elementary set theory - How come $f^{-1}(Q cap R)$ = $f^{-1}(Q) cap f^{-1}(R)$ is true?

$f(S\cap T) \neq f(S) \cap f(T)$

but

$f^{-1}(Q \cap R)=f^{-1}(Q) \cap f^{-1}(R)$

Can you explain it in simple terms, so I understand why and develop the intuition to see if a statement is true or false just by looking at it?