I am trying to find a function such that
- $x$ reaches 0 at a set point $x^*$ such that $x* > 0$
- as $x \rightarrow 0 \implies y \rightarrow 1.$
- The curvature of the function between $x = 0$ and $x = x*$ can be changed such that it looks like the red lines in the figure.
I'm thinking the function has to be piecewise but I'm unsure of what to use for the part between $x=0$ and $x = x*$.
I've attached a picture of the kind of function I want, where the red lines represent the curve.
I've tried a negative exponential however I can't seem to get a curve such as the top red line of the figure.
Sorry this feels like a daft question. Any ideas? Thanks!!!