Let X be the set of all reals except 0 and 1. Find all real valued functions f(x) on X which satisfy f(x) + f(1 - 1/x) = 1 + x for all x in X.

**Solution**

The trick is that x → 1 - 1/x → 1/(1-x) → x. Thus we have:

f(x) + f(1 - 1/x) = 1 + x (1);

f(1 - 1/x) + f(1/(1-x) ) = 2 - 1/x (2);

f(1/(1-x) ) + f(x) = 1 + 1/(1-x) (3).

Now (1) - (2) + (3) gives 2 f(x) = x + 1/x + 1/(1-x) or f(x) = (x^{3} - x^{2} - 1) / (2x^{2} - 2x). It is easily checked that this does indeed satisfy the relation in the question.

© John Scholes

jscholes@kalva.demon.co.uk

27 Jan 2001