To express the function $h(x) = \frac{1}{x+2}$ in the form $h(x) = f(g(x))$, given that $g(x) = x + 2$, we follow these steps:

Step-by-Step Breakdown:

1. Understand the function form: The function $h(x)$ is to be expressed as a composition of two functions, $f(x)$ and $g(x)$, where: $h(x) = f(g(x))$ and $g(x) = x + 2$.

2. Substitute $g(x)$ into $h(x)$: Since $g(x) = x + 2$, the expression $h(x) = \frac{1}{x+2}$ can be written as: $h(x) = \frac{1}{g(x)}$

3. Identify the function $f(x)$: From the equation above, we see that $f(g(x)) = \frac{1}{g(x)}$. Therefore, the function $f(x)$ is: $f(x) = \frac{1}{x}$

Conclusion:

Thus, the function $f(x) = \frac{1}{x}$.

Would you like to go over more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the domain of the function $h(x) = \frac{1}{x+2}$?
2. How would you find the inverse of the function $f(x) = \frac{1}{x}$?
3. What is the composition of $f(g(x))$ if $g(x) = 2x$?
4. How do you differentiate $h(x) = \frac{1}{x+2}$?
5. What transformations does the function $f(x) = \frac{1}{x}$ undergo to become $h(x) = \frac{1}{x+2}$?

Tip: When composing functions, always check if any domain restrictions carry over from the inner function to the overall composition.