The equation of a curve which is almost similar to Gaussian curve (normal distribution curve) with $2$ asymptotes

Question

We know that the equation of the following curve could be of the form $y=A+Be^{ax^2+bx+c}$ where $A,B,a,b,c$ are constants.

Which has one horizontal asymptote.

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I have (almost) same kind of curves (Maybe same as the above one but in more general form). The following has two horizontal asymptotes. What could be its equation?

Please suggest me an explicit (& not piecewise) function.

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Any help would be appreciated. THANKS!

Answer 1

You can add an s-shaped function.

E.g.

$$5e^{-x^2}+\tanh(2x+x^3).$$

Make sure that it tends to its asymptotes faster than the Gaussian, to avoid a local minimum.

Answer 2

Here is a solution obtained by adding to a gaussian function a modified $\arctan \ (= \tan^{-1})$ function, namely

$$f(x)=e^{-x^2}+a \arctan(b(x+c))$$

with $a=0.05;b=10;c=0.7.$

I have featured in red the derivative of this function.