Sigmoid functions similar to atan but with different upper and lower bound

Question

Do there exist smooth sigmoid functions similar to atan but with different upper and lower bounds? This function must go through origin. Something like on the picture: Function If not, why? Where can I get more info about this?

Answer 1

Welcome to math.stackexhange. You can transform any standard sigmoidal function $f$ such as $f = \arctan$ or the hyperbolic tangent $f = \tanh$ by

• Multiplying $f$ with a constant, $f(x)$ becomes $cf(x)$
• Shifting the graph to the left or the right so that is still passes through the origin, $f(x)$ becomes $f(x + b) - f(b)$

Combining these, new functions $g(x) = c(f(x + b) - f(b))$ can be made that are increasing, have desired limits at $\pm \infty$, and satisfy $g(0) = 0$.

For example, $g(x) = 4(\arctan(x + 1) - \arctan(1))$ has limits $- 3\pi$ at $- \infty$, $\pi$ at $\infty$, and $g(0) = 0$. See the graph.