Find the joint density, given the conditional one

Question

Let $X$ be uniform on $[1,2]$. Conditional on $X$, $Y$ is exponential of parameter $X$. Find the joint density. The joint density should be $f(x,u) = xe^{−xy} I_{[1,2]× [0,+inf]}(x,y)$. I know how to get $xe^{−xy}$, but how do I know that the range of $Y$ is $[0,+inf]$, so that the support is $S(f(x,y))=[1,2]× [0,+inf]$?