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In a problem I have been working on for a while now I have found that I want to find the function satisfying the functional relation

f(x)

^{n}f(a - x) = 1

for n = 1 I believe I have proven that f(x) = x/(a - x). On this page is an answer I do not quite understand. One of the prerequisits for f(x) in my problem was that f(0) = 0, f(a/2) = 1 and f(a) = ∞. I have difficulties seeing that the answer provided will satisfy f(0) = 0.

Apart from this, I have made no progress what so ever in adressing the case where n ≠ 1. In particular I have failed to find solutions for when n = 2.

The problem is this: I want to make a scale mapping where f(x) represents the value and x represents the position of the scale such that the graph of 1/x

^{n}is a straight line from positions (0, a) through ((a/2, a/2) to (a, 0), representing the values (0, ∞) through (1, 1) to (∞, 0).

How can I go about finding f(x) for n ≠ 1?

Oh, and I don't have any problem with not getting to solve it myself - I you know the answer, please just tell me!