College Algebra, Chapter 3, 3.1, Section 3.1, Problem 60

Determine the domain of the function $g(x) = \sqrt{x^2 - 2x - 8}$
The function is not defined when the radicand is a negative valuie, So...

$
\begin{equation}
\begin{aligned}
x^2 - 2x - 8 &\geq 0 \\
\\
(x+2)(x-4) &\geq 0
\end{aligned}
\end{equation}
$


The factors on the left hand side of the inequality are $x+2$ and $x-4$. These factors are zero when $x$ is $-2$ and $4$, respectively. These numbers divide the number line into intervals
$(-\infty,-2],[-2,4],[4,\infty)$
By testing some points on the interval...



Thus, the domain of $g(x)$ is $(-\infty, -2]\bigcup[4,\infty)$