For the given function and sets $A = \{0,1\}, B = (0,1), C = [1, + ∞)$ find counterimages

Question

For the given function and sets $A = \{0,1\}, B = (0,1), C = [1, + ∞)$ find $ f^{-1}(A), f^{-1}(B), f^{-1}(C)$.

Function: $ f:[0,1]\times[0,3]\rightarrow R,\\f(x,y)=\max(x,y)$

How can I do it? Any tips?

Answer 1

What gives you $A$ when plugged into $f$?

$f(0,0) = 0 \in A$, so the point $(0,0)$ is in the preimage of $A$. $f(1,[0,1]) = 1$, so the set of points $\{1\}\times[0,1]$ is in the preimage of $A$.

Keep asking this question until you've exhausted $[0,1]\times[0,3]$ for each of the sets $A,B,C$