\text{the original function. It must be evaluated to determine whether it has }\cr\text{one-to-one correspondence with the original function, }\textcolor{#000099}{f(x) = ") * f) * "}.")) → text6
  19. If(a ≟ 1, LaTeX["f(x)"], If[a ≟ -1, LaTeX["-f(x)"], LaTeX[(a * "f(x)")]]) → inversefronta
  20. If(b ≟ 1, LaTeX["f(x)"], If[b ≟ -1, LaTeX["-f(x)"], LaTeX[(b * "f(x)")]]) → inversemida
  21. LaTeX(((((inversefronta * "(") * inversemida) * inverseend) * ")")) → inversefunca
  22. LaTeX("\text{Solve the possible inverse for }\textcolor{#006633}{f^{-1}(x)}.") → text7
  23. If(ab ≟ -1, LaTeX["-[f^{-1}(x)]^2"], If[ab ≟ 1, LaTeX["[f^{-1}(x)]^2"], LaTeX[(ab * "[f^{-1}(x)]^2")]]) → quadfront
  24. If(ac ≟ -1, LaTeX["+f^{-1}(x)"], If[ac ≟ 1, LaTeX["-f^{-1}(x)"], If[ac < 0, LaTeX[(("+" * abs(ac)) * "f^{-1}(x)")], LaTeX[(("-" * ac) * "f^{-1}(x)")]]]) → quadmid
  25. LaTeX("\text{Determine the domain of }\textcolor{#006633}{f^{-1}(x)}.") → text9
  26. LaTeX("\text{Determine the range of }\textcolor{#006633}{f^{-1}(x)}.") → text13
  27. LaTeX("\text{We need to compare the range of }\textcolor{#006633}{f^{-1}(x)}\text{ to the domain of }\textcolor{#000099}{f(x)}.") → text14
  28. Simplify(f) → fsimp
  29. LaTeX("\text{The function }\textcolor{#000099}{f(x)}\text{ can only have an inverse over the parts of its domain where it exhibits}\cr\text{one-to-one correspondence.}") → text19
  30. LaTeX((((("\text{We can determine that the function }\textcolor{#000099}{f(x) =" * f) * "}\text{ does not show one-to-one}\cr\text{correspondence for all domain values by looking at it in standard form: }\textcolor{#000099}{f(a) = ") * fsimp) * "}.\cr\text{We can now more easily see that }\textcolor{#000099}{f(x)}\text{ is a quadratic function, and that its graph is a parabola.}")) → text20
  31. LaTeX("\text{Determine the parts of its domain over which }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text21
  32. LaTeX("\text{Because, }\textcolor{#000099}{f(x)}\text{ is quadratic, the parts of the graph of }\textcolor{#000099}{f(x)}\text{ that show a one-to-one correspondence}\cr\text{lie on either side of its axis of symmetry. This axis divides the graph of the function into two}\cr\text{halves. If }\textcolor{#000099}{f(x)}\text{ is written as }\textcolor{#000099}{ax^2 + bx + c}\text{, the equation of its axis of symmetry is given by }\cr\textcolor{#000099}{x = −\frac{b}{2a}}.\text{ The domains of the halves of the graph of }\textcolor{#000099}{f(x)}\text{ are defined by this value of }\textcolor{#000099}{x}.") → text22
  33. LaTeX((((("\textcolor{#000099}{x = −\frac{b}{2a}=\frac{" * ac) * "}{2(ab)}=") * Simplify[ac / ((2 * ab))]) * "}")) → text22a
  34. TableText({{LaTeX[(("\textcolor{#000099}{f(x)=" * f) * "}")], LaTeX["\text{Original function}"]}, {LaTeX[(("\textcolor{#000099}{x=" * inversefunca) * "}")], LaTeX["\text{Switch }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}."]}}) → text3
  35. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text31
  36. LaTeX((("\text{In other words, because of the square root, the domain of }\textcolor{#006633}{f^{-1}(x)} \text{ is }\textcolor{#006633}{[" * Simplify[((-ac^(2))) / ((4 * b))]) * ",+∞)}.")) → text12
  37. LaTeX((((("\text{There are two restricted domains where }\textcolor{#000099}{f(x)}\text{ is one-to-one: }\textcolor{#000099}{(-\infty," * Simplify[ac / ((2 * ab))]) * "]}\text{ and }\textcolor{#000099}{[") * Simplify[ac / ((2 * ab))]) * ",+\infty)}.")) → text22b
  38. LaTeX((((("\text{The range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{[" * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}\text{. The domain of }\textcolor{#006633}{f^{-1}(x)}\text{ is }\textcolor{#006633}{[") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}.")) → text28
  39. LaTeX((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * f) * "}\text{ on the restricted domain }\textcolor{#000099}{[") * Simplify[ac / ((2 * ab))]) * ",+\infty)}\text{, is }\textcolor{#006633}{f^{-1}(x)=") * g_1) * "}.")) → text29
  40. LaTeX((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * f) * "}\text{ on the restricted domain }\textcolor{#000099}{(-\infty,") * Simplify[ac / ((2 * ab))]) * "]}\text{, is }\textcolor{#006633}{f^{-1}(x)=") * g_2) * "}.")) → text30
