To determine the domain of the function

$f(x) = \frac{5x - 8}{5x + 15},$

we need to identify the values of $x$ for which the function is defined. A rational function like this is undefined when its denominator is equal to zero. So, we solve the equation:

$5x + 15 = 0.$

Solving for $x$:

$5x = -15 \quad \Rightarrow \quad x = -3.$

Thus, the function is undefined at $x = -3$. Therefore, the domain of the function is all real numbers except $x = -3$.

The domain of $f(x)$ is:

$\boxed{(-\infty, -3) \cup (-3, \infty)}.$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the domain of a square root function?
2. What is the range of this function $f(x)$?
3. How do you simplify rational functions?
4. How would the domain change if the denominator had multiple factors?
5. How can vertical asymptotes be related to domain restrictions?

Tip: Always check for values that make the denominator zero when finding the domain of a rational function.