jmc

We iterate the function to find $g$:

$$\begin{array}{rl}f(f(x))& =3(3x-2)-2=9x-8\\ f(f(f(x)))& =3(9x-8)-2=27x-26\\ f(f(f(f(x))))& =3(27x-26)-2=81x-80\end{array}$$

This is an increasing, continuous function. The minimum in the domain is at $0$, where it equals $-80$, and the maximum is at $2$, where it equals $-80+2(81)=82$. It covers all values between these, so the range is $\boxed{-80\le g(x)\le 82}$.