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 $f^{-1}(b) = a$.