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Inverse of a Radical Function and Its Domain

Inverse of a Radical Function and Its Domain
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2014-08-06 01:39
FunctionCoordinate GeometryGraphing
In collections UCSS Secondary Math III Unit 4A Lesson 1
This example demonstrates how to find the inverse of the radical function f(x)=√[3]{x+2} and determine its domain and the domain of the inverse function. Key math concepts include function inversion, domain determination, and cube root operations. GeoGebra commands and objects like LaTeX, Simplify, If, TableText, and objects such as function f, inverse function g, slider a, and text objects are used to construct the demonstration dynamically, enhancing comprehension. Suitable for high school mathematics teaching and function learning.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. Simplify(((a * x) + b)^(1 / 3)) → f
  3. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
  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{Determine the domain of }\textcolor{#000099}{f(x)}.") → text16
  6. LaTeX("\text{Determine whether the function }\textcolor{#000099}{f(x)}\text{ exhibits one-to-one correspondence.}") → text18
  7. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}.") → text21
  8. LaTeX("\text{Match domains to ranges to find the inverse(s) of }\textcolor{#000099}{f(x)}.") → text23
  9. LaTeX("\text{Check that the range of }\textcolor{#000099}{f(x)}\text{ matches the domain of }\textcolor{#006633}{g(x)}.") → text25
  10. Simplify((x^(3) - b) / a) → g
  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. If(a ≟ 1, LaTeX["f^{-1}(x)"], If[a ≟ -1, LaTeX["-f^{-1}(x)"], LaTeX[(a * "f^{-1}(x)")]]) → inversefront
  14. If(b ≟ 0, LaTeX[""], If[b < 0, LaTeX[("-" * abs(b))], LaTeX[("+" * abs(b))]]) → inversemid
  15. If(a ≟ 1, LaTeX[(("\sqrt[3]{f^{-1}(x)" * inversemid) * "}")], If[a ≟ -1, LaTeX[(("\sqrt[3]{-f^{-1}(x)" * inversemid) * "}")], LaTeX[(((("\sqrt[3]{" * a) * "f^{-1}(x)") * inversemid) * "}")]]) → inversefunc
  16. 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
  17. LaTeX("f(x)") → inversefronta
  18. If(b ≟ 1, LaTeX["f(x)"], If[b ≟ -1, LaTeX["-f(x)"], LaTeX[(b * "f(x)")]]) → inversemida
  19. If(a ≟ 1, LaTeX[(("\sqrt[3]{f(x)" * inversemid) * "}")], If[a ≟ -1, LaTeX[(("\sqrt[3]{-f(x)" * inversemid) * "}")], LaTeX[(((("\sqrt[3]{" * a) * "f(x)") * inversemid) * "}")]]) → inversefunca
  20. LaTeX("\text{Solve the possible inverse for }\textcolor{#006633}{f^{-1}(x)}.") → text7
  21. LaTeX("\text{Determine the domain of }\textcolor{#006633}{f^{-1}(x)}.") → text9
  22. LaTeX("\text{Determine the range of }\textcolor{#006633}{f^{-1}(x)}.") → text13
  23. LaTeX("\text{We need to compare the range of }\textcolor{#006633}{f^{-1}(x)}\text{ to the domain of }\textcolor{#000099}{f(x)}.") → text14
  24. 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
  25. LaTeX("\text{The function }\textcolor{#006633}{f^{-1}(x)}\text{ is a polynomial function so its domain is all real numbers.}") → text11
  26. LaTeX("\text{The range of }\textcolor{#000099}{f(x)}\text{ is }\textcolor{#000099}{(-\infty,+\infty)}\text{. The domain of }\textcolor{#006633}{f^{-1}(x)}\text{ is }\textcolor{#006633}{(-\infty,+\infty)}.") → text26
  27. LaTeX("\text{Check that }\textcolor{#000099}{f(x)}\text{ is one-to-one with the horizontal line test.}") → text27
  28. LaTeX("\text{Check that }\textcolor{#006633}{f^{-1}(x)}\text{ is a function using the vertical line test.}") → text28
  29. LaTeX((((("\text{The inverse of }\textcolor{#000099}{f(x)=" * f) * "}\text{ is }\textcolor{#006633}{f^{-1}(x)=") * g) * "}.")) → text29
  30. LaTeX((("\text{We can determine that the function }\textcolor{#000099}{f(x) =" * f) * "}\text{ does show one-to-one}\cr\text{correspondence for all domain values by noting that cube roots, and other odd roots,}\cr\text{preserve negative signs so that no two values of }\textcolor{#000099}{x}\text{ can produce the same }\textcolor{#000099}{f(x)}\text{ value.}")) → text20
  31. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text31
  32. 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
  33. 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
  34. LaTeX("\text{In other words, the domain of }\textcolor{#006633}{f^{-1}(x)} \text{ is }\textcolor{#006633}{(-\infty,+\infty)}.") → text12
  35. TableText({{LaTeX[(("\textcolor{#006633}{x=" * inversefunc) * "}")], LaTeX["\text{Possible inverse}"]}, {LaTeX[(((("\textcolor{#006633}{x^3=(" * a) * ")f^{-1}(x)+(") * b) * ")}")], LaTeX["\text{Cube both sides.}"]}, {LaTeX[(((("\textcolor{#006633}{" * Simplify[x^(3) - b]) * "=(") * a) * ")f^{-1}(x) }")], LaTeX[(("\text{Subtract }\textcolor{#006633}{" * b) * "}\text{ from both sides.}")]}, {LaTeX[(((("\textcolor{#006633}{\frac{" * Simplify[x^(3) - b]) * "}{") * a) * "}=f^{-1}(x)}")], LaTeX[(("\text{Divide both sides by }\textcolor{#006633}{" * a) * "}.")]}, {LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g) * "}")], LaTeX["\text{Simplify and use the Symmetric Property of Equality.}"]}}) → text8
  36. LaTeX("\text{We need to know where }\textcolor{#006633}{f^{-1}(x)}\text{ is defined in order to know its domain, which we}\cr\text{want to compare to the range of }\textcolor{#000099}{f(x)}.") → text10
  37. 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)}.") → text15
  38. 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
  39. LaTeX("\text{There is no limitation on the values }\textcolor{#000099}{f(x)}\text{ can have, so its range is }\textcolor{#000099}{(-\infty,+\infty)}.") → text22
  40. TableText({{LaTeX["\text{Domain of }\textcolor{#000099}{f(x)}\text{:}"], LaTeX["\text{Inverse function with matching range:}"]}, {LaTeX["\textcolor{#000099}{(-\infty,+\infty)}"], LaTeX[(("\textcolor{#006633}{f^{-1}(x)=" * g) * "}")]}}) → text24

AI Drawing Prompt Reference

Draw the graph of a cube root function and its inverse, showing the function f(x)=√[3]{x+2} and its inverse g(x). First, plot the graph of f(x) and label its domain and range. Then, plot the graph of g(x) and label its domain and range. Finally, verify the one-to-one correspondence of f(x) and g(x) using the horizontal and vertical line tests.
The prompt above was compiled from the original drawing commands and is for reference only.