Let's work through each part of the problem step-by-step.

Part (a)

We have:

• $f(x) = -3x$
• $g(x) = -\frac{x}{3}$

To determine if $f$ and $g$ are inverses, we need to check if $f(g(x)) = x$ and $g(f(x)) = x$.

1. Calculate $f(g(x))$: $f(g(x)) = f\left(-\frac{x}{3}\right) = -3 \left(-\frac{x}{3}\right) = x$

2. Calculate $g(f(x))$: $g(f(x)) = g(-3x) = -\frac{-3x}{3} = x$

Since both $f(g(x)) = x$ and $g(f(x)) = x$, we conclude that $f$ and $g$ are inverses of each other for part (a).

Part (b)

We have:

• $f(x) = 2x + 5$
• $g(x) = \frac{x - 5}{2}$

Again, we need to check if $f(g(x)) = x$ and $g(f(x)) = x$.

1. Calculate $f(g(x))$: $f(g(x)) = f\left(\frac{x - 5}{2}\right) = 2\left(\frac{x - 5}{2}\right) + 5 = (x - 5) + 5 = x$

2. Calculate $g(f(x))$: $g(f(x)) = g(2x + 5) = \frac{(2x + 5) - 5}{2} = \frac{2x}{2} = x$

Since both $f(g(x)) = x$ and $g(f(x)) = x$, we conclude that $f$ and $g$ are inverses of each other for part (b) as well.

Summary of Results

• Part (a): $f$ and $g$ are inverses of each other.
• Part (b): $f$ and $g$ are inverses of each other.

Would you like further details on any of these calculations?

Here are 5 related questions to expand on this topic:

1. What are the general conditions for two functions to be inverses of each other?
2. How can you find the inverse of a function algebraically?
3. What role does the identity function $f(g(x)) = x$ play in verifying inverse functions?
4. How do domain and range affect whether two functions are inverses?
5. Can every function have an inverse? If not, what conditions must be met?

Tip: When calculating compositions, always simplify step-by-step to avoid algebraic mistakes.