General equation for the change of variable (random variables' functions)

I've never worked with dirac delta function, but here is my stab at it. Under the stipulations given here, let $h(x) = g(x)-y$, then $h'(x) = g'(x)$, and \begin{align*} f_Y(y) &= \int_{-\infty}^{\infty}\delta (h(x))f_X(x)dx\\ &= \int_{-\infty}^\infty\sum_i \frac{\delta(x-x_i)}{|h'(x_i)|}f_X(x)\,dx\\ &= \sum_i\int_{-\infty}^\infty \frac{\delta(x-x_i)}{|h'(x_i)|}f_X(x)\,dx\\ &= \sum_i\frac{1}{|h'(x_i)|}f_X(x_i)\tag{$\star$}, \end{align*} where $i$ is such that $x_i$ is a root of $h(x)$.

This $(\star)$ is the many-to one formula and is particularly useful when we have to make a many-to-one transform. For example, say $X\sim \text{unif}(-1,1)$, and I would like to find the density of $Y = X^2$. Notice that over $(-1,1)$, $Y$ is not one-to-one but two-to-one. Thus we apply the many-to-one formula. $X = \pm \sqrt Y$, and $$f_Y(y)=\frac{f_X(-\sqrt y)}{\left|\frac{dy}{dx}\right|_{-\sqrt y}}+\frac{f_X(\sqrt y)}{\left|\frac{dy}{dx}\right|_{\sqrt y}} = \frac{1/2}{|2(-\sqrt y)|}+\frac{1/2}{|2\sqrt y|} = \frac{1}{2\sqrt y}.$$

As a final note, I'm sure $$f_Y(y) = \int_{-\infty}^{\infty}\delta (g(x)-y)f_X(x)dx$$ covers more case when the stipulations I mentioned earlier are not met, but I wouldn't know about it