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.