To find the domain of the function $r(x) = \frac{2x}{x^2 - 1}$, we need to determine the values of $x$ for which the function is defined. The function is undefined when the denominator is zero.

Step-by-step:

1. Set the denominator equal to zero: $x^2 - 1 = 0$

2. Solve for $x$: $x^2 = 1$ $x = \pm 1$

Conclusion:

The function $r(x)$ is undefined at $x = 1$ and $x = -1$. Thus, the domain of $r(x)$ is all real numbers except $x = 1$ and $x = -1$.

Domain:

$\text{Domain of } r(x): \quad x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty)$

Would you like further clarification or more examples related to finding the domain of functions? Here are some related questions for practice:

1. What is the domain of $f(x) = \frac{3x + 2}{x - 4}$?
2. Find the domain of $g(x) = \sqrt{x^2 - 4}$.
3. What is the domain of $h(x) = \frac{5x}{x^2 + x - 6}$?
4. Determine the domain of $p(x) = \frac{1}{x^2 + 3x + 2}$.
5. Find the domain of $q(x) = \ln(x^2 - 4)$.

Tip: Always check for zeroes in the denominator and the domain restrictions when dealing with rational functions.