https://file.aiquickdraw.com/mathbot/file/1725988829770cypobchb.jpg

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 \( x \).

For each pair of functions f and g, find f(g(x)) and g(f(x)). Then determine whether f and g are inverses of each other. Simplify your answers as much as possible.

Learn how to determine if two functions are inverses by finding f(g(x)) and g(f(x)). This problem involves key concepts of function composition and inverse functions, suitable for high school math students in Grades 9-12.

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