I am trying to evaluate the integral
$$\int dx \ \ f(g(x)) \ \delta(\alpha-g(x))$$
where $\alpha$ is a constant and $g$ some invertible function.
Here's what I did: a change of variables
$$g(x)=y$$
in order to obtain
$$\int \frac{dy}{\left| g'(g^{-1}(y))\right|} \ \ f(y) \ \delta(\alpha-y)$$
so that the result should be
$$\int dx \ \ f(g(x)) \ \delta(\alpha-g(x)) =\frac{f(\alpha)}{\left| g'(g^{-1}(\alpha))\right|}$$
Is my reasoning correct? If not, how should I proceed?
Update:
As Semiclassical pointed out, there are issues with my argument when $g(x)=\alpha$ has more than one solution, but I don't know how to proceed in that case.