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?

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.