Is it true that for any function $f$, and sets $S_1,S_2$ such that $f:S_1\rightarrow S_2$, if $g$ is the inverse of sets in $S_2$, then $f(g[S_2])\subseteq S1\subseteq g(f[S_1])$? If yes, is there a proof for this statement and if no, is there some counter example?
Hints: $f$ must one-to-one for the inverse function to exist. Consider what happens if it is onto $S_2$ or not. (See Bijection, injection, surjection)