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In this question, you're given a function \( h(x) = \frac{2}{(-9x + 8)^8} \) and asked to decompose it into two component functions \( f(x) \) and \( g(x) \) such that \( h(x) = f(g(x)) \).

### Step-by-Step Decomposition:

1. **Look at the structure of \( h(x) \):**
   - The denominator is raised to the power of 8, which suggests that one of the functions is responsible for this power.

2. **Choose the inside function \( g(x) \):**
   - A natural choice for \( g(x) \) would be the expression inside the parentheses, because that's the part being manipulated before raising it to a power. So let’s define:
     \[
     g(x) = -9x + 8
     \]

3. **Determine the outside function \( f(x) \):**
   - Now that we’ve expressed the inner part as \( g(x) \), the remaining operation is raising \( g(x) \) to the 8th power and putting it in the denominator. Therefore, the outer function \( f(x) \) should be:
     \[
     f(x) = \frac{2}{x^8}
     \]

### Final Decomposition:
- \( f(x) = \frac{2}{x^8} \)
- \( g(x) = -9x + 8 \)

So the function \( h(x) \) can be written as \( h(x) = f(g(x)) \).

Would you like more details or have any questions about the steps? Here are 5 related questions that can expand on this concept:

1. What is the process for finding composite functions in general?
2. How do you verify that your decomposition of \( h(x) \) into \( f(x) \) and \( g(x) \) is correct?
3. What types of functions can be easily decomposed into simpler functions?
4. How would you approach the reverse process of combining two functions into a single one?
5. What are the practical applications of function composition in calculus?

**Tip:** When decomposing functions, try to look for nested operations like powers, fractions, or other transformations to identify inner and outer functions!

Given the function h(x) = 2 / (-9x + 8)^8, decompose into component functions f(x) and g(x) so that h(x) = f(g(x)).

Decompose the function h(x) = 2 / (-9x + 8)^8 into f(x) and g(x) for composition. This problem involves function composition and algebraic manipulation, making it suitable for students in Grades 11-12 or college-level precalculus.

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