Answer
Domain: $(-\infty,\ -1)\cup (1,\ \infty)$.
Work Step by Step
We need to find the domain of the given function:
$g(x)=\log_{3}(x^{2}-1)$
We know that we can't take the log of a negative number, so:
$x^{2}-1\gt 0$
$x^{2}\gt 1$
$x\lt -1$ or $x\gt 1$
Thus the domain is: $(-\infty,\ -1)\cup (1,\ \infty)$.