https://file.aiquickdraw.com/mathbot/file/1726545091203z0r1j31w.jpg

Let's go through the questions one by one:

### (a) If \( f(2) = 7 \), find \( f^{-1}(7) \).

For a one-to-one function, the inverse function \( f^{-1} \) "reverses" the operation of the original function. Specifically:
- If \( f(a) = b \), then \( f^{-1}(b) = a \).

Given:
- \( f(2) = 7 \), this implies that \( f^{-1}(7) = 2 \).

So, the answer to (a) is: 
\[
f^{-1}(7) = 2
\]

### (b) If \( f^{-1}(-7) = -6 \), find \( f(-6) \).

From the inverse property, we know that:
- If \( f^{-1}(b) = a \), then \( f(a) = b \).

Given:
- \( f^{-1}(-7) = -6 \), this implies that \( f(-6) = -7 \).

So, the answer to (b) is: 
\[
f(-6) = -7
\]

Let me know if you would like more details or have any questions.

### Related questions:
1. What is the relationship between a function and its inverse?
2. How do you verify if a function is one-to-one?
3. Can a function have an inverse if it's not one-to-one?
4. What is the graphical interpretation of the inverse of a function?
5. How do you compute the inverse of a function algebraically?

### Tip:
When dealing with inverse functions, remember that the input and output are reversed. If \( f(a) = b \), then \( f^{-1}(b) = a \).

Assume that the function f is a one-to-one function.
(a) If f(2) = 7, find f^(-1)(7).
(b) If f^(-1)(-7) = -6, find f(-6).

Learn how to find the inverse of a function and use the inverse function theorem. This example covers one-to-one functions and shows how to compute f^(-1)(7) and f(-6) using function properties. Suitable for high school students.

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