I wish to have a function that maps [0,1] to [0,1]. I also require that f(0) = 0 and f(1) = 1.
Also, I would like the function to be of sigmoidal shape, such as this:
The above function is ok, but I would like the function to be flatter at the bottom and the top, and have a steeper ascent.
Ideally I would be able to adjust the parameters of the function to control how "flat" it is.
Optional: If I could also adjust the parameters so that I can choose where the inflection occurs that would be great. (e.g. in the above function the inflection seems to occur at x = 0.5, but ideally I would be able to shift this to say x = 0.1, or x = 0.05 etc.)