Find the domain of \( f \) \[ \begin{array}{l}f(x)=\frac{\sqrt{5 x-8}}{x^{2}-13 x+12} \\ (-\infty, 12) \cup(12, \infty) \\ (-8,-5) \cup(12,13) \\ {\left[-\frac{5}{8}, 12\right) \cup(13, \infty)} \\ {\left[\frac{8}{5}, 12\right) \cup(12, \infty)} \\ \left(-\infty, \frac{8}{5}\right) \cup(12, \infty)\end{array} \]

To find the domain of the function \( f(x) = \frac{\sqrt{5x - 8}}{x^2 - 13x + 12} \), we need to ensure two things: 1. The expression under the square root, \( 5x - 8 \), must be greater than or equal to zero, leading to the condition \( 5x - 8 \geq 0 \) or \( x \geq \frac{8}{5} \). 2. The denominator \( x^2 - 13x + 12 \) must not be zero. This can be factored to \( (x - 12)(x - 1) \), which means \( x \neq 12 \) and \( x \neq 1 \). Combining these conditions, we can establish the intervals for the domain: - From \( x \geq \frac{8}{5} \), we consider numbers greater than or equal to \( \frac{8}{5} \). - We must exclude \( x = 12 \). Thus, the domain of \( f \) is \( \left[\frac{8}{5}, 12\right) \cup (12, \infty) \).