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The problem asks to express the function \( h(x) = \frac{1}{x + 8} \) in the form \( f(g(x)) \), where \( g(x) = x + 8 \).

Here’s how we can break it down:

1. We are given that \( g(x) = x + 8 \). Therefore, we want to find a function \( f(x) \) such that \( f(g(x)) = h(x) = \frac{1}{x + 8} \).

2. Since \( g(x) = x + 8 \), the expression \( h(x) = \frac{1}{x + 8} \) becomes \( f(g(x)) = f(x + 8) \).

3. Therefore, \( f(u) \), where \( u = g(x) \), must be \( f(u) = \frac{1}{u} \).

Thus, the function \( f(x) \) is \( f(x) = \frac{1}{x} \).

To summarize:
- \( g(x) = x + 8 \)
- \( f(x) = \frac{1}{x} \)

Would you like further clarification on this? Let me know if you have more questions. 

Here are five follow-up questions that might help extend your understanding:
1. What would \( f(g(x)) \) look like if \( g(x) = x + 5 \) instead of \( x + 8 \)?
2. How would the composition of functions change if \( f(x) = \frac{1}{x^2} \)?
3. Can you express \( h(x) = \frac{1}{x - 3} \) in the form \( f(g(x)) \) and determine \( f(x) \) and \( g(x) \)?
4. How does function composition work when one function is a quadratic and the other is a linear function?
5. Can you generalize the process for any rational function in the form \( h(x) = \frac{1}{x + a} \)?

**Tip:** In function composition, always check the inner and outer functions carefully, as they determine the transformation applied to \( x \) at each step.

The function h(x) = 1 / (x + 8) can be expressed in the form f(g(x)) where g(x) = (x + 8) and f(x) is defined as f(x) = 1 / x.

This problem focuses on expressing the function h(x) = 1/(x+8) in the form f(g(x)), where g(x) = x+8 and f(x) = 1/x. Learn how to decompose functions using the function composition theorem, suitable for high school students (Grades 9-12).

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