The problem involves a function from R to R defined by the equation x + f(x) = f(f(x)). The original poster seeks to understand why the condition f(x) = f(y) leads to the conclusion that x = y, particularly in the context of finding solutions to f(f(x)) = 0.
Several participants are actively engaging with the original poster's question, offering different perspectives on the implications of the function's definitions. Some have provided reasoning related to substituting values into the original equation, while others are questioning the assumptions made about the relationships between f(x) and f(f(x)). There is no explicit consensus, but the discussion is progressing with various interpretations being explored.
Participants note that the original problem may involve constraints related to the definitions of the function and the nature of injectivity. There is an emphasis on understanding the implications of the function's behavior rather than arriving at a definitive solution.
Hypersphere said:Have you tried comparing f(f(x)) and f(f(y))?
HallsofIvy said:No, he meant nothing of the sort. And f(f(x))= 0 does NOT imply f(f(y))= 0.
Wiz14 said:but why does f(x) = f(y) imply that x = y?
Office_Shredder said:If f(x) = f(y), then f(f(x)) = f(f(y)). Use this to prove that x=y. This has nothing to do with anything equaling zero
Wiz14 said:If f(y) = f(x) then substituting f(y) into the original equation gives f(f(y) = f(x) + y = f(x) + x, then subtracting the f(x) from the last equation gives x = y, is this correct? Thanks for the help.