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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.

Find f(g(x)) for f(x) = 4/x and g(x) = 13/x^2 + 1. Which of the following options is correct?

Solve the function composition problem where f(x) = 4/x and g(x) = 13/x^2 + 1. This problem tests your understanding of function composition and simplifying rational expressions. Step-by-step solution included. Suitable for Grades 10-12.

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