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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.

Express the given function h as a composition of two functions f and g so that h(x) = (f ∘ g)(x). h(x) = 1/(8x-8)

Learn how to express a rational function as a composition of two functions, f and g. This example, where h(x) = 1/(8x-8), demonstrates the process of decomposing a function. Suitable for students in Grades 9-11.

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