Trying to find a function like this:
- 2 y-axis asymptotes: -1 / 1
- x values range from -infinity to infinity.
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nubela
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2looks like $\arctan$, or maybe $\tanh$? – Angina Seng Jun 16 '17 at 17:56
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seems right. thanks! can you answer it? – nubela Jun 16 '17 at 17:58
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It might as well be $y = \operatorname{erf}(x)$, or any S-shaped (sigmoid) function. – Dmoreno Jun 16 '17 at 18:00
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You can also study $f_n(x) = x(1+x^{2n})^{-\frac{1}{2n}}$ for different $n \in \mathbb N$. – md2perpe Jun 16 '17 at 18:18
2 Answers
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There are many functions having these properties. Here, for example, is the graph of
$$ f(x)=\frac{2}{\pi}\arctan(x) $$
Here is
$$ y=\frac{1-e^{-ax}}{1+e^{-ax}}\text{ for }a=1 $$
but you can let $a>0$ for any $a\in\mathbb{R}$
John Wayland Bales
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Another which immediately comes to mind for me is a modified logistic function $f(x)=\frac{2}{1+e^{-x}}-1$. It looks like: 
MathTrain
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