MathPlotAI Logo MathPlotAI

Function Composition Example

Function Composition Example
Loading the math board and drawing, please wait…
2014-07-29 03:21
FunctionCoordinate GeometryTransformationSlider AnimationGraphing
In collections CCGPS AA Unit 6 Lesson 7
This example demonstrates how to calculate the composite functions f(g(x)) and g(f(x)) for the given functions f(x) = 2x³ - 1 and g(x) = (x+1/2)^(1/3), and verify if they are inverse functions of each other. Key mathematical concepts include functions, inverse functions, and composite functions. It uses GeoGebra commands such as LaTeX, If, and Max for construction. Suitable for high school mathematics education and learning.

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. Polynomial((a * x^(c)) + b) → f
  7. If(IsInteger[c / 2], LaTeX["\pm"], "") → evenodd
  8. LaTeX("\textcolor{#CC0000}{f(g(x))=x}") → text10
  9. LaTeX("\textcolor{#CC0000}{g(f(x))=x}") → text21
  10. Max(Zip[If[Defined[x(a)], abs(x(a)), 555], a, list1] / {555}) → maxzeros
  11. LaTeX((((((((((("\textcolor{#CC0000}{f(g(x))=" * a) * "\left(") * evenodd) * "") * g) * "\right)^{") * c) * "}+(") * b) * ")}")) → text6
  12. LaTeX("\text{Simplify the powers.}") → text7
  13. LaTeX("\text{Simplify the powers.}") → text20
  14. LaTeX((((((("\text{Find }\textcolor{#CC0000}{f(g(x))}\text{ using }\textcolor{#000099}{f(x)=" * f) * "}\text{ and }\textcolor{#006633}{g(x) = ") * evenodd) * "") * g) * "}.")) → text1
  15. LaTeX("\text{Original function }\textcolor{#000099}{f(x)}") → text3
  16. LaTeX((("\text{Substitute }\textcolor{#006633}{" * g) * "}\text{ for }\textcolor{#000099}{x}\text{ and }\cr\text{name the new function }\textcolor{#CC0000}{f(g(x))}.")) → text5
  17. LaTeX("\text{Simplify.}") → text9
  18. LaTeX((((((("\text{Therefore }\textcolor{#CC0000}{f(g(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * f) * "}\text{ and }\textcolor{#006633}{g(x)=") * evenodd) * "") * g) * "}.")) → text11
  19. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → text4
  20. LaTeX((((((("\text{Find }\textcolor{#CC0000}{g(f(x))}\text{ using }\textcolor{#000099}{f(x)=" * f) * "}\text{ and }\textcolor{#006633}{g(x) = ") * evenodd) * "") * g) * "}.")) → text12
  21. LaTeX("\text{Original function }\textcolor{#006633}{g(x)}") → text14
  22. LaTeX("\text{We have determined that }\textcolor{#CC0000}{f(g(x)) = x}\text{ and }\textcolor{#CC0000}{g(f(x)) = x}\text{. }\cr\text{By substitution, }\textcolor{#CC0000}{f(g(x)) = g(f(x))};\text{ therefore, }\textcolor{#006633}{g(x)}\text{ is the inverse of }\textcolor{#000099}{f(x)}.") → text24
  23. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  24. LaTeX("\text{State a conclusion about whether }\textcolor{#006633}{g(x)}\text{ is the inverse of }\textcolor{#000099}{f(x)}\text{ based on the }\cr\text{results for }\textcolor{#CC0000}{f(g(x))}\text{ and }\textcolor{#CC0000}{g(f(x))}.") → text23
  25. 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)}\text{ with the }\cr\text{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
  26. LaTeX((((((((("\textcolor{#CC0000}{f(g(x))=" * a) * "\left(\frac{") * Simplify[x - b]) * "}{") * a) * "}\right)+(") * b) * ")}")) → text8
  27. 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)}\text{ with }\cr\text{the value of the function }\textcolor{#000099}{f(x)}\text{. Therefore, replace }\textcolor{#006633}{x}\text{ with }\textcolor{#000099}{" * f) * "}\text{ and then solve}\cr\text{the resulting composition of functions.}")) → text13
  28. LaTeX((("\textcolor{#006633}{g(x)=" * g) * "}")) → text15
  29. LaTeX((("\text{Substitute }\textcolor{#000099}{" * f) * "}\text{ for }\textcolor{#006633}{x}\text{ and }\cr\text{name the new function }\textcolor{#CC0000}{g(f(x))}.")) → text16
  30. LaTeX((((((((("\textcolor{#CC0000}{g(f(x))=\left(\frac{(" * f) * ")-(") * b) * ")}{") * a) * "}\right)^{1/") * c) * "}}")) → text17
  31. LaTeX((((((("\textcolor{#CC0000}{g(f(x))=" * evenodd) * "\left(x^") * c) * "\right)^{1/") * c) * "}}")) → text19
  32. LaTeX("\text{Simplify.}") → text18
  33. LaTeX((((((("\text{}\textcolor{#CC0000}{g(f(x))=x}\text{ when }\textcolor{#000099}{f(x)= " * f) * "}\text{ and }\textcolor{#006633}{g(x)=") * evenodd) * "") * g) * "}.")) → text22

AI Drawing Prompt Reference

Plot the functions f(x) = 2x³ - 1 and g(x) = (x+1/2)^(1/3), and calculate their composite functions f(g(x)) and g(f(x)). Verify if they are inverse functions of each other. First, plot f(x) and g(x), then calculate f(g(x)) and g(f(x)), simplify the expressions, and check if the results are equal to x.
The prompt above was compiled from the original drawing commands and is for reference only.