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The problem involves finding the domains of the given functions. Let's go through each part of the problem.

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

### (a) Domain of \( f \)

For \( f(x) = \frac{x}{x^2 - 1} \), the function is undefined where the denominator is zero. Therefore:
\[
x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1
\]
The domain of \( f \) is all real numbers except \( x = 1 \) and \( x = -1 \). In interval notation:
\[
\text{Domain of } f = (-\infty, -1) \cup (-1, 1) \cup (1, \infty)
\]

### (b) Domain of \( g \)

For \( g(x) = x + 1 \), this is a linear function, which is defined for all real numbers. So:
\[
\text{Domain of } g = (-\infty, \infty)
\]

### (c) Domain of \( f \circ g \) (Composition of \( f \) and \( g \))

To find the domain of \( f \circ g \), we need to evaluate \( f(g(x)) \). The function \( f(g(x)) = \frac{g(x)}{(g(x))^2 - 1} = \frac{x + 1}{(x + 1)^2 - 1} \). 

We need to find when the denominator is zero:
\[
(x + 1)^2 - 1 = 0
\]
\[
(x + 1 - 1)(x + 1 + 1) = 0
\]
\[
x(x + 2) = 0
\]
This results in \( x = 0 \) or \( x = -2 \).

So, the domain of \( f \circ g \) is all real numbers except \( x = 0 \) and \( x = -2 \). In interval notation:
\[
\text{Domain of } f \circ g = (-\infty, -2) \cup (-2, 0) \cup (0, \infty)
\]

### Summary of Answers
- (a) Domain of \( f \): \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \)
- (b) Domain of \( g \): \( (-\infty, \infty) \)
- (c) Domain of \( f \circ g \): \( (-\infty, -2) \cup (-2, 0) \cup (0, \infty) \)

Would you like further details or have any questions?

Here are 5 related questions that expand on the above:
1. What is the range of \( f(x) \)?
2. How would the domain change if \( g(x) = x^2 + 1 \)?
3. Can you find the domain of \( g \circ f \) instead of \( f \circ g \)?
4. How does adding a constant to the numerator of \( f(x) \) affect its domain?
5. If \( g(x) \) were a trigonometric function, how would you approach finding the domain of \( f \circ g \)?

**Tip:** Always start by finding points where the denominator is zero when working with rational functions to determine the domain.

Determine the domains of the following functions: f(x) = x/(x^2 - 1) and g(x) = x + 1. Find the domains of f, g, and the composition f(g).

This problem involves finding the domains of two functions, f(x) = x/(x^2 - 1) and g(x) = x + 1, along with their composition. Learn how to calculate the domain by analyzing the functions and their restrictions.

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