Define a function for a curve

Question

I am stuck with this problem longer than I'd like to admit. I wanna define a function which looks like this:

It starts at $(0,0)$, then increases up to a maximum and then decreases, never reaching $0$.

I would like to change the rate of increase, the maximum point, and the rate of decrease.

Answer 1

You could start by taking two positive real numbers $a$ and $b$ and then define $f(x)=\dfrac{ax}{x^2+b}$ ($x\in\mathbb{R}^+$).

Answer 2

You could try:

$$f(x)=\frac{\log(x+1)}{x+1}$$

for $x\in[0,+\infty)$

http://www.wolframalpha.com/input/?i=log(x%2B1)%2F(x%2B1)