https://file.aiquickdraw.com/mathbot/file/1727229856940v47rxdj7.jpg

Let's break down and solve the two parts of the question.

The problem assumes that \( f \) is a one-to-one function, which means it has an inverse function \( f^{-1} \).

### Part (a)
We are given that:
- \( f(2) = 4 \)

We need to find \( f^{-1}(4) \).

By the definition of inverse functions, \( f^{-1}(y) \) is the value of \( x \) such that \( f(x) = y \). In this case, since \( f(2) = 4 \), it means that:
\[
f^{-1}(4) = 2
\]

So, the answer for (a) is:
\[
f^{-1}(4) = 2
\]

### Part (b)
We are given that:
- \( f^{-1}(-4) = -8 \)

We need to find \( f(-8) \).

Since \( f^{-1}(y) = x \) means that \( f(x) = y \), the statement \( f^{-1}(-4) = -8 \) means:
\[
f(-8) = -4
\]

So, the answer for (b) is:
\[
f(-8) = -4
\]

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Do you have any questions or want further details?

Here are five related questions to deepen your understanding:
1. How does the one-to-one property of a function help in defining its inverse?
2. Can all functions have an inverse? Why or why not?
3. What is the graphical relationship between a function and its inverse?
4. How would you verify if a function is one-to-one?
5. If \( f(x) = x^2 \), why doesn't it have an inverse function on its entire domain?

**Tip:** Inverse functions essentially "reverse" the original function's input-output relationship. Understanding how to manipulate and work with inverses is essential for mastering function operations.

Assume that the function f is a one-to-one function. (a) If f(2) = 4, find f^{-1}(4). (b) If f^{-1}(-4) = -8, find f(-8).

This detailed solution addresses inverse function problems involving a one-to-one function. Learn how to find f^{-1}(4) given f(2) = 4, and f(-8) given f^{-1}(-4) = -8. Suitable for students in Grades 9-12.

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