Answer
Domain: $(-\infty,-1)\cup(2,\infty)$
Work Step by Step
Logarithmic functions are defined for positive arguments only.
$ f(x)=\ln(x^{2}-x-2)$ is defined when
$ x^{2}-x-2 \gt 0\quad $
Factor the trinomial: find factors of $-2$ whose sum is $-1$; here, we find $-2$ and $+1.$
$(x-2)(x+1) \gt 0$
The graph of $ y=x^{2}-x-2=(x-2)(x+1)$
is a parabola opening upwards, intersecting the x-axis at $-1$ and $+2.$ We know tht $ y $ is positive where the graph is above the x-axis.
The graph is above the x-axis before the left x-intercept $(x \lt -1) $ and to the right of the right intercept $(x \gt 2)$
Domain: $(-\infty,-1)\cup(2,\infty)$
