Suppose there is a function $f(\vec{x})$ which can be given as a perturbative expansion of the form $$f(x) = f_0 (x) + f_1 (x) + f_2 (x)+\cdots$$ where $f_n$ represents a function of order some $\epsilon^n$, where $\epsilon\ll 1$.
Now suppose I introduce a new variable $y$ such that $$x= g_0(y) + g_1(y)+\cdots$$ where $g_n$ represents a monomial of $y$ with coefficient of order $\epsilon^n$ .
I want to obtain the perturbative expansion of $f(y)$, such that $$f(y)= \hat{f}_0 (y) + \hat{f}_1 (y) + \hat{f}_2 (y)+\cdots.$$
I presume one does this by expanding $$f(y)= f_0 \left( g_0(y) + g_1(y)+\cdots\right) + f_1 \left( g_0(y) + g_1(y)+\cdots\right) +\cdots\\\\ $$
where by $f_0\left( g_0(y) + g_1(y)+\cdots\right)$ I mean using the exact expression of $f_0(x)$ but substitute in $x=\left( g_0(y) + g_1(y)+\cdots\right)$.. Is this right?
[UPDATE:] What should one do when the relation between the coordinates is given in terms of $$y= h_0(x) + h_1(x)+\cdots$$ instead?