What formula to chose a nonlinear formula? Suppose I have a formula 
$$f(x) = x,$$ 
where $0 \leq x \leq 255$
Now I want to have a formula 
$$f(x) = y,$$ 
where $0 \leq x \leq 255$, where $f(0) = 0$, $f(255) = 255$ and e.g., $f(128) = 150$ (the value of $150$ might vary).
All other values should be interpolated.
So actually I want a function that is nonlinear (starts with $0$ and goes up to $255$, starting increasing fast and finishes increasing slow); opposite as a parabolic function.
What (kind of) function should I use?

To satisfy the requirements:
$$f(128)=150, f(255)=255$$
However, $f(0)=0$ can't be satisfied by this method. I assumed that f(1)=1, however 
you can change the numbers but not use zero, otherwise, there would be no inverse.

An example of the curve looks like this Curve
I can provide more info about the derivation if you want.
