Increasing, continuous, concave downward function normalised between 0.5 and 1

Question

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.

Answer 1

Try

$$f(x,a)=\frac{ax+1}{ax+2},\text{ }(a>0)$$

Answer 2

How do you like $f_k(x) = \dfrac{1}{1+e^{-kx}}$, where $k\in(0,\infty)$ is a parameter?

Answer 3

Let $f(x)=A(1-B e^{-k x})$ for $k>0$. Then \begin{align} \lim_{x\to\infty}f(x) &= A =1, \\ \lim_{x\to 0} f(x) &= A(1-B) = 1/2\end{align} and so $f(x) = 1-e^{-k x}/2$. This leaves $k$ as a free parameter which we can use to control the concavity at some point, e.g. $f''(0) = -k^2/2$. (I don't think we find an increasing function where the concavity is constant for $x>0$).

Credit where credit is due: An earlier poster gave this as a functional form but with $k=1$ (and therefore no parametric dependence for concavity.)

Answer 4

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