 Solution

Complete the square in the transformation:Since $0<X<2$ almost surely, $0<Y\le 1$. For $0\le y<1$, put $r=\sqrt{1-y}$. The event $Y>y$ is equivalent to $1-r<X<1+r$. Integrating the given probability density function over both branches at once gives the survival functionThus, including the tails,The cumulative distribution function on $(0,1)$ is $1-\sqrt{1-y}$. Differentiation givesBoth inverse branches matter: $x=1\pm \sqrt{1-y}$. The endpoint divergence is integrable, and $\mathbb{P}(Y=1)=\mathbb{P}(X=1)=0$, so no endpoint atom is needed.