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We are given the following functions:

\[
f(x) = \frac{x^2 + 1}{x^2 - 1}, \quad g(x) = \frac{1}{x+1}
\]

We are asked to find:

1. \( g(x+1) \)
2. \( f(g(x)) \)
3. \( g(f(x)) \)

### 1. Finding \( g(x+1) \)

Given \( g(x) = \frac{1}{x+1} \), we substitute \( x+1 \) for \( x \) in \( g(x) \):

\[
g(x+1) = \frac{1}{(x+1) + 1} = \frac{1}{x + 2}
\]

### 2. Finding \( f(g(x)) \)

We need to substitute \( g(x) \) into \( f(x) \). Since \( g(x) = \frac{1}{x+1} \), we compute \( f\left(\frac{1}{x+1}\right) \):

\[
f(g(x)) = f\left(\frac{1}{x+1}\right) = \frac{\left(\frac{1}{x+1}\right)^2 + 1}{\left(\frac{1}{x+1}\right)^2 - 1}
\]

Simplify the numerator and denominator separately:

- Numerator:
\[
\left(\frac{1}{x+1}\right)^2 + 1 = \frac{1}{(x+1)^2} + 1 = \frac{1 + (x+1)^2}{(x+1)^2} = \frac{x^2 + 2x + 2}{(x+1)^2}
\]

- Denominator:
\[
\left(\frac{1}{x+1}\right)^2 - 1 = \frac{1}{(x+1)^2} - 1 = \frac{1 - (x+1)^2}{(x+1)^2} = \frac{1 - (x^2 + 2x + 1)}{(x+1)^2} = \frac{-x^2 - 2x}{(x+1)^2}
\]

Thus, we have:
\[
f(g(x)) = \frac{\frac{x^2 + 2x + 2}{(x+1)^2}}{\frac{-x^2 - 2x}{(x+1)^2}} = \frac{x^2 + 2x + 2}{-x^2 - 2x}
\]

### 3. Finding \( g(f(x)) \)

We need to substitute \( f(x) \) into \( g(x) \). Since \( g(x) = \frac{1}{x+1} \), we compute \( g(f(x)) \):

\[
g(f(x)) = \frac{1}{f(x) + 1} = \frac{1}{\frac{x^2 + 1}{x^2 - 1} + 1}
\]

Simplify the expression inside the denominator:

\[
\frac{x^2 + 1}{x^2 - 1} + 1 = \frac{x^2 + 1 + (x^2 - 1)}{x^2 - 1} = \frac{2x^2}{x^2 - 1}
\]

Thus:
\[
g(f(x)) = \frac{1}{\frac{2x^2}{x^2 - 1}} = \frac{x^2 - 1}{2x^2}
\]

### Final answers:

1. \( g(x+1) = \frac{1}{x+2} \)
2. \( f(g(x)) = \frac{x^2 + 2x + 2}{-x^2 - 2x} \)
3. \( g(f(x)) = \frac{x^2 - 1}{2x^2} \)

Would you like any further details or clarifications?

Here are 5 related questions:
1. How can you determine the domain of the composite function \( f(g(x)) \)?
2. What happens when \( x = -1 \) for the function \( g(x) \)?
3. How do you find the inverse of \( f(x) \)?
4. What are the asymptotes of the functions \( f(x) \) and \( g(x) \)?
5. How do composite functions behave as \( x \to \infty \)?

**Tip:** Always check the domains of the individual functions before combining them, as certain values may cause undefined expressions in the composite functions.

6. f(x) = (x^2 + 1) / (x^2 - 1), g(x) = 1 / (x + 1). Find g(x + 1), f(g(x)), g(f(x)).

Learn how to solve a function composition problem involving rational functions f(x) and g(x). This exercise covers the steps to compute g(x+1), f(g(x)), and g(f(x)), along with simplifying complex rational expressions.

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