First I find the cdf of $X$ --> $$F_{X}(x) = \int_{0}^{x}{1} \;dw = w|_{w=0}^{w=x}=x$$ for $0<x<1$. Then the cdf of $Y$ --> $$F_{Y}(y) = Pr(Y\leq y) \\= Pr(\sqrt{X} \leq y) \\=Pr(X \leq y^2) \\=F_{X}(y^2) = y^2$$ My question is this: why is $0<y<1$ the domain of $F_{Y}$ and not $|y| < 1$?
I know the former will satisfy $\int f_{Y}(y) = 1$ where $f_{Y}(y) = F^{'}_{Y}(y) = 2y$, but what's the actual (mathematical) reason we attain the former domain?