For any relation, the graph of the relation's inverse can be formed by reflecting the graph of the relation about the line [math]y=x[/math]. [br][br]Recall that all functions are relations, but not all relations are functions. (What causes a relation to be a function?)[br][br]In this interactive figure, you can enter any function [i]f[/i] and restrict its natural domain, if you choose, to [math]x[/math]-values between -10 and 10. You also have the option to graph the function over its natural domain.

[color=#000000][b]Exploration: [br][br][/b]1) Choose the [b]"Graph f on its natural domain"[/b] option.[br]2) Enter in the [/color][color=#980000][b]function [math]f\left(x\right)=x^2[/math][/b][/color][color=#000000] [br]3) Choose [/color][color=#38761d][b]"Show Inverse Relation"[/b].[br][/color][color=#000000]4) Is the [/color][color=#38761d][b]graph of this inverse relation[/b][/color][color=#000000] the graph of a function? Explain why or why not. [br]5) If your answer to (4) above was "no," uncheck the [b]"Default to Natural Domain of f"[/b] checkbox.[br]6) Now, can you come up with a set of [i]Xmin [/i]and [i]Xmax [/i]values so that the function shown has an inverse [br] that is a function? Explain. [br][br]7) What is the domain of [i]f[/i]? [br]8) What is the range of [i]f[/i]? [br]9) What is the domain of [math]f^{-1}[/math]? [br]10) What is the range of [math]f^{-1}[/math]? [br][br]11) Do you notice anything interesting about any set of answers for (8) - (11)? If so, explain. [br][br]Repeat steps (1) - (11) again, this time for different functions [i]f[/i] (for step 2) provided to you by your instructor. [br][br][i]Developed for use with Thomas' Calculus, published by Pearson.[/i][/color]