Sigmoid function increasing for large values of variable x

Question

I am looking for a function that involves the sigmoid function but for large values of variable $x$ increases. Maybe sth like this, but there is a part missing: $f(x)= A +B\ \frac{1}{(1+e^{-x})}\ +\ \textbf{?}$

where A,B are constants. Any suggestion?

Answer 1

These are functions that "live" between $c$ and $1$. But you can scale them to arbitrary domains. $\Phi$ is the cumulative normal distribution.

$$f_1=\frac{1-c}{\pi}\arctan(a(x-b))+\frac{1+c}{2}$$ $$f_2=(1-c)\Phi(a(x-b))+c$$ $$f_3=\frac{1-c}{2}\frac{a(x-b)}{\sqrt{a^2(x-b)^2+1}}+\frac{1+c}{2}$$

You can also use other cumulative probability functions and rescale them as you wish.