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Finding the Inverse Function and Domain Example by Walch Education

Finding the Inverse Function and Domain Example by Walch Education
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2014-08-06 01:33
FunctionCoordinate GeometryOptimization
In collections UCSS Secondary Math III Unit 4A Lesson 1
This example demonstrates how to find the inverse function g(x) of f(x) = 4x^2 and determine its domain. Key math concepts include inverse functions, domain, completing the square, and piecewise functions. The construction utilizes GeoGebra commands like LaTeX, If, TableText, stepMax, stepper, steps, counter, and text objects. Suitable for high school mathematics students learning about inverse functions and their solutions.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
  3. LaTeX("\text{Switch the domain and function variables, and then rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x).}") → text1
  4. 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
  5. LaTeX("\text{Next, rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x)}\text{, since this new relation represents the inverse of the}\cr\text{original function.}") → text4
  6. LaTeX((((("\text{The resulting inverse relation, }\textcolor{#006633}{x = " * a) * "[g(x)]^2}\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) = ") * a) * "x^2}.")) → text6
  7. LaTeX("\text{Solve the possible inverse for }\textcolor{#006633}{g(x)}.") → text7
  8. LaTeX("\text{Determine the domain of }\textcolor{#006633}{g(x)}.") → text9
  9. LaTeX("\text{Determine the range of }\textcolor{#006633}{g(x)}.") → text13
  10. LaTeX("\text{We need to compare the range of }\textcolor{#006633}{g(x)}\text{ to the domain of }\textcolor{#000099}{f(x)}.") → text14
  11. LaTeX("\text{Determine the domain of }\textcolor{#000099}{f(x)}.") → text16
  12. LaTeX("\text{Determine whether the function }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text18
  13. LaTeX("\text{Determine the parts of its domain over which }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one}\cr\text{correspondence.}") → text21
  14. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}.") → text23
  15. LaTeX("\text{Match domains to ranges to find the inverse(s) of }\textcolor{#000099}{f(x)}.") → text25
  16. LaTeX("\text{Check that the range of }\textcolor{#000099}{f(x)}\text{ matches the domain of }\textcolor{#006633}{g(x)}.") → text28
  17. If(a ≥ 0, "", "-") → signtext
  18. If(a ≥ 0, "non-negative", "negative") → numberwords
  19. If(a ≥ 0, LaTeX["[0, + \infty)"], LaTeX["(-\infty,0]"]) → domaing
  20. LaTeX((("\text{Because }\textcolor{#000099}{f(x)}\text{ is a quadratic function and }\textcolor{#000099}{x^2}\text{ can never be negative, the}\cr\text{range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{" * domaing) * "}.")) → text24
  21. If(a ≥ 0, sqrt(x) / asqrt, sqrt((-x)) / asqrt) → g_1
  22. If(a ≥ 0, ((-sqrt(x))) / asqrt, ((-sqrt((-x)))) / asqrt) → g_2
  23. LaTeX((((("\text{The range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{" * domaing) * "}\text{. The domain of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{") * domaing) * "}.")) → text29
  24. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text31
  25. TableText({{LaTeX[(("\textcolor{#000099}{f(x)=" * a) * "x^2}")], LaTeX["\text{Original function}"]}, {LaTeX[(("\textcolor{#000099}{x=" * a) * "[f(x)]^2}")], LaTeX["\text{Switch }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{x}."]}}) → text3
  26. TableText({{LaTeX[(("\textcolor{#000099}{x=" * a) * "[f(x)]^2}")], LaTeX["\text{Equation with variables switched}"]}, {LaTeX[(("\textcolor{#006633}{x=" * a) * "[g(x)]^2}")], LaTeX["\text{Rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x)}."]}}) → text5
  27. LaTeX("\text{The function }\textcolor{#000099}{f(x)}\text{ can only have an inverse over the part(s) of its domain}\cr\text{for which it exhibits one-to-one correspondence.}") → text19
  28. LaTeX((((((((("\text{We can determine that the function }\textcolor{#000099}{f(x) = " * a) * "x^2}\text{ does not show}\cr\text{one-to-one correspondence for all domain values because }\textcolor{#000099}{f(a) = ") * a) * "a^2}\text{ and }\cr\textcolor{#000099}{f(–a) = ") * a) * "(–a)^2 = ") * a) * "a^2}\text{. Since }\textcolor{#000099}{f(x)}\text{ has the same value for two different}\cr\text{values of }\textcolor{#000099}{x}\text{, }\textcolor{#000099}{f(x)}\text{ is not one-to-one.}\cr\text{An equivalant graphical test is the horizontal line test. If any horizontal}\cr\text{line touches the graph of }\textcolor{#000099}{f(x)}\text{ more than once, then }\textcolor{#000099}{f(x)}\text{ is not one-to-one.}")) → text20
