Step function approximation with explicit and symmetric limits?

Question

I can approximate step function with

This will describe a function, stepping from $y_{\min}=0$ to $y_{\max}=1$ at $x=0$.

I can rescale this function to any y-range, but failed to reach a form, where $y_{\min}$ and $y_{\max}$ stands symmetrically.

I got, for example

$\frac{y_{\max}+y_{\min} e^{-2kx}}{1+e^{-2kx}}$

which is not symmetric.

How to express symmetrically?

Answer 1

How about $$ \frac{y_{\max}}{1+e^{-2kx}}+\frac{y_\min}{1+e^{2kx}}?$$ Or simply $$ \frac{y_\max+y_\min}{2}+\frac{y_\max-y_\min}{2}\tanh(kx).$$