How to perform a transformation on a number so that it is between a high and low value?

Question

This is probably very simple for some of you, but I can't for the life of me get something that works reliably. Given any positive number, $x$, and a positive high and low value $(h, l)$ what kind of functions $f$ are there such that $l\leq f(x)\leq h$?

Answer 1

I guess what you want is actually bijection between $(0,\infty)$ and $(l,h)$. Here's a possible way. For $x\in (0,\infty)$, Clearly $\frac{\arctan x}{\pi/2}(h-l)+l\in (l,h)$ is a bijection.

Answer 2

As an easier-to-compute alternative to caozhu's arctan solution: $$ x\mapsto h - \frac{h-l}{x+1} $$

Answer 3

Say $l=0, h=1$ then $0\le \sin^2(x)\le 1$.