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I'm stuck on a mathematical problem in a small feature I'm building for a website. I need a function that starts out with exponential growth and than levels out as x grows. I've drawn this wonderful picture to communicate what I'm looking for since my math lingo isn't the best.

Any help appreciated.


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up vote 20 down vote accepted

I think you want a sigmoid function.

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Thanks, solved my problem :) – Oscar Kilhed Jun 15 '11 at 12:44
Use these all the time for programming, easiest to use are logistic and atan – BlueRaja - Danny Pflughoeft Jun 15 '11 at 17:35
You say "starts out with exponential growth", so you want the logistic function. – starblue Jun 15 '11 at 20:13

Perhaps you are looking for something like ${\rm erf}(x-2)+1$, where erf is the error function, ${\rm erf}(x) = \frac{2}{{\sqrt \pi }}\int_0^x {e^{ - t^2 } dt} $. You can use WolframAlpha to watch the graph: plot $f(x)={\rm erf}(x-2)+1$, say from $0$ to $5$. Further see Wikipedia's plot of the error function, here, to understand the reason for choosing ${\rm erf}(x-2)+1$.

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