Let's go over each part of the problem in the image.

Problem (a)

The functions are:

• $f(x) = -\frac{4}{x}$, where $x \neq 0$
• $g(x) = -\frac{4}{x}$, where $x \neq 0$

You are asked to find $f(g(x))$ and $g(f(x))$, and determine whether $f$ and $g$ are inverses.

Step 1: Find $f(g(x))$

Substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(-\frac{4}{x}\right) = -\frac{4}{\left(-\frac{4}{x}\right)} = x$

Step 2: Find $g(f(x))$

Substitute $f(x)$ into $g(x)$: $g(f(x)) = g\left(-\frac{4}{x}\right) = -\frac{4}{\left(-\frac{4}{x}\right)} = x$

Since both $f(g(x)) = x$ and $g(f(x)) = x$, the functions $f$ and $g$ are inverses of each other.

Problem (b)

The functions are:

• $f(x) = -x + 3$
• $g(x) = x + 3$

Step 1: Find $f(g(x))$

Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x + 3) = -(x + 3) + 3 = -x - 3 + 3 = -x$

Step 2: Find $g(f(x))$

Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(-x + 3) = (-x + 3) + 3 = -x + 6$

Since $f(g(x)) = -x$ and $g(f(x)) = -x + 6$, the functions are not inverses of each other.

Would you like further details on any part of this? Here are 5 related questions to explore further:

1. How do you determine if two functions are inverses using their compositions?
2. What is the domain restriction for inverse functions, and why is it important?
3. Can a function have more than one inverse?
4. How does the algebraic process of simplifying compositions reveal inverse relationships?
5. What are common pitfalls in determining if two functions are inverses?

Tip: Inverse functions always "undo" each other, meaning their compositions $f(g(x))$ and $g(f(x))$ must equal $x$.