Alrighty, math noob here, so be nice :P. We're building an app and need an exponential function that exists between zero and one, and, depending on the importance we give it, will fluctuate between rising near the beginning or end of the graph.
Lemme know if you have any ideas or need clarification :P
On
Sounds like you want the Logistic function. The values are between 0 and 1, though the growth is centered around $0$ and it has an exponential in it. You can shift it horizontally by adding a constant to $x$ and change the steepness by changing $e^{-x}$ to $e^{-ax}$ for your choice of $a$. Does that work?
Maybe you mean power law rather than exponential. In that case @fedja's suggestion is great: $x^a$. $a<1$ for low importance and $a>1$ for high importance.