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Finding Function Values Using Inverse Graphs

Finding Function Values Using Inverse Graphs
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2014-09-13 03:42
VectorFunctionTrigonometryCoordinate GeometryGraphing
In collections UCSS Secondary Math III Unit 4A Lesson 1
This example demonstrates how to find function values and inverse function values by graphing the inverse function. Key math concepts include inverse functions, symmetry of function graphs, and coordinate point correspondence. GeoGebra commands like LaTeX, Corner, If, Vector, and objects such as stepMax, stepper, etc., are used to dynamically display and calculate. Suitable for high school students learning about function properties and inverse functions.

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. If(set ≟ 1, If[0 ≤ x ≤ 6,-6 / (x - 6)], If[set ≟ 2, If[6 ≤ x ≤ 12,-6 / (x - 6)], If[set ≟ 3, If[-6 ≤ x ≤ 0,6 / (x + 6)], If[-12 ≤ x ≤ -6,6 / (x + 6)]]]) → inverse
  6. If(set ≟ 1, -1, If[set ≟ 2, -7, If[set ≟ 3, -1, -7]]) → ymin
  7. LaTeX("\textcolor{#000099}{f^{-1}(x)}") → label
  8. If(set ≟ 1, 1, If[set ≟ 2, -6, If[set ≟ 3, 6, -2]]) → v1
  9. If(set ≟ 1, 2, If[set ≟ 2, -1, If[set ≟ 3, 2, -3]]) → v2
  10. If(set ≟ 1, 3, If[set ≟ 2, 8, If[set ≟ 3, -3, -12]]) → v3
  11. If(set ≟ 1, 4, If[set ≟ 2, 9, If[set ≟ 3, -4, -9]]) → v4
  12. If(set ≟ 1, 0, If[set ≟ 2, 7, If[set ≟ 3, -5, -9]]) → m1
  13. If(set ≟ 1, 3, If[set ≟ 2, 12, If[set ≟ 3, -3, -8]]) → m2
  14. If(set ≟ 1, 2, If[set ≟ 2, -3, If[set ≟ 3, 2, -1]]) → m3
  15. If(set ≟ 1, 3, If[set ≟ 2, 2, If[set ≟ 3, 3, -2]]) → m4
  16. If(set ≟ 1, (0, 1), If[set ≟ 2, (0, -6), If[set ≟ 3, (0, 6), (0, -2)]]) → pstart1
  17. If(set ≟ 1, (0, 2), If[set ≟ 2, (0, -1), If[set ≟ 3, (0, 2), (0, -3)]]) → pstart2
  18. If(set ≟ 1, (3, 0), If[set ≟ 2, (8, 0), If[set ≟ 3, (-3, 0), (-12, 0)]]) → pstart3
  19. If(set ≟ 1, (4, 0), If[set ≟ 2, (9, 0), If[set ≟ 3, (-4, 0), (-9, 0)]]) → pstart4
  20. If(set ≟ 1, (0, 0), If[set ≟ 2, (7, 0), If[set ≟ 3, (-5, 0), (-9, 0)]]) → pend1
  21. If(set ≟ 1, (3, 0), If[set ≟ 2, (12, 0), If[set ≟ 3, (-3, 0), (-8, 0)]]) → pend2
  22. If(set ≟ 1, (0, 2), If[set ≟ 2, (0, -3), If[set ≟ 3, (0, 2), (0, -1)]]) → pend3
  23. If(set ≟ 1, (0, 3), If[set ≟ 2, (0, -2), If[set ≟ 3, (0, 3), (0, -2)]]) → pend4
  24. If(set ≟ 1, (0, 1), If[set ≟ 2, (7, -6), If[set ≟ 3, (-5, 6), (-9, -2)]]) → p1
  25. If(set ≟ 1, (3, 2), If[set ≟ 2, (12, -1), If[set ≟ 3, (-3, 2), (-8, -3)]]) → p2
  26. If(set ≟ 1, (3, 2), If[set ≟ 2, (8, -3), If[set ≟ 3, (-3, 2), (-12, -1)]]) → p3
  27. If(set ≟ 1, (4, 3), If[set ≟ 2, (9, -2), If[set ≟ 3, (-4, 3), (-9, -2)]]) → p4
  28. Vector(pstart1, p1) → trace1a
  29. Vector(pstart2, p2) → trace2a
  30. Vector(pstart3, p3) → trace3a
  31. Vector(pstart4, p4) → trace4a
  32. Vector(p1, pend1) → trace1b
  33. Vector(p2, pend2) → trace2b
  34. Vector(p3, pend3) → trace3b
  35. Vector(p4, pend4) → trace4b
  36. LaTeX("\text{Move left or right to the graph of }\textcolor{#000099}{f^{-1}(x)}.") → text5
  37. If(set ≟ 1, 0, If[set ≟ 2, 7, If[set ≟ 3, -5, -9]]) → m1_1
  38. LaTeX("\text{Move left or right to the graph of }\textcolor{#000099}{f^{-1}(x)}.") → text11
