The given function is:

$f(x) = \frac{5x}{x^2 - 16}$

To find the domain of the function, we need to determine where the function is defined. Specifically, the function will be undefined when the denominator is zero.

The denominator is $x^2 - 16$, so we solve for when it equals zero:

$x^2 - 16 = 0$

$x^2 = 16$

$x = \pm 4$

Thus, the function is undefined at $x = 4$ and $x = -4$. Therefore, the domain of the function is all real numbers except $x = 4$ and $x = -4$.

The domain of $f(x)$ is:

$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How would the domain change if the function was $\frac{5x}{x^2 + 16}$?
2. What is the range of the given function $f(x)$?
3. How would you find the vertical asymptotes of this function?
4. What would the derivative of $f(x)$ tell you about the behavior of the function?
5. How do you find horizontal asymptotes for rational functions like this?

Tip: When finding the domain of a function, always look for values that make the denominator zero or create undefined operations like taking the square root of a negative number.