Question: Find the domain of the function $f(x) = \frac{x^2 - 25}{x^2 - 16}$.
1. **State the problem:** We need to find the domain of the function $$f(x) = \frac{x^2 - 25}{x^2 - 16}$$.
2. **Recall the domain rule for rational functions:** The domain includes all real numbers except where the denominator is zero because division by zero is undefined.
3. **Set the denominator equal to zero and solve:**
$$x^2 - 16 = 0$$
4. **Solve for $x$:**
$$x^2 = 16$$
$$x = \pm 4$$
5. **Exclude these values from the domain:** The function is undefined at $x = 4$ and $x = -4$.
6. **Write the domain:** All real numbers except $x = \pm 4$, which in interval notation is:
$$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$
**Final answer:** The domain of $f(x)$ is all real numbers except $x = \pm 4$.