jmc

We have $g(x)=f(-2x)$, which is defined if and only if $-2x$ is in the domain of $f$, that is, if $$-8\le -2x\le 4.$$ Dividing all expressions in this chain of inequalities by $-2$ compels us to reverse the direction of the inequalities: $$4\ge x\ge -2.$$ Thus $g(x)$ is defined if and only if $-2\le x\le 4$. In other words, the domain of $g(x)$ is $\boxed{[-2,4]}$.