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To determine if two functions \( f \) and \( g \) are inverses of each other, we need to check if:

1. \( f(g(x)) = x \)
2. \( g(f(x)) = x \)

If both of these conditions hold, then \( f \) and \( g \) are inverses of each other. Let's evaluate each pair given in the image.

### Part (a)

Given:
- \( f(x) = \frac{x}{3} \)
- \( g(x) = 3x \)

**Step 1: Find \( f(g(x)) \)**

Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(3x) = \frac{3x}{3} = x
\]

**Step 2: Find \( g(f(x)) \)**

Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{x}{3}\right) = 3 \cdot \frac{x}{3} = x
\]

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

### Part (b)

Given:
- \( f(x) = \frac{x - 1}{2} \)
- \( g(x) = 2x + 1 \)

**Step 1: Find \( f(g(x)) \)**

Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(2x + 1) = \frac{(2x + 1) - 1}{2} = \frac{2x}{2} = x
\]

**Step 2: Find \( g(f(x)) \)**

Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{x - 1}{2}\right) = 2 \cdot \frac{x - 1}{2} + 1 = (x - 1) + 1 = x
\]

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

### Final Answers:

- **(a)** \( f \) and \( g \) are inverses of each other.
- **(b)** \( f \) and \( g \) are inverses of each other.

Would you like more details on any part of the solution?

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### Follow-Up Questions

1. How can we determine if a function is invertible?
2. What is the general process for finding the inverse of a function?
3. Can all functions have inverses?
4. How does the composition of functions relate to verifying inverses?
5. How would the answer change if one of the compositions didn't equal \( x \)?

### Tip

When checking if two functions are inverses, always confirm that both \( f(g(x)) \) and \( g(f(x)) \) return \( x \); this ensures that the inverse relationship holds in both directions.

For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other.

Learn how to determine if two functions are inverses of each other by using composition. This solution explains the steps to find f(g(x)) and g(f(x)) and shows how to verify if both compositions equal x, confirming the inverse relationship. Ideal for high school students in Grades 10-12 studying algebra and functions.

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