Function whose shape is similar to logistic function but without horizontal asymptotes

Question

I'm looking for a function whose shape is similar to the logistic function but without any horizontal aymptotes. Instead, it should decrease without bound as $x$ approaches negative infinity and should increase without bound as $x$ approaches positive infinity. Its shape should look just like a logistic function such as $\frac{10}{1+100e^{-x}}-5$, but without asymptotes at $y=5$ and $-5$.

Thanks

Answer 1

The asymptotes are what makes a logistic function a logistic function.

If you want to preserve the rapid increase that occurs towards the center of the function, keep the body, that is $\displaystyle \frac{L}{1+ce^{-k(x-x_0)}}$, but consider adding a term $+kx$, where $k$ is small as to not disrupt the center shape.

Here is a screenshot from Desmos.