To solve the problem:

We are given $f(3) = 9$, and we need to find $f^{-1}(9)$.

Explanation:

• $f(3) = 9$ means that the function $f$ maps $3$ to $9$.
• By definition, the inverse function $f^{-1}$ reverses this mapping. So, $f^{-1}(9)$ will give the value $3$, since $f(3) = 9$.

Final Answer:

$f^{-1}(9) = 3$

Would you like a more detailed explanation or further examples?

Here are 5 questions to expand on this topic:

1. How do we verify that a function has an inverse?
2. What are the conditions for a function to be one-to-one?
3. How do you graphically interpret $f(x)$ and $f^{-1}(x)$?
4. Can all functions have inverses? Why or why not?
5. How do we find the inverse of a function algebraically?

Tip: Always check if the given function is one-to-one (passes the horizontal line test) before determining its inverse!