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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.

Let f(x) = 1/(x - 4) and g(x) = 2/x + 4. Find the following functions: f(g(x)) and g(f(x)). Simplify your answers.

Learn how to find and simplify the composite functions f(g(x)) and g(f(x)) for f(x) = 1/(x - 4) and g(x) = 2/x + 4. This problem involves key concepts in function composition and algebra. Suitable for students in Grades 10-12.

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