jmc

$$h(x)=f(g(x))=7(x-1)+5=7x-2.$$Let's replace $h(x)$ with $y$ for simplicity, so $$y=7x-2.$$In order to invert $h(x)$ we may solve this equation for $x$. That gives $$y+2=7x$$or $$x=\frac{y+2}{7}.$$Writing this in terms of $x$ gives the inverse function of $h$ as $$h^{-1}(x)=\boxed{\frac{x+2}{7}}.$$