To find the restriction on the domain of the function:

$f(x) = \frac{x^2}{x^2 - 1}$

We need to ensure that the denominator does not equal zero because division by zero is undefined.

Step 1: Set the denominator equal to zero and solve for $x$.

$x^2 - 1 = 0$

Step 2: Solve for $x$.

$x^2 = 1$

$x = \pm 1$

Conclusion:

The function is undefined when $x = 1$ or $x = -1$ because these values would make the denominator zero.

Therefore, the domain of $f(x)$ is all real numbers except $x = 1$ and $x = -1$. In interval notation, this is:

$\boxed{(-\infty, -1) \cup (-1, 1) \cup (1, \infty)}$

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Would you like a more detailed explanation of this process?

Here are some related questions you might find interesting:

1. How do you handle rational functions with square roots in the denominator?
2. What is the domain of a rational function with a cubic denominator?
3. How does the domain of a function affect its graph?
4. What happens if you try to graph this function near the points $x = 1$ and $x = -1$?
5. Can you simplify the function $\frac{x^2}{x^2 - 1}$ before finding its domain?

Tip: Always check the denominator first when determining the domain of rational functions to avoid division by zero errors.