For a schematic of a real-world system ($x$ axis is time and $y$ axis is a sudden deterioration of the state of some physical system), I can almost model this system as $f(x) = -e^x$. However, this simple function has no inflection and/or undulation point. For the (non-mathematically inclined) audience for this specific application, I need a "simple" (not piecewise, not imaginary, not differential) function that introduces a "kink" into this function, representing a sudden decay in the state of this system.
The function below is almost exactly what I want (for this audience, "minor" issues, such as the discontinuity, are irrelevant). However, as $x \to -\infty$, I would like this (arbitrary) value to also be $y_\max$. In the case of the function below, a single inflection point is introduced (the sort of "kink" I am looking for), but $y_\max$ is now "right" of the inflection point, which does not make sense for this particular physical system.
Thoughts on a better, "simple" function? Ideally, I still want $y_\max$ at $-\infty$, but $y_\max$ "left" of the inflection point below would be acceptable.
Note: I say "kink" because I do not care if this sudden deterioration of the state of some physical system is technically an inflection/undulation point, or even a smooth function. As long as it is not piecewise...
Thank you.
