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I need the expression of a function that looks like the following:

enter image description here

The expression preferably needs to be simple e.g., comprised of as few elementary functions as possible. It doesn't matter what the function looks like for $x<0$, but as $x \rightarrow \infty$, the function should approach $0$. It seems like it could be a polynomial.

Also, the expression preferably should not be segmented, i.e., one expression for small $x$, another for large $x$, but be the same for all $x>0$.

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  • $\begingroup$ wouldn't be $e^{-x^2}$? $\endgroup$ – janmarqz Oct 15 '15 at 1:25
  • $\begingroup$ No, becuase $e^{-x^2}$ would not decrease gradually for small $x$, and there would not be the inflexion point. It would only approach $0$ for large $x$. $\endgroup$ – user173690 Oct 15 '15 at 2:34
  • $\begingroup$ It is hard to tell what exactly the shape is you are looking for. Would a sigmoid function, shifted and reflected as necessary, match your requirements? $\endgroup$ – njuffa Oct 16 '15 at 21:15
  • $\begingroup$ Yes, this looks promising! Is there a way to determine certain parameters to control the curve? For example, what features of the sigmoid function do the terms $A,B,C$ in $S(t) = \frac{A}{B+C e^{-t}}$ control? $\endgroup$ – user173690 Oct 18 '15 at 3:09
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    $\begingroup$ @u5609110 Negate the input to flip the function about the y-axis. I don't know the answer to your other question. I have never encountered such a "generalized" sigmoid function before, but you could find out yourself by graphing the function for different values of $A, B, C$. It is easier to think about further transformation by manipulation of argument or result. The result can be multiplied with a scale factor to modify the amplitude, and an offset can be added to shift the function horizontally. The argument can be multiplied with a scale factor to stretch the function along the x-axis. $\endgroup$ – njuffa Oct 19 '15 at 14:50
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To flip the function about the x-axis use f(-x). Therefore, for example, if the sigmoid is $$Y(t) = \frac{1}{1+exp(-t)}$$

the analytical expression for the reverse sigmoid is

$$Y(-t) = \frac{1}{1+exp(t)}$$

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