Let f be an involutive (f(f(x)) = x) function from R>0  to R>0 such that f(1) = c, (then f(c) = 1).

I have obtained the following f(f(x)f(y))/xy = f(xy)/f(x)f(y), by setting f(y) = 1, we get f(xy) = f(x)/y. How to prove that this is valid for all y>0?

I have used the injectivity of f. Note that f is a bijection, and f(y) = 1 for y = c.

The function in the question is f(x) = c/x.