Preimages of intersections/unions Let $f(x) = x^2$ and suppose that $A$ is the closed interval $[0, 4]$ and $B$ is the closed interval $[−1, 1]$. In this case find $f^{−1}(A)$ and $f^{−1}(B)$. 
Does $f^{−1}(A\cap B) = f^{−1}(A) \cap f^{−1}(B)$ in this case? 
Does $f^{−1}(A \cup B) = f^{−1}(A) \cup f^{−1}(B)$?

Hint: 
The statements $f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B)$ and $f^{-1}(A\cap B)=f^{-1}(A)\cap f^{-1}(B)$ are always true. Also if you are dealing with infinite unions/intersections (and much more). 
It can be proved on base of: $$x\in f^{-1}(U)\iff f(x)\in U$$
Prove this in general and you do not have to bother about 'cases' anymore.
