Prove $f(A \cup B)=f(A) \cup f(B)$ and $f(A \cap B) \subseteq f(A) \cap f(B)$

Question

Where $S$ and $T$ are metric spaces with metrics $d_S$ and $d_T$. Given a function $f: S \to T$ and 2 arbitrary subsets $A \subseteq S$ and $B \subseteq S$. Prove $f(A \cup B)=f(A) \cup f(B)$ and $f(A \cap B) \subseteq f(A) \cap f(B)$.