A formula that yields a particular graph shape

Question

I would like a formula for a function whose graph has the following properties:

1. $f(0) = 0$.

2. $\lim\limits_{x\to\infty}f(x) = y$.

3. The shape of the function is approximately the following:

   

4. It should have an exponential or a logarithm in the formula.

Any function like this?

Answer 1

$$y = 5 - 5 \exp(-\alpha x)$$ where $\alpha >0$ will do the job. You can control the rate of growth by playing around with $\alpha$.

Answer 2

The graph of $y=5-5e^{-x}$ has the desired characteristics. More generally, if you want $\lim\limits_{x\to\infty}f(x)=a>0$, the function $f(x)=a-ae^{-x}$ works.

Answer 3

Also your graph doesn't have anything shown for x<0, so presumably its equal to zero there?

i.e. Use a piecewise defined function:

f(x) = a-a*exp(-x) : for positive x and f(x)=0 for non-positive x.