We are given the functions:

• $f(x) = \frac{1}{x - 4}$
• $g(x) = \frac{2}{x} + 4$

To solve for $f(g(x))$ and $g(f(x))$:

1. $f(g(x))$:

Substitute $g(x)$ into $f(x)$:

$f(g(x)) = f\left( \frac{2}{x} + 4 \right) = \frac{1}{\left( \frac{2}{x} + 4 \right) - 4}$

Simplifying the expression inside the denominator:

$f(g(x)) = \frac{1}{\frac{2}{x} + 4 - 4} = \frac{1}{\frac{2}{x}} = \frac{1}{\frac{2}{x}} = \frac{x}{2}$

So, $f(g(x)) = \frac{x}{2}$.

2. $g(f(x))$:

Substitute $f(x)$ into $g(x)$:

$g(f(x)) = g\left( \frac{1}{x - 4} \right) = \frac{2}{\frac{1}{x - 4}} + 4$

Simplifying the first part of the expression:

$g(f(x)) = 2(x - 4) + 4 = 2x - 8 + 4 = 2x - 4$

So, $g(f(x)) = 2x - 4$.

Final answers:

• $f(g(x)) = \frac{x}{2}$
• $g(f(x)) = 2x - 4$

Would you like any further details or explanations? Here are some related questions:

1. What is the difference between function composition and regular function evaluation?
2. How do you find the domain of a composite function?
3. Can you compose functions in reverse order? Does the result change?
4. What happens when $f(x)$ or $g(x)$ have restricted domains?
5. How would these function compositions change if we had different expressions for $f(x)$ or $g(x)$?

Tip: Always check the domain of both functions before composing them, as some values might be excluded due to undefined operations like division by zero.