  39. If(set ≟ 1, -1, If[set ≟ 2, -1, If[set ≟ 3, -7, -13]]) → xmin
  40. If(set ≟ 1, 7, If[set ≟ 2, 13, If[set ≟ 3, 1, 1]]) → xmax
  41. If(set ≟ 1, (6, inverse(6)), If[set ≟ 2, (7, inverse(7)), If[set ≟ 3, ((-5), inverse((-5))), (-9, -2)]]) → C
  42. If(set ≟ 1, 6, If[set ≟ 2, 1, If[set ≟ 3, 7, 1]]) → ymax
  43. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  44. LaTeX((("\text{Find the function value }\textcolor{#000099}{f(" * v1) * ")}\text{ from the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}.")) → text1
  45. LaTeX((((((((("\text{Use the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{ to find }\textcolor{#000099}{f(" * v1) * ")}\text{, }\textcolor{#000099}{f(") * v2) * ")}\text{, }\textcolor{#000099}{f^{–1}(") * v3) * ")}\text{, and }\textcolor{#000099}{f^{–1}(") * v4) * ")}.")) → Problem
  46. LaTeX((((("\text{To find the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x = " * v1) * "}\text{ from the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)},\cr\text{locate the value }\textcolor{#000099}{") * v1) * "}\text{ on the }\textcolor{#000099}{y}\text{-axis of the graph.}")) → text4
  47. LaTeX("\text{Now move up or down from the point on the graph to the }\textcolor{#000099}{x}\text{-axis.}") → text6
  48. LaTeX((((("\text{The }\textcolor{#000099}{x}\text{-coordinate, }\textcolor{#000099}{" * m1) * "}\text{, where this vertical line hits the }\textcolor{#000099}{x}\text{-axis represents the output }\cr\text{value of }\textcolor{#000099}{f(x)}\text{ at an input value of }\textcolor{#000099}{x=") * v1) * "}.")) → text7
  49. LaTeX((("\text{Find the function value }\textcolor{#000099}{f(" * v2) * ")}\text{ from the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}.")) → text9
  50. LaTeX((((("\text{To find the value of }\textcolor{#000099}{f(x)}\text{ at }\textcolor{#000099}{x = " * v2) * "}\text{ from the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)},\cr\text{locate the value }\textcolor{#000099}{") * v2) * "}\text{ on the }\textcolor{#000099}{y}\text{-axis of the graph.}")) → text10
  51. LaTeX("\text{Now move up or down from the point on the graph to the }\textcolor{#000099}{x}\text{-axis.}") → text12
  52. LaTeX((((("\text{The }\textcolor{#000099}{x}\text{-coordinate, }\textcolor{#000099}{" * m2) * "}\text{, where this vertical line hits the }\textcolor{#000099}{x}\text{-axis represents the output }\cr\text{value of }\textcolor{#000099}{f(x)}\text{ at an input value of }\textcolor{#000099}{x=") * v2) * "}.")) → text13
  53. LaTeX((("\text{Find the inverse function value }\textcolor{#000099}{f^{-1}(" * v3) * ")}\text{ from the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}.")) → text15
  54. LaTeX("\text{Remember that this is a graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{, and not a graph of the }\cr\text{function }\textcolor{#000099}{f(x)}\text{. That means the values of the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{ are read from the }\cr\text{graph in the usual way, with the }\textcolor{#000099}{x}\text{-axis as the input.}") → text16
  55. LaTeX((((("\text{To find the value of }\textcolor{#000099}{f^{–1}(x)}\text{ at }\textcolor{#000099}{x = " * v3) * "}\text{ from the graph, locate }\textcolor{#000099}{") * v3) * "}\text{ on the }\textcolor{#000099}{x}\text{-axis.}")) → text17
