The given function is $h(x) = \frac{1}{x^2 + 1}$, and we know that this function is the result of the composition $h(x) = f(g(x))$, where $g(x) = x^2 + 1$.

We need to find $f(x)$, such that $h(x) = f(g(x))$.

1. Since $g(x) = x^2 + 1$, we can rewrite the function $h(x)$ as: $h(x) = f(g(x)) = f(x^2 + 1).$

2. We are given that $h(x) = \frac{1}{x^2 + 1}$. So, comparing this with $f(g(x))$, we can conclude that: $f(g(x)) = \frac{1}{g(x)}.$

3. Therefore, the function $f(x)$ is: $f(x) = \frac{1}{x}.$

Thus, $f(x) = \frac{1}{x}$ is the function you're looking for.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do we verify that $h(x) = f(g(x))$ holds true with the given $f(x)$?
2. Can you explain what happens if $g(x)$ is a different function?
3. How would the derivative of $h(x)$ change if $g(x)$ were more complex?
4. What is the domain of the composed function $h(x)$?
5. How would you solve for $f(x)$ if the given $h(x)$ were a different rational function?

Tip: When solving for composed functions, carefully trace each part of the function through substitution to avoid errors.