Taylor series of composition of function with polynomial

Question

Say I have a taylor series around $0$ of some function $f(x) = a_0 + a_1x + a_2x^2 + \cdots$
Say $g(x)$ is a polynomial. Then is it true that taylor series of $h(x) = f(g(x))$ is term by term equal to $a_0 + a_1g(x)+a_2g(x)^2+....$?
The examples I've tried so far seem to work (on wolfram alpha)
Though I have no idea how to go about the proof

Answer 1

$ h(x)=f(g(x))$, since $f(x)=a_0+a_1 x+a_2 x^2$, $h(x)=a_0+a_1 g(x)+a_2 g(x)^2$,also since $g(x)$ is already polynomial, there is no need for further taylor expansion and the equality is prooven if the limit of $g(x)=0$ for $x->0$