Example 8. Find the domain of the function

$$
f(x)=\frac{x-8}{x^{2}-7 x+12}
$$
Solution. The given function is not defined for those values of $x$ for which the denominator is zero (since division by zero is meaningless). By equating the denominator to zero, we get $x^{2}-7 x+12=0$, from which $x_{1}=3, x_{2}=4$. Thus, the function is defined on the entire real line $-\infty<x<+\infty$, except for the points $x_{1}=3, x_{2}=4$. In other words, the domain of definition is the union of three intervals: $(-\infty, 3),(3,4),(4,+\infty)$.

Remark. A rational function $f(x)=\frac{P_{m}(x)}{Q_{n}(x)}$, where $P_{m}(x)$ and $Q_{n}(x)$ are polynomials of degree $m$ and $n$ respectively, is defined for all $x$, except for the zeros of the function $Q_{n}(x)$.