Change-of-variables formula for a density

A random variable $X$ has density $f_X$, and $Y=g(X)$ for a given function $g$. How do you find the density of $Y$?

For strictly monotone, differentiable $g$: $f_Y(y)=f_X\!\left(g^{-1}(y)\right)\left|\tfrac{d}{dy}g^{-1}(y)\right|$. This follows by differentiating $\mathbb{P}(Y\le y)$, which equals $F_X(g^{-1}(y))$ or $1-F_X(g^{-1}(y))$. If $g$ is not monotone, sum this expression over every root $x$ of $g(x)=y$, i.e. $f_Y(y)=\sum f_X(x)/|g'(x)|$.