jmc

Since $(1,2)$ is on the graph of $y=\frac{f(x)}{2}$, we know that $$2=\frac{f(1)}{2},$$which implies that $f(1)=4$. Therefore, $f^{-1}(4)=1$, which implies that $\left(4,\frac{1}{2}\right)$ is on the graph of $y=\frac{f^{-1}(x)}{2}$. The sum of this point's coordinates is $\boxed{\frac{9}{2}}$.