I'm looking for a function that creates the blue curve shown in the picture. The point is that within a certain range (e.g. x=0 to x=0.8) the output of the function should be relatively close to 1.0. If possible it would be exactly 1 but "close" to 1 (within 10% or so) would be good enough. However, the first value produced by the function should be exactly 1. eventually, the output should drop sharply to y=0.
The available input value is x which linearly increases along the x-axis from 0 to 1. The function could rely on any other parameter necessary to achieve the illustrated curve.
Any ideas?
Best Answer
The first candidate that comes to mind is the function of the form
\begin{align} y(x)&=(1-x^n)^{1/n} . \end{align}
This is how the graph of this function looks for $n=5\dots12$: