The graph of any function and the graph of its inverse are symmetric with respect to the

A. y-axis
B. x-axis
C. graph of the equation y = x
D. graph of the equation y = -x

The graph of a function and the graph of its inverse are symmetric with respect to the:

C. graph of the equation \( y = x \)

This symmetry occurs because the inverse of a function \( f \), denoted as \( f^{-1} \), essentially swaps the roles of \( x \) and \( y \) in the function \( f \). Graphically, this means reflecting the graph of \( f \) across the line \( y = x \).

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Here are 8 related questions to expand your understanding:

1. How do you determine if a function has an inverse?
2. What are the steps to find the inverse of a function algebraically?
3. Can all functions have inverses? Why or why not?
4. How do you verify that two functions are inverses of each other?
5. What is the relationship between the domains and ranges of a function and its inverse?
6. How does the derivative of a function relate to the derivative of its inverse?
7. Can the inverse of a quadratic function be a function? Explain.
8. What is the graphical interpretation of the inverse of a function that is not one-to-one?

**Tip:** When verifying if two functions are inverses, compose them (i.e., \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \)) and check if both compositions yield \( x \).

Explore why the graphs of functions and their inverses are symmetric with respect to the line y = x. Learn how this symmetry is depicted graphically and algebraically. Suitable for high school students studying functions and their inverses.

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