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Verify Inverse Functions f(x)=±√(x+3) and g(x)=x²-3

Verify Inverse Functions f(x)=±√(x+3) and g(x)=x²-3
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2014-07-29 03:09
FunctionTransformationGraphing
In collections CCGPS AA Unit 6 Lesson 7
① This example demonstrates how to verify that two functions are inverses by evaluating the compositions f(g(x)) and g(f(x)) both algebraically and graphically. ② Core mathematical concepts include function composition, the definition of inverse functions, domain‑range correspondence, and the handling of the positive and negative branches of the square‑root function. ③ Construction highlights: the applet uses GeoGebra commands LaTeX, Corner, If, Simplify, Max together with objects such as stepMax, stepper, steps, counter, text0, Original, Randomize, ZoomFit, button1, A, B, txmin, txmax, tymin, tymax, Problem, a, b, c, f, evenodd, f2, g, d, text4, text6, text8, text9, text10, text21, text23, text24, list1, maxzeros, range, xmax, xmin, ymax, ymin, text3 to create a step‑by‑step interactive verification. ④ Suitable for high‑school “Functions and Derivatives” units or college‑prep courses as a teaching demo or self‑study resource for inverse‑function verification.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Move the slider to solve the problem.}") → text0
  3. Corner(1) → A
  4. Corner(3) → B
  5. LaTeX("\text{}") → Problem
  6. If(IsInteger[c / 2], LaTeX["\pm"], "") → evenodd
  7. Simplify(Polynomial[x^(c) - b] / a) → g
  8. LaTeX((((((((("\textcolor{#000099}{f(x)=" * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}")) → text4
  9. LaTeX((((((((("\textcolor{#CC0000}{f(g(x))=" * evenodd) * "((") * Simplify[(a * g(x))]) * ")+(") * b) * "))^{1/") * c) * "}}")) → text6
  10. LaTeX((((((("\textcolor{#CC0000}{f(g(x))=" * evenodd) * "\left(x^") * c) * "\right)^{1/") * c) * "}}")) → text8
  11. LaTeX("\text{Simplify the powers.}") → text9
  12. LaTeX("\textcolor{#CC0000}{f(g(x))=x}") → text10
  13. LaTeX("\textcolor{#CC0000}{g(f(x))=x}") → text21
  14. LaTeX("\text{State a conclusion about whether }\textcolor{#006633}{g(x)}\text{ is the inverse}\cr\text{of }\textcolor{#000099}{f(x)}\text{ based on the results for }\textcolor{#CC0000}{f(g(x))}\text{ and }\textcolor{#CC0000}{g(f(x))}.") → text23
  15. LaTeX("\text{We have determined that }\textcolor{#CC0000}{f(g(x)) = x}\text{ and }\cr\textcolor{#CC0000}{g(f(x)) = x}\text{. By substitution, }\textcolor{#CC0000}{f(g(x)) = g(f(x))};\cr\text{therefore, }\textcolor{#006633}{g(x)}\text{ is the inverse of }\textcolor{#000099}{f(x)}.") → text24
  16. Max(Zip[abs(x(a)), a, list1]) → maxzeros
  17. LaTeX("\text{Original function }\textcolor{#000099}{f(x)}") → text3
  18. LaTeX((("\text{Substitute }\textcolor{#006633}{" * g) * "}\text{ for }\textcolor{#000099}{x}\text{ and name the }\cr\text{new function }\textcolor{#CC0000}{f(g(x))}.")) → text5
  19. LaTeX((("\text{The notation }\textcolor{#CC0000}{f(g(x))}\text{ means that the input for }\textcolor{#000099}{f(x)}\text{ is equal to the output of }\textcolor{#006633}{g(x)}\text{.}\cr\text{In other words, to find }\textcolor{#CC0000}{f(g(x))}\text{, replace }\textcolor{#000099}{x}\text{ in the value of the function }\textcolor{#000099}{f(x)}\cr\text{with the value of the function }\textcolor{#006633}{g(x)}\text{. Therefore, replace }\textcolor{#000099}{x}\text{ with }\textcolor{#006633}{" * g) * "}\text{ and then}\cr\text{solve the resulting composition of functions.}")) → text2
  20. LaTeX((((((((((("\text{Find }\textcolor{#CC0000}{f(g(x))}\text{ using }\textcolor{#000099}{f(x)=" * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\text{ and }\textcolor{#006633}{g(x) = ") * g) * "}.")) → text1
  21. LaTeX("\text{Original function }\textcolor{#006633}{g(x)}") → text14
  22. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  23. LaTeX("\text{Simplify.}") → text20
  24. LaTeX((((((((((("\text{Find }\textcolor{#CC0000}{g(f(x))}\text{ using }\textcolor{#000099}{f(x)=" * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\text{ and }\textcolor{#006633}{g(x) = ") * g) * "}.")) → text12
  25. LaTeX((((((((("\text{The notation }\textcolor{#CC0000}{g(f(x))}\text{ means that the input for }\textcolor{#006633}{g(x)}\text{ is equal to the output of }\textcolor{#000099}{f(x)}\text{.}\cr\text{In other words, to find }\textcolor{#CC0000}{g(f(x))}\text{, replace }\textcolor{#006633}{x}\text{ in the value of the function }\textcolor{#006633}{g(x)}\cr\text{with the value of the function }\textcolor{#000099}{f(x)}\text{. Therefore, replace }\textcolor{#006633}{x}\text{ with }\textcolor{#000099}{" * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\cr\text{and then solve the resulting composition of }\text{functions.}")) → text13
  26. LaTeX((("\textcolor{#006633}{g(x)=" * g) * "}")) → text15
  27. LaTeX((((((((((((((("\textcolor{#CC0000}{g(f(x))=\frac{\left(" * evenodd) * "((") * Simplify[(a * x)]) * ")+(") * b) * "))^{1/") * c) * "}\right)^{") * c) * "}-(") * b) * ")}{") * a) * "}}")) → text17
  28. LaTeX((((((((("\text{Substitute }\textcolor{#000099}{" * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\text{ for }\textcolor{#006633}{x}\text{ and }\cr\text{name the new function }\textcolor{#CC0000}{g(f(x))}.")) → text16
  29. LaTeX((((((((("\textcolor{#CC0000}{g(f(x))=\frac{\left((" * Simplify[(a * x)]) * ")+(") * b) * ")\right)-") * b) * "}{") * a) * "}}")) → text19
  30. LaTeX("\text{Simplify the powers.}") → text18
  31. LaTeX((((((((((("\text{}\textcolor{#CC0000}{g(f(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\text{ and }\textcolor{#006633}{g(x)=") * g) * "}.")) → text22
  32. LaTeX((((((((((("\text{}\textcolor{#CC0000}{f(g(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * evenodd) * "(") * Simplify[(a * x)]) * "+(") * b) * "))^{1/") * c) * "}}\text{ and }\textcolor{#006633}{g(x)=") * g) * "}.")) → text11
  33. LaTeX("\text{Simplify.}") → text7

AI Drawing Prompt Reference

Draw the graphs of functions f(x) = ±√(x+3) and g(x) = x²-3, and verify if they are inverse functions of each other. First, draw the positive and negative branches of f(x). Then, draw g(x). Next, calculate f(g(x)) and g(f(x)), simplify the expressions, and check if the results are equal to x to verify if they are inverse functions.
The prompt above was compiled from the original drawing commands and is for reference only.