Let $X$ be a real valued random variable with cumulative distribution function (CDF) $F_X$ and probability density function (DF) $f_X$. Suppose $g\colon\Bbb{R}\to\Bbb{R}$ is a differentiable, strictly monotonic function. Then it can be proved that $Y=g(X)$ is a continuous random variable with CDF $$F_Y(y)=F_X(g^{-1}(y)),$$ and DF $$f_Y(y)=\bigg|\frac{1}{g^\prime(g^{-1}(y))}\bigg|f_X(g^{-1}(y)).$$
Is there a similar formula, in general, for the moment generating function (MGF) of $Y$?
I thought we might find it as follows:
$$M_Y(t)=E[e^{tg(X)}]\stackrel{?}{=}\int_\Bbb{R}e^{tg(x)}f_Y(x)\,dx,$$ but my attempts to simplify this integral (in general) and get it in the form of $M_X(t)$ have not succeeded, and only left me more confused.