I'm wondering if there are homomorphisms $f$ between unitary rings $R,S$, such that $f(r)$ is invertible but it's inverse doesn't equal $f(r^{-1})$. That is only possible if the inverse isn't in the range of $f$. Nonetheless I couldn't find example of such rings and homomorphisms.
(Notice that of course we know that if $r$ is invertible, we have $f(r)^{-1}= f(r^{-1})$, but I only assume that $f(r)$ is invertible)