S-shaped Reverse Logistic Curve

Question

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve.

Here's raw idea how it should look like (drawn with paint)

And here it is compared with logistic curve

Answer 1

The function \begin{align*} f: (-6,6) &\to \mathbb{R} \\ x &\mapsto \frac{\log(\frac{1}{2}\left(1+x/6)\right)}{1-\frac{1}{2}\left(1+x/6)\right)}, \end{align*}

which is just an adaptation from the one found here, seems to satisfy your needs. (The adaptation is done in order to conform with the interval in your image.) From this you may be able to make further modifications which are relevant to you, if needed.

Looking for the same thing too. This looks promising.