Creating a custom exponential function

Question

I'm trying to come up with an exponential function that starts at zero, rises quickly (in $y$) between $x = 0$ and $x = 100$ and then slowly levels off as $x$ continues into infinity.

Something that looks like....

Answer 1

Try

$$ y=\mu \frac{1}{e^{-\alpha x} + \epsilon}+\beta$$

Adjust $\alpha$, $\beta$ $\mu$ and $\epsilon$ to meet your needs.

Note:

• $\mu$ scales the function
• $\beta$ shifts it up
• $\alpha$ controls the rate of growth for small $x$
• $\epsilon$ controls when the fuction starts to level off

Answer 2

If you want a function going from 0 to 100, may be you could use

y = 100 (1 - Exp[-a x])

parameter "a" defines the slope at the origin.