jmc

We have $x$ in the domain of $f(x)$ as long as the denominator, $(3x-9)(3x+6)$, is nonzero. This is true for all $x$ except the solutions to the equations $3x-9=0$ and $3x+6=0$. These solutions are $x=3$ and $x=-2$, respectively.

Therefore, the domain of $f(x)$ is all real numbers except $3$ and $-2$. Expressed as a union of intervals, the domain is $\boxed{(-\infty ,-2)\cup (-2,3)\cup (3,\infty )}$.