 Solution

Since a square is nonnegative, $F_Y(y)=0$ for $y<0$. For $y\ge 0$, the event $X^2\le y$ is $-\sqrt{y}\le X\le \sqrt{y}$. As $X$ has no point masses,Differentiating for $y>0$ gives the contributions of the two inverse branches:The value assigned to the probability density function at $y=0$ is arbitrary. It can be unbounded near $0$, but there is no atom there because $\mathbb{P}(X=0)=0$. Both positive and negative preimages must be included; keeping just one would generally lose probability.