jmc

Substituting $f^{-1}(x)$ into our expression for $f$ we get $$f(f^{-1}(x))=3f^{-1}(x)-8.$$Since $f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$, we have $$x=3f^{-1}(x)-8.$$or $$f^{-1}(x)=\frac{x+8}{3}.$$We want to solve the equation $f(x)=f^{-1}(x)$, so $$3x-8=\frac{x+8}{3}.$$or $$9x-24=x+8.$$Solving for $x$, we find $x=\boxed{4}$.