In my research, I am dealing with functions$$f(x) = \sum_{k=1}^N \frac{A_k}{B_k} \left(1- e^{-B_k x} \right), \quad g(x) = \sum^N_{k=1} \frac{A_k}{B_k}\left( \frac{1}{2}- \frac{1}{1 + e^{-B_k x}} \right),$$ where $A_k, B_k \in \mathbb{R}$ and $x \in [0, \infty)$. Clearly, $f(0) = g(0)=0$. Furthermore, we are interested in roots $f(x^*)=0$.
The question I want to solve is:
do $f(x)$ and $g(x)$ have the same amount of roots in the domain $(0, \infty)$?
It would be even better to show that each root of $f(x)$ maps uniquely to a zero of $g(x)$ (pigeonhole principle style).
I have been trying to exploit some structure or symmetries in the problem. For example, each term $-e^{- B_k x}$ in $f(x)$ turns up as a geometric series in $g(x)$.
Coming up with a counterexample, or giving me a hint towards a body of math literature that deals with these kind of questions would be greatly appreciated!
Feel free to ask more context of the problem if you need.