To express the function $h(x) = \frac{1}{x+9}$ in the form $f(g(x))$, where $g(x) = x+9$, we need to determine the function $f(x)$.

We know that $g(x) = x + 9$, so when we substitute this into $h(x)$, we get:

$h(x) = \frac{1}{g(x)}$

This means that $f(x) = \frac{1}{x}$, since we are applying the function $f$ to $g(x)$ to get $h(x)$.

Therefore, $f(x) = \frac{1}{x}$.

Summary:

$h(x) = f(g(x)) = \frac{1}{g(x)}, \text{ where } f(x) = \frac{1}{x} \text{ and } g(x) = x + 9$

Do you want more details, or have any questions?

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Here are 5 related questions to expand this concept:

1. How would the composition change if $g(x) = x - 9$?
2. What happens if $f(x) = x^2$? How does that affect $h(x)$?
3. Can the composition $f(g(x))$ be used to represent more complex functions?
4. How do you decompose a function into compositions of two functions in general?
5. What is the derivative of $h(x)$ in terms of $f(x)$ and $g(x)$?

Tip: Function composition can simplify the analysis of more complex expressions, breaking them into smaller, more manageable parts!