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Let $f(x)$ be a function on the positive real line. Suppose that for all nonnegative reals $A$, $Ae^{x}$ intersects $f(x)$ exactly once. Is there a simple description of the set of functions $f$ which satisfy this property?

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up vote 5 down vote accepted

I presume that by a function on the positive real line you mean a function from the positive real line to the positive real line.

The condition is equivalent to $f(x)\mathrm e^{-x}$ taking every value $A$ exactly once. If $f$ is continuous, this is equivalent to $f(x)\mathrm e^{-x}$ either increasing monotonically with $\lim_{x\to0}f(x)=0$ and $\lim_{x\to\infty}f(x)=\infty$, or decreasing monotonically with $\lim_{x\to0}f(x)=\infty$ and $\lim_{x\to\infty}f(x)=0$.

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