Motivation:
Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k x^2 + \ldots), \quad |k|\ll1$$ This can be seen for instance in the distortion corrections for lens, or in the equation for the force applied by a spring.
A problem arises when we want to invert this equation - the inverse of the cubic function $y = x(1+kx^2)$ is a mess, and adding higher terms makes an analytic solution for the inverse impossible.
Here's the question:
Do there exist non power series expansions that lead to simpler inversion formula? Specifically, I'm looking for a sequence of non-power functions $f_n(x)$, that when given a finite series such as: $$y(x) = x\left(\sum_{i=0}^m a_i f_i(x)\right)$$ Then we should also be able to expand the inverse of $y$ in a finite number of these functions $f_n$: $$y^{-1}(x) = x\left(\sum_{i=0}^m b_i f_i(x)\right)$$
So, do such $f_n$ exist? I'm looking for non trivial $f_n$, preferably that can be used to approximate a wide variety of smooth functions.
You can use the following
$$ x = y - k x^3 = y - k(y- k x^3)^3 = y - ky ^3 - 3 k^2 y^2 x^3 + 3 k^3 y x^6 - k^4 x^9 $$ You can extend this to any accuracy you want.
By contraction mapping results, you can estimate the error.
Note that the expansion is a contraction map of $|3 k x^2| <1$ and $\rho=|3 k x^2| is useful in bounding the error.
Based on the edit provided by OP
To keep the symbols straight, I will write $$x(y) = y\left(\sum_{i=0}^m b_i f_i(y)\right)$$
Note that this can never be exact. However to the accuracy in the original equation:
$$ y = x + k x^3$$ we have $$ x = y - k x^3$$
For this inversion to work in general we need $$ \lim_{x\to 0} \sum_{i=0}^m a_i f_i(x) \neq 0$$