MathPlotAI Logo MathPlotAI

Determining Function Invertibility and Values

Determining Function Invertibility and Values
Loading the math board and drawing, please wait…
2014-09-13 03:34
FunctionCalculusProbability & Statistics
In collections UCSS Secondary Math III Unit 4A Lesson 1
This example demonstrates how to determine if a function represented in a table is invertible and to find specific function values using the table. Key math concepts include function invertibility and its determination criteria. GeoGebra commands and objects like Element, IndexOf, LaTeX, Corner, TableText, If are used to construct and manipulate the table, function representation, and conditional logic. Suitable for high school students learning about function properties and GeoGebra applications.

GeoGebra Commands & Parameters

  1. Element(common, 3) → v3
  2. IndexOf(v3, functionvalues) → p3
  3. Element(common, 2) → v2
  4. IndexOf(v2, xvalues) → p2
  5. Element(common, 1) → v1
  6. IndexOf(v1, xvalues) → p1
  7. LaTeX(steps) → counter
  8. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  9. Corner(1) → A
  10. Corner(3) → B
  11. TableText({Append["x", xvalues], Append["f(x)", functionvalues]}, "c|_") → data
  12. LaTeX("\text{Determine whether the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{ exists.}") → text1
  13. LaTeX("\text{To determine whether a function is invertible, check to see if any }\textcolor{#000099}{x}\text{- or }\textcolor{#000099}{y}\text{-values}\cr\text{have been used more than once.}") → text2
  14. LaTeX("\text{The function represented in the table repeats an }\textcolor{#000099}{x}\text{- value, so the function is not}\cr\text{“one-to-one” and is not invertible. The inverse }\textcolor{#000099}{f^{-1}(x)}\text{ does not exist.}") → text3b
  15. If(Length[xvalues] ≟ Length[Unique[xvalues]], "y", "n") → xunique
  16. LaTeX("\text{The function represented in the table repeats a }\textcolor{#000099}{y}\text{- value, so the function is not}\cr\text{“one-to-one” and is not invertible. The inverse }\textcolor{#000099}{f^{-1}(x)}\text{ does not exist.}") → text3c
  17. LaTeX("\text{The function represented in the table repeats an }\textcolor{#000099}{x}\text{- and a }\textcolor{#000099}{y}\text{-value, so the}\cr\text{function is not “one-to-one” and is not invertible. The inverse }\textcolor{#000099}{f^{-1}(x)}\text{ does not}\cr\text{exist.}") → text3d
  18. If(Length[functionvalues] ≟ Length[Unique[functionvalues]], "y", "n") → inverseunique
  19. LaTeX("\text{The function represented in the table does not repeat any }\textcolor{#000099}{x}\text{-values or any }\cr\textcolor{#000099}{y}\text{-values, so the function is “one-to-one” and is invertible. The inverse }\textcolor{#000099}{f^{–1}(x)}\cr\text{exists.}") → text3a
  20. If(xunique ≟ "y", If[inverseunique ≟ "y", LaTeX[text3a], LaTeX[text3c]], If[inverseunique ≟ "y", LaTeX[text3b], LaTeX[text3d]]) → text3
  21. Element(common, 4) → v4
  22. Element(functionvalues, p1) → m1
  23. LaTeX((("\text{Find the function value }\textcolor{#000099}{f(" * v1) * ")}.")) → text4
  24. Element(xvalues, p3) → m3
  25. Element(functionvalues, p2) → m2
  26. LaTeX((("\text{Find the function value }\textcolor{#000099}{f(" * v2) * ")}.")) → text7
  27. LaTeX((((((((((((((("\text{To find the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x=" * v1) * "}\text{, locate the value }\textcolor{#000099}{") * v1) * "}\text{ in the }\textcolor{#000099}{x}\text{ row of the table.}\cr\text{The value }\textcolor{#000099}{") * m1) * "}\text{ directly below }\textcolor{#000099}{x = ") * v1) * "}\text{ is the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x = ") * v1) * "}\text{. That is, }\textcolor{#000099}{f(") * v1) * ") = ") * m1) * "}.")) → text5
  28. LaTeX((((((((((((((("\text{To find the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x=" * v2) * "}\text{, locate the value }\textcolor{#000099}{") * v2) * "}\text{ in the }\textcolor{#000099}{x}\text{ row of the table.}\cr\text{The value }\textcolor{#000099}{") * m2) * "}\text{ directly below }\textcolor{#000099}{x = ") * v2) * "}\text{ is the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x = ") * v2) * "}\text{. That is, }\textcolor{#000099}{f(") * v2) * ") = ") * m2) * "}.")) → text8
  29. LaTeX((("\text{Find the inverse function value }\textcolor{#000099}{f^{-1}(" * v3) * ")}.")) → text10
  30. IndexOf(v4, functionvalues) → p4
  31. Element(xvalues, p4) → m4
