Is there a name for the function $(1 - e^{ct})/(1 - e^{c})$?

Question

$$f(t) = \frac{1 - e^{ct}}{1 - e^{c}}$$

This is a function which is somehow a streched exponential which is zero at $t = 0$, and one at $t = 1$, where $c$ determines the curvature (with $c = 0$, it is a line segment).

Is there a common name for this function?

Answer 1

There's no name, in particular/specifically for your function; as ZettoSuro noted, it's "just a regular exponential function," just shifted down, vertically, and inverted.

It's stretched, depending on the value of $c$: if $0 < c < 1$ it is stretched (less curvature). For $c>1$, it's curvature increases as $c$ increases. But the curve is flipped for negative values of $c$.