Let $X$ be a continuous random variable with continuous density $f_X$ and $\phi$ a bijection that behaves well (i.e. with compatible assumptions). The density of $f_{\phi(X)}$ is given by : $$f_{\phi(X)}(x)=\frac{1}{\lvert \phi'(\phi^{-1}(x))\rvert} f_X(\phi^{-1}(x)) $$
However, I am sometimes interested in knowing a density for a $\phi$ that is not bijective for example $Z^2$ where $Z \approx N(0,1)$. In this link, Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$, the first answer uses a formula that I never saw before. Why is his claim true ? Is there such formulas for $\phi$ not bijective ?