elementary set theory - How do I prove the following: $f(Scup T) = f(S) cup f(T)$

I have a question.

How do I prove the following identity?

$$f(S\cup T) = f(S) \cup f(T)$$

Answer

Element chasing is a promising method here.

$y\in f(s\cup t)$ if and only if there is some $x\in s\cup t$ such that $f(x)=y$. If $x\in s$ then $y\in f(s)$, if $x\in t$ then $y\in f(t)$. Therefore $y\in f(s)\cup f(t)$.

I leave the second inclusion to you.