Probability - Finding the Support of a Joint Transformation

Question

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$

Let $Z = XY^2$ and $W=Y$ be a joint transformation of $(X,Y)$.

Find the support of $(Z,W)$.

I always have a hard time finding this support.

Is there a general method?

After analyzing the graphs of the joint pdfs, I'm thinking it might be $$0<z<w^2,0<w<1$$ although I'm very unsure.

Answer 1

Maybe this helps:

We know that: $0<x<1$, $0<y<1$, $Z=XY^2$ and $W=Y$

first for $Z$: $$0<y<1 \implies 0<y^2<1 \implies 0<xy^2<1 \implies 0<z<1$$

for $W$, just replace $y$ by $w$: $0<y<1 \implies 0<w<1$