To find the domain of the function $f(x) = \frac{x}{5x - 8}$, we need to determine the values of $x$ for which the function is defined. The function will be undefined wherever the denominator is equal to zero, because division by zero is undefined.

1. Set the denominator equal to zero: $5x - 8 = 0$

2. Solve for $x$: $5x = 8$ $x = \frac{8}{5}$

This means the function is undefined at $x = \frac{8}{5}$.

3. Domain: The domain of the function is all real numbers except $x = \frac{8}{5}$. In interval notation, the domain is: $(-\infty, \frac{8}{5}) \cup (\frac{8}{5}, \infty)$

So, the domain of $f(x) = \frac{x}{5x - 8}$ is $\boxed{(-\infty, \frac{8}{5}) \cup (\frac{8}{5}, \infty)}$.

Would you like a detailed explanation on any part of this solution, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How do you find the domain of a function involving a square root?
2. What is the domain of the function $f(x) = \frac{1}{x^2 - 9}$?
3. How does the domain change if the function is $f(x) = \frac{x + 3}{5x - 8}$?
4. Can the domain of a function ever be all real numbers?
5. How do you determine the domain of a logarithmic function?

Tip: Remember, to find the domain of a rational function, always check where the denominator equals zero. These points are excluded from the domain.