D14. Flip Functions

Contributed by Yagub Aliyev, ADA University, Azerbaijan

Call a function f mapping the real numbers to themselves a sign-flipping function if for all x , y R , ( x y ) ( f ( x ) + f ( y ) ) = ( x + y ) f ( x y ) . For example, the identity function i ( x ) = x is a sign-flipping function because ( x y ) ( x + y ) = ( x + y ) ( x y ) by commutativity of multiplication. Are there any others? Find all sign-flipping functions.

This problem originally appeared in the Prisoner’s Dilemma in the 2023 Fall issue of the PMP Newsletter. Solutions will be accepted through 2024 Mar 1.