  41. TableText({{LaTeX[(("\textcolor{#006633}{x=" * inversefunc) * "}")], LaTeX["\text{Possible inverse}"]}, {LaTeX[((("\textcolor{#006633}{x=" * quadfront) * quadmid) * "}")], LaTeX["\text{Multiply out the right side.}"]}, {LaTeX[((("\textcolor{#006633}{0=" * quadfront) * quadmid) * "-x}")], LaTeX["\text{Subtract }\textcolor{#006633}{x}\text{ from both sides.}"]}, {LaTeX[(((((((("\textcolor{#006633}{f^{-1}(x)=\frac{" * ac) * "\pm\sqrt{(") * ac) * ")^2-4(") * ab) * ")(-x)}}{2(") * ab) * ")}}")], LaTeX["\text{Apply the quadratic formula. Treat }\textcolor{#006633}{x}\text{ as a constant and }\textcolor{#006633}{f^{-1}(x)}\text{ as the variable.}"]}, {LaTeX[(((("\textcolor{#006633}{f^{-1}(x)=" * gcon) * "\pm") * gfunc) * "}")], LaTeX["\text{Simplify.}"]}}) → text8
  42. LaTeX("\text{We need to know where }\textcolor{#006633}{f^{-1}(x)}\text{ is defined in order to know its domain,}\cr\text{which we want to compare to the range of }\textcolor{#000099}{f(x)}.") → text10
  43. LaTeX((((("\text{The function }\textcolor{#006633}{f^{-1}(x)}\text{ is a radical function; specifically, it is a square root function.} \cr\text{Recall that the value of the expression inside the square root must be positive}\cr\text{in order for the function to be defined. The expression under the square root is }\textcolor{#006633}{" * Simplify[acsq + ((4 * ab) * x)]) * "};\cr\text{therefore, the values of }\textcolor{#006633}{x}\text{ are limited to }\textcolor{#006633}{x\ge ") * Simplify[((-ac^(2))) / ((4 * b))]) * "}.")) → text11
  44. LaTeX((((((((("\text{There are no limitations on the output values; therefore, the range of }\textcolor{#006633}{f^{-1}(x)}\text{ is }\cr\textcolor{#006633}{(−\infty, +\infty)}\text{. However, }\textcolor{#006633}{f^{-1}(x)}\text{ is made of two pieces, and each piece has some}\cr\text{limitation on its range. The range of }\textcolor{#006633}{" * g_1) * "}\text{ is }\textcolor{#006633}{[") * Simplify[ac / ((2 * ab))]) * ",+∞)}\text{. The range of }\cr\textcolor{#006633}{") * g_2) * "}\text{ is }\textcolor{#006633}{(-∞,") * Simplify[ac / ((2 * ab))]) * "]}.")) → text15
  45. LaTeX("\text{There are no limitations on the input values; therefore, the domain }\text{of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{(−\infty, +\infty)}.") → text17
  46. TableText({{LaTeX[(("\textcolor{#000099}{x=" * inversefunca) * "}")], LaTeX["\text{Equation with variables switched}"]}, {LaTeX[(("\textcolor{#006633}{x=" * inversefunc) * "}")], LaTeX["\text{Rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{f^{-1}(x)}."]}}) → text5
  47. TableText({{LaTeX["\text{Restricted domain of }\textcolor{#000099}{f(x)}\text{:}"], LaTeX["\text{Inverse function with matching range:}"]}, {LaTeX[(("\textcolor{#000099}{[" * Simplify[ac / ((2 * ab))]) * ",+\infty)}")], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_1) * "}")]}, {LaTeX[(("\textcolor{#000099}{(-\infty," * Simplify[ac / ((2 * ab))]) * "]}")], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g_2) * "}")]}}) → text26
  48. LaTeX("\text{Check that the range of }\textcolor{#000099}{f(x)}\text{ matches the domain of }\textcolor{#006633}{f^{-1}(x)}.") → text27
  49. LaTeX((((((("\text{Because }\textcolor{#000099}{f(x)}\text{ is a parabolic function, its range is limited. Since the coefficient of the quadratic term is positive, }\cr\text{the parabola opens up and the minimum of the range is the y-coordinate of the minimum of the parabola.}\cr\text{The x-coordinate of the minimum is known; it is the axis of symmetry, }\textcolor{#000099}{x=" * Simplify[ac / ((2 * ab))]) * "}\text{. To determine the y-coordinate,}\cr\text{substitute the equation of the axis of symmetry into the original function, }\textcolor{#000099}{f(x).}\cr\text{The result is }\textcolor{#000099}{y=") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * "}\text{. Therefore, the range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{[") * Simplify[((-((a * c^(2))))) / ((4 * b))]) * ",+\infty)}.")) → text24

AI Drawing Prompt Reference

Draw the graph of the function f(x) = x(x-1) and show the process of finding its inverse. First, plot the graph of f(x) and label its expression. Then, by swapping variables and solving, obtain the expression for the inverse function and plot its graph. Also, analyze the domain and range of f(x) to determine the domain and range of the inverse function, using different colors to distinguish the original function and its inverse. Finally, label the monotonic intervals of f(x) and the expressions of the inverse function over these intervals.
The prompt above was compiled from the original drawing commands and is for reference only.