Which analytically-given function could allow one to independently tune the power-law behavior of its left and right tails?

Question

I would like to find the analytic expression of a function $f: \mathbb{R}^{+} \to \mathbb{R}^{+} $ which initially grows as a power-law, then achieves its maximum in a smooth bell-shaped way and then decreases as a power-law. In summary:

i) $f(x \to 0) \propto x^\alpha$ with $\alpha > 0$

ii) $f(x \to \infty) \propto x^\beta$ with $\beta < 0$

iii) The function has a bell-like shape somewhere between both extremes

Thanks.

Answer 1

A simpler function that works is $$f(x) = \frac{x^\alpha}{C+x^{\alpha-\beta}},$$ where $C$ is a positive constant.

Answer 2

Take $f(x)=\frac{e^{(\ln{x})^3}x^{\beta}+e^{-(\ln{x})^3}x^{\alpha}}{\cosh{\ln^3{x}}}$.