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

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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I'm not sure that I understand the question. Are you looking for an expression that can be converted to a boolean value indicating whether or not x falls within the specified interval? – Andrew Jun 15 '12 at 15:43
What are the conditions on the function you are loking for? The answer gives one possible solution; another would be $f(x) = (l-h)e^{-x} +l$. Technically, the constant function $f(x) = l$ would work too... The right function depends on the exact problem you are trying to solve. – Johannes Kloos Jun 15 '12 at 15:49
@Andrew no, I'm trying to perform a function on x so that it falls within the interval. – scottm Jun 15 '12 at 15:55
@scottm: I have edited your post. Does it match what you are trying to ask? – Cameron Buie Jun 15 '12 at 16:30
@CameronBuie yes, thanks – scottm Jun 15 '12 at 16:31

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.
As an easier-to-compute alternative to caozhu's arctan solution: $$x\mapsto h - \frac{h-l}{x+1}$$
Say $l=0, h=1$ then $0\le \sin^2(x)\le 1$.
And of course, you can scale and shift this for arbitrary $l,h$.... – Cameron Buie Jun 15 '12 at 16:27