jmc

$$h(x)=f(g(x))=3(2x-3)+4=6x-5.$$Let's replace $h(x)$ with $y$ for simplicity, so $$y=6x-5.$$In order to invert $h(x)$ we may solve this equation for $x$. That gives $$y+5=6x$$or $$x=\frac{y+5}{6}.$$Writing this in terms of $x$ gives the inverse function of $h$ as $$h^{-1}(x)=\boxed{\frac{x+5}{6}}.$$