To solve this problem, we need to express the function $h(x) = \frac{1}{8x - 8}$ as a composition of two functions $f(x)$ and $g(x)$ such that:

$h(x) = (f \circ g)(x) = f(g(x))$

This means we need to find two functions $f$ and $g$ where:

1. $g(x)$ is some function that transforms $x$ into an intermediate expression.
2. $f(x)$ takes the result of $g(x)$ and produces the final expression $h(x) = \frac{1}{8x - 8}$.

Analyzing the Function Composition

Given $h(x) = \frac{1}{8x - 8}$, let's identify possible forms of $f(x)$ and $g(x)$:

1. Consider the expression inside the denominator: $8x - 8$. Let's try setting $g(x) = 8x - 8$. Then, $g(x)$ simplifies the denominator of $h(x)$ directly.

2. With $g(x) = 8x - 8$, we have $g(x)$ producing the expression $8x - 8$. For $h(x) = \frac{1}{8x - 8}$, we want:

   $f(g(x)) = f(8x - 8) = \frac{1}{8x - 8}$

   This suggests $f(u) = \frac{1}{u}$ where $u = g(x)$.

Identifying the Correct Answer

Based on the options provided, we find:

• $f(x) = \frac{1}{x}$, $g(x) = 8x - 8$ matches our decomposition.

Conclusion

The correct answer is:

$\boxed{f(x) = \frac{1}{x}, \, g(x) = 8x - 8}$

This corresponds to the second option in the provided choices.

Would you like more details or have any questions? Here are some related questions to explore:

1. How does function composition work mathematically?
2. What are other examples of functions that can be decomposed into compositions?
3. Why does selecting the correct $g(x)$ simplify finding $f(x)$?
4. How can we apply function composition to solve real-world problems?
5. What are common patterns for identifying function compositions?

Tip: When decomposing functions, focus on finding an inner function that simplifies the structure of the outer function.