Answer
The domain of this function is $(\infty,-2)\cup(6,\infty)$
Work Step by Step
$f(x)=\ln(x^{2}-4x-12)$
For this function to be defined, the expression inside the logarithm must always be positive and cannot be zero.
Solve the following inequality to obtain the domain of this function:
$x^{2}-4x-12\gt0$
Factor the expression on the left:
$(x-6)(x+2)\gt0$
Set both factors equal to $0$ and solve each individual equation for $x$ to obtain the critical points:
$x-6=0$
$x=6$
$x+2=0$
$x=-2$
The critical points are $x=-2$ and $x=6$. These point divide the real line in the following intervals:
$(-\infty,-2)$ $,$ $(-2,6)$ and $(6,\infty)$
Elaborate a sign diagram (shown below) using test points within each of the intervals found and evaluating them in each factor.
From the diagram, it can be seen that the inequality is satisfied in the intervals $(\infty,-2)$ and $(6,\infty)$. Since the endpoints of these intervals make the factors $0$, they do not satisfy the inequality and are to be left open.
The domain of this function is $(\infty,-2)\cup(6,\infty)$
