Let $X$ have the p.d.f $f(x)= e^{-x}$, $ x > 0$, $0 $ otherwise,find the pdf of $Y = X^2$ and space range $Y$.
I use the change of variable formula
The inverse is
$${x} = \sqrt{{y}} $$
The derivative is $$f(x)= \frac{1} {2\sqrt{y}}$$
The pdf is $$ \frac{e^{-\sqrt{{y}}}} {2\sqrt{y}}$$
My question is how do I get the range? and how do I check that I am correct? I saw that I can use integration, but what limits should I use?