We are given that $h(x) = (f \circ g)(x) = f(g(x))$, where $h(x) = \frac{3}{(x - 8)^7}$. We need to find functions $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$.

Step-by-step approach:

1. Examine $h(x)$: $h(x) = \frac{3}{(x - 8)^7}$ It looks like the expression inside the denominator can be split into two functions. The $(x - 8)$ part suggests one function, while raising to the power of 7 and dividing by 3 suggests another.

2. Identify $g(x)$: Since $h(x)$ involves $(x - 8)$, we can try setting $g(x) = x - 8$. This would simplify $h(x)$ to a form that depends on $f(x)$.

3. Identify $f(x)$: Now, substitute $g(x) = x - 8$ into $h(x)$: $h(x) = \frac{3}{(g(x))^7}$ Therefore, we can set $f(x) = \frac{3}{x^7}$.

So, we have:

• $f(x) = \frac{3}{x^7}$
• $g(x) = x - 8$

Now, let's match this with the options:

• A. $f(x) = \frac{3}{7x}$ and $g(x) = x - 8$: Incorrect, since the form of $f(x)$ doesn't match.
• B. $f(x) = \frac{7}{x}$ and $g(x) = x - 8$: Incorrect, as the power and constant don't match.
• C. $f(x) = \frac{3}{x}$ and $g(x) = (x - 8)^7$: Incorrect, since $g(x)$ is not raised to the seventh power before applying $f$.
• D. $f(x) = \frac{7}{x^3}$ and $g(x) = (x - 8)^7$: Incorrect as the powers and constants are not correct.

Correct Answer:

None of the provided options match the correct solution, which is $f(x) = \frac{3}{x^7}$ and $g(x) = x - 8$.

Do you want further explanation or have any other questions?

Related questions:

1. How does function composition work in general?
2. How can you identify potential $f(x)$ and $g(x)$ in composite functions?
3. Why is $g(x) = x - 8$ the correct choice here?
4. How does the order of operations impact function composition?
5. What are other ways to decompose complex functions?

Tip:

When working with function composition, try simplifying the outer function first and look for common patterns like shifts or powers to identify the inner function.