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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 $l\leq f(x)\leq h$?

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    Draw a sketch of the graph you have in mind, and the community will provide you with an analytical expression for an $f$ that approximates your graph.2012-06-15

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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.

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

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

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    And of course, you can scale and shift this for arbitrary $l,h$....2012-06-15