other functions the look similar to the hyperbolic sine function

Question

This is probably an easy question, but I am looking for other simple functions that behave similar to the hyperbolic sine function. I have a chart below. Basically I am looking for other functions whose derivatives increase in either the positive or negative directions as the value of $x$ gets farther from the origin. Further, the domain should be from positive infinity to negative infinity and the codomain should also be from negative infinity to positive infinity. I am not sure if this class of functions has any special names.

Note that I am not looking for cubic equations which can have 3 zeros. I am looking for something that passes through the origin and behaves in this manner.

Any suggestions? I am doing some modelling of a particular phenomena and some of the pieces behave in this way.

Answer 1

Take any function $f(x) > 0$, any $a \ge 0$, and let your function be $$F(x) = a x + \int_0^x (x-t) f(t)\; dt$$ Note that $F(0) = 0$, $F'(0)=a$, and $F''(x) = f(x)$.

Answer 2

Consider a cubic of the form $$ y = ax^3 + bx $$ with positive $a$ and $b$.

The only root is at the origin and the derivative behaves as you wish. You can vary the coefficients to get a shape that makes sense in your model.