  29. LaTeX("\text{We need to know where }\textcolor{#006633}{g(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
  30. LaTeX((((("\text{The function }\textcolor{#006633}{g(x)}\text{ is a radical function; specifically, it is a square root} \cr\text{function. Recall that the value of the expression inside the square root} \cr\text{must be positive in order for the function to be defined. The expression} \cr\text{under the square root is }\textcolor{#006633}{" * signtext) * "x}\text{; therefore, the values of }\textcolor{#006633}{x}\text{ are limited to}\cr\text{") * numberwords) * " numbers.}")) → text11
  31. LaTeX("\text{There are no limitations on the output values; therefore, the range}\cr\text{of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{(−\infty, +\infty)}.") → text15
  32. LaTeX("\text{There are no limitations on the input values; therefore, the domain}\cr\text{of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{(−\infty, +\infty)}.") → text17
  33. LaTeX("\text{ The restricted domains of }\textcolor{#000099}{f(x)}\text{, on which }\textcolor{#000099}{f(x)}\text{ is one-to-one, are }\textcolor{#000099}{(−\infty, 0]}\cr\text{ and }\textcolor{#000099}{[0, +\infty)}\text{. Only the point }\textcolor{#000099}{(0, 0)}\text{ is not restricted, because all of}\cr\text{ the other points do not have one-to-one correspondence. The horizontal}\cr\text{ line crosses the graph at two points for all other points on the graph.}") → text22
  34. LaTeX("\text{In order for a function to have an inverse, it must be one-to-one. Therefore, }\cr\text{match the restricted domains of }\textcolor{#000099}{f(x)}\text{ to the range of }\textcolor{#006633}{g(x)}.") → text26
  35. TableText({{LaTeX["\text{Restricted domain of }\textcolor{#000099}{f(x)}\text{:}"], LaTeX["\text{Inverse function with matching range:}"]}, {LaTeX["\textcolor{#000099}{[0,+\infty)}"], LaTeX[(("\textcolor{#006633}{g(x)=" * g_1) * "}")]}, {LaTeX["\textcolor{#000099}{(-\infty,0]}"], LaTeX[(("\textcolor{#006633}{g(x)=" * g_2) * "}")]}}) → text27
  36. LaTeX((((((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * a) * "x^2}\text{ on the restricted domain }\textcolor{#000099}{0≤x<+\infty}\text{ is }\textcolor{#006633}{g(x)=") * g_1) * "}.\cr \text{The inverse of }\textcolor{#000099}{f(x)=") * a) * "x^2}\text{ on the restricted domain }\textcolor{#000099}{-\infty<x≤0}\text{ is }\textcolor{#006633}{g(x)=") * g_2) * "}.")) → text30
  37. LaTeX((("\text{Thus, the domain of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{" * domaing) * "}\text{ because }\textcolor{#006633}{x}\text{ is raised to the }\textcolor{#006633}{\frac{1}{2}}\text{ power}\cr\text{ (recall that }\textcolor{#006633}{\sqrt{x} = x^\frac{1}{2}}\text{).}")) → text12
  38. TableText({{LaTeX[(("\textcolor{#006633}{x=" * a) * "[g(x)]^2}")], LaTeX["\text{Possible inverse}"]}, {LaTeX[(("\textcolor{#006633}{\frac{x}{" * a) * "}=[g(x)]^2}")], LaTeX[(("\text{Divide both sides by }\textcolor{#006633}{" * a) * "}.")]}, {LaTeX[(("\textcolor{#006633}{[g(x)]^2}=\frac{x}{" * a) * "}")], LaTeX["\text{Use Symmetric Property of Equality.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\sqrt{\frac{" * signtext) * "x}{") * abs(a)) * "}}")], LaTeX["\text{Find both roots of the equation. (Recall }(-1)^2=(1)^2\text{.)}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{\sqrt{" * signtext) * "x}}{\sqrt{") * abs(a)) * "}}")], LaTeX["\text{Distribute the square root operation.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{\sqrt{" * signtext) * "x}}{") * asqrt) * "}")], LaTeX["\text{Simplify.}"]}, {LaTeX[(((("\textcolor{#006633}{g(x)}=\pm\frac{1}{" * asqrt) * "}\sqrt{") * signtext) * "x}")], LaTeX["\text{Separate out the coefficient.}"]}}) → text8

AI Drawing Prompt Reference

Draw the graph of the function f(x) = 4x^2 and show the process of finding its inverse function g(x). First, switch x and f(x) in f(x) to get x = 4y^2, then solve for y to find g(x). Highlight the graph of f(x) and show g(x) in two different colors over the intervals [0, +∞) and (-∞, 0], corresponding to the monotonic intervals of f(x). Finally, use text to explain the domain and range of g(x), and the correspondence between the monotonic intervals of f(x) and the domain of g(x).
The prompt above was compiled from the original drawing commands and is for reference only.