Sorry if I'm asking a stupid question, please bear with me :)
I have this function:
$$f(x) = x + A e^{Bx}$$
where A and B are known constants
Is it possible to find an inverse to this function?
(Even if in an iterative way, like using Netwon-Raphson method for example)
We can write it in terms of the Lambert W function.
If $y=x+Ae^{Bx},$ then:
$$(y-x)e^{-Bx}=A.$$ Multiply both sides by $Be^{By}$ gives: $$B(y-x)e^{B(y-x)}=ABe^{By},$$ giving $$B(y-x)=W\left(ABe^{By}\right)$$ or $$x=y-\frac1B W\left(ABe^{By}\right).$$
Since the W function can’t be written in terms of the so-called elementary functions, your inverse cannot, either, unless $AB=0.$