We are asked to find $f(g(x))$ where:

• $f(x) = \frac{4}{x}$
• $g(x) = \frac{13}{x^2} + 1$

To compute $f(g(x))$, we substitute $g(x)$ into $f(x)$.

1. Start with $f(x) = \frac{4}{x}$.
2. Substitute $g(x)$ into $f(x)$: $f(g(x)) = \frac{4}{g(x)} = \frac{4}{\frac{13}{x^2} + 1}$

Now, simplify the expression:

$f(g(x)) = \frac{4}{\frac{13 + x^2}{x^2}} = \frac{4x^2}{13 + x^2}$

Thus, the correct answer is:

$f(g(x)) = \frac{4x^2}{13 + x^2}$

This matches the third option in the provided choices.

Would you like further details on any step?

Here are 5 related questions:

1. How would the solution change if $f(x)$ were a different function, such as $f(x) = \frac{5}{x+1}$?
2. How do you determine the domain of $f(g(x))$?
3. What happens if $g(x)$ is a linear function instead of a rational function?
4. Can you expand the expression $\frac{4x^2}{13 + x^2}$ using partial fractions?
5. How can the composition of functions be applied to real-life scenarios?

Tip: When working with function compositions, always substitute the inner function first and simplify carefully step by step.