  32. TableText({Join[{{LaTeX["\textcolor{#000099}{x}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, 1, p1 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[xvalues, p1]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, p1 + 1, 6]}], Join[{{LaTeX["\textcolor{#000099}{f(x)}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, 1, p1 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[functionvalues, p1]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, p1 + 1, 6]}]}, "c|_") → text6
  33. TableText({Join[{{LaTeX["\textcolor{#000099}{x}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, 1, p2 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[xvalues, p2]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, p2 + 1, 6]}], Join[{{LaTeX["\textcolor{#000099}{f(x)}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, 1, p2 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[functionvalues, p2]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, p2 + 1, 6]}]}, "c|_") → text9
  34. TableText({Join[{{LaTeX["\textcolor{#000099}{x}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, 1, p3 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[xvalues, p3]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, p3 + 1, 6]}], Join[{{LaTeX["\textcolor{#000099}{f(x)}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, 1, p3 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[functionvalues, p3]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, p3 + 1, 6]}]}, "c|_") → text12
  35. LaTeX((("\text{Find the inverse function value }\textcolor{#000099}{f^{-1}(" * v4) * ")}.")) → text13
  36. TableText({Join[{{LaTeX["\textcolor{#000099}{x}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, 1, p4 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[xvalues, p4]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[xvalues, i]) * "}")], i, p4 + 1, 6]}], Join[{{LaTeX["\textcolor{#000099}{f(x)}"]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, 1, p4 - 1], {LaTeX[(("\textcolor{#CC0000}{" * Element[functionvalues, p4]) * "}")]}, Sequence[LaTeX[(("\textcolor{#000099}{" * Element[functionvalues, i]) * "}")], i, p4 + 1, 6]}]}, "c|_") → text15
  37. LaTeX((((((((((((((("\text{To find the value of }\textcolor{#000099}{f^{-1}(" * v4) * ")}\text{, locate the value }\textcolor{#000099}{") * v4) * "}\text{ in the }\textcolor{#000099}{f(x)}\text{ row of the table.}\cr\text{The value }\textcolor{#000099}{") * m4) * "}\text{ directly above }\textcolor{#000099}{f(x) = ") * v4) * "}\text{ is the value of }\textcolor{#000099}{f^{-1}(") * v4) * ")}\text{. That is, }\textcolor{#000099}{f^{-1}(") * v4) * ") = ") * m4) * "}.")) → text14
  38. LaTeX("\text{Summarize your findings.}") → text16
  39. LaTeX((((((((((((((((("\text{As found using the table for }\textcolor{#000099}{f(x)}\text{, }\textcolor{#000099}{f(" * v1) * ")= ") * m1) * "}\text{, }\textcolor{#000099}{f(") * v2) * ") = ") * m2) * "}\text{, }\textcolor{#000099}{f^{–1}(") * v3) * ") = ") * m3) * "}\text{, and }\textcolor{#000099}{f^{–1}(") * v4) * ") = ") * m4) * "}.")) → text17
  40. If(inverseunique ≟ "y" ∧ xunique ≟ "y", stepMax, 4) → trigger
  41. LaTeX("\text{Move the slider to solve the problem.}") → text0
  42. LaTeX((("\textcolor{#000099}{f(x)}\text{ consists of the following values:}\cr\textcolor{#000099}{" * data) * "}")) → Problem
  43. LaTeX((((((((((((((("\text{There are multiple ways to find an inverse function value. One way is to use the existing table.}\cr\text{To find the value of }\textcolor{#000099}{f^{-1}(" * v3) * ")}\text{, locate the value }\textcolor{#000099}{") * v3) * "}\text{ in the }\textcolor{#000099}{f(x)}\text{ row of the table.}\cr\text{The value }\textcolor{#000099}{") * m3) * "}\text{ directly above }\textcolor{#000099}{f(x) = ") * v3) * "}\text{ is the value of }\textcolor{#000099}{f^{-1}(") * v3) * ")}\text{. That is, }\textcolor{#000099}{f^{-1}(") * v3) * ") = ") * m3) * "}.")) → text11

AI Drawing Prompt Reference

Draw a function table showing x-values and corresponding f(x) values, highlighting a specific x and f(x) with a different color. Add text above the table explaining whether the function is invertible and why, such as repeating x or y values. Include text explaining how to find the f(x) value for a specific x, and how to find the inverse function value for a specific f(x). Finally, summarize the found function and inverse function values.
The prompt above was compiled from the original drawing commands and is for reference only.