  56. LaTeX((("\text{Find the inverse function value }\textcolor{#000099}{f^{-1}(" * v4) * ")}\text{ from the graph of the {inverse function }\textcolor{#000099}{f^{–1}(x)}.")) → text20
  57. LaTeX((((("\text{To find the value of }\textcolor{#000099}{f^{–1}(x)}\text{ at }\textcolor{#000099}{x = " * v4) * "}\text{ from the graph, locate }\textcolor{#000099}{") * v4) * "}\text{ on the }\textcolor{#000099}{x}\text{-axis.}")) → text21
  58. LaTeX("\text{Notice that this is a graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{, and not a graph of the}\cr\text{function }\textcolor{#000099}{f(x)}\text{. Also remember that the inverse of }\textcolor{#000099}{f^{–1}(x)}\text{ is the original function }\textcolor{#000099}{f(x)}.") → text2
  59. LaTeX("\text{Because the inverse relationship between }\textcolor{#000099}{f(x)}\text{ and }\textcolor{#000099}{f^{–1}(x)}\text{ goes both ways, you can use}\cr\text{the graph of the inverse function }\textcolor{#000099}{f^{–1}(x)}\text{ to find values of the original function }\textcolor{#000099}{f(x)}.") → text3
  60. LaTeX((((("\text{Therefore, }\textcolor{#000099}{f(" * v1) * ")=") * m1) * "}.")) → text8
  61. LaTeX((((("\text{That is, }\textcolor{#000099}{f(" * v2) * ")=") * m2) * "}.")) → text14
  62. LaTeX((((((("\text{Then find the point on the function directly above or below }\textcolor{#000099}{x = " * v3) * "}\text{;}\cr\text{in this case, }\textcolor{#000099}{(") * v3) * ", ") * m3) * ")}.")) → text18
  63. LaTeX((((((((("\text{The }\textcolor{#000099}{y}\text{-coordinate, }\textcolor{#000099}{" * m3) * "}\text{, of this point is the value of }\textcolor{#000099}{f^{–1}(x)}\text{ at }\textcolor{#000099}{x = ") * v3) * "}\text{.}\cr\text{Therefore, }\textcolor{#000099}{f^{–1}(") * v3) * ") = ") * m3) * "}.")) → text19
  64. LaTeX((((((("\text{Then find the point on the function directly above or below }\textcolor{#000099}{x = " * v4) * "}\text{;}\cr\text{in this case, }\textcolor{#000099}{(") * v4) * ", ") * m4) * ")}.")) → text22
  65. LaTeX((((((((("\text{The }\textcolor{#000099}{y}\text{-coordinate, }\textcolor{#000099}{" * m4) * "}\text{, of this point is the value of }\textcolor{#000099}{f^{–1}(x)}\text{ at }\textcolor{#000099}{x = ") * v4) * "}\text{.}\cr\text{Therefore, }\textcolor{#000099}{f^{–1}(") * v4) * ") = ") * m4) * "}.")) → text23

AI Drawing Prompt Reference

Draw a function and its inverse function to demonstrate how to find function values using the inverse graph. Key objects include the function graph, inverse function graph, coordinate axes, and several key points. First, plot the function graph, then the inverse function graph, and label the inverse function. Next, use several key points to show how to find corresponding function values from the inverse function graph, including intersections with the coordinate axes and the position of vertical lines. Finally, dynamically display a slider and text explanations to clarify how to find function values using the inverse function graph.
The prompt above was compiled from the original drawing commands and is for reference only.