To determine whether $f$ and $g$ are inverses of each other, we need to compute the compositions $f(g(x))$ and $g(f(x))$ for each pair of functions. If both compositions yield $x$, then $f$ and $g$ are inverses.

Let's go through each part.

Part (a)

1. Given Functions:

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

2. Find $f(g(x))$: $f(g(x)) = f(x + 3) = (x + 3) - 3 = x$

3. 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.

Answer for (a): $f$ and $g$ are inverses of each other.

Part (b)

1. Given Functions:

   • $f(x) = \frac{1}{4x}$, $x \neq 0$
   • $g(x) = \frac{1}{4x}$, $x \neq 0$

2. Find $f(g(x))$: $f(g(x)) = f\left(\frac{1}{4x}\right) = \frac{1}{4 \cdot \frac{1}{4x}} = \frac{1}{\frac{4}{4x}} = \frac{1}{\frac{1}{x}} = x$

3. Find $g(f(x))$: $g(f(x)) = g\left(\frac{1}{4x}\right) = \frac{1}{4 \cdot \frac{1}{4x}} = x$

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

Answer for (b): $f$ and $g$ are inverses of each other.

Summary of Results

Both pairs of functions, (a) and (b), are inverses of each other as both satisfy $f(g(x)) = x$ and $g(f(x)) = x$.

Would you like further clarification on any step?

Here are 5 related questions you might explore:

1. How would the process change if $f$ and $g$ were not one-to-one functions?
2. What are the properties of inverse functions in terms of their graphs?
3. How can inverse functions be found for functions that are not linear?
4. What role does the domain and range play in determining if two functions are inverses?
5. How do we verify inverses for functions with more complex compositions?

Tip: Remember that for two functions to be inverses, both compositions $f(g(x))$ and $g(f(x))$ must simplify to $x$.