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The function is the result of the composition *f*(*g*(*x*)). If *g*(*x*) = *x*2 + 1, what is *f*(*x*)?

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.

The function is the result of the composition f(g(x)). If g(x) = x^2 + 1, what is f(x)?

Learn how to solve the function composition problem: If g(x) = x^2 + 1 and h(x) = f(g(x)), find f(x). This problem covers key algebraic concepts in function composition and rational functions. Suitable for students in Grades 10-12.

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