Assume f(x) = ax

Then a*ax = 12x - ax

a = 3 or -4

but f(x) = -4x does not map positives to positives. Thus, a must = 3.

Assume f(x) = b/x

Then x = 12x - b/x

There is no value of b such that this equation works for all x. Therefore, f(x) is not of the form b/x.

I think that the same is true if f(x) = ax^b where b <> 1

I have not exhausted all functions that map R^{+}**→** R^{+} ,

but I suspect that the only solution is f(x) = 3x