elementary set theory - Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective

Let $f: A \rightarrow B$ be a function. How can we show that for all subsets $S$ of $A$, $S \subseteq f^{-1}(f(S))$? I think this is a pretty simple problem but I'm new to this so I'm confused.

Also, how can we show that $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective?