Increasing, continuous, concave downward function normalised between 0.5 and 1 What would be a good function which is increasing, continuous, concave downward with 

$$\lim_{x \to 0} f(x) = 0.5$$

and 

$$\lim_{x \to \infty} f(x) = 1.$$

It should be concave downward whose concavity can be parametrically controlled.
How do you like $f_k(x) = \dfrac{1}{1+e^{-kx}}$, where $k\in(0,\infty)$ is a parameter?
Consider the function $$f (t)=1-\frac {1}{2} e^{-t}, \forall x\in \mathbb {R} $$ which is continuous, increasing, concave downward and satisfies the limits