Let's solve each part of the problem step by step.

Part (a)

Given:

• $f(x) = x - 3$
• $g(x) = x + 3$

Find $f(g(x))$:

$f(g(x)) = f(x + 3) = (x + 3) - 3 = x$

Find $g(f(x))$:

$g(f(x)) = g(x - 3) = (x - 3) + 3 = x$

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

Part (b)

Given:

• $f(x) = \frac{1}{2x}$, $x \neq 0$
• $g(x) = -\frac{1}{2x}$, $x \neq 0$

Find $f(g(x))$:

$f(g(x)) = f\left(-\frac{1}{2x}\right) = \frac{1}{2 \cdot \left(-\frac{1}{2x}\right)} = \frac{1}{-\frac{1}{x}} = -x$

Find $g(f(x))$:

$g(f(x)) = g\left(\frac{1}{2x}\right) = -\frac{1}{2 \cdot \left(\frac{1}{2x}\right)} = -\frac{1}{\frac{1}{x}} = -x$

Since neither $f(g(x))$ nor $g(f(x))$ equals $x$, $f$ and $g$ are not inverses of each other.

Final Answers:

• (a): $f(g(x)) = x$ and $g(f(x)) = x$. Therefore, $f$ and $g$ are inverses of each other.
• (b): $f(g(x)) = -x$ and $g(f(x)) = -x$. Therefore, $f$ and $g$ are not inverses of each other.

Do you want further details or have any questions?

Here are some related questions you might find helpful:

1. What are the conditions for two functions to be inverses of each other?
2. Can a function be its own inverse? If yes, provide examples.
3. How do the compositions of functions $f(g(x))$ and $g(f(x))$ differ for non-inverse functions?
4. What is the significance of the domain in determining whether two functions are inverses?
5. How does the behavior of inverse functions relate to their graphs?

Tip: When verifying if two functions are inverses, remember to check both $f(g(x))$ and $g(f(x))$ to ensure they both simplify to $x$.