Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)
Problem
Source: Tournament of towns, Senior A-Level paper, Fall 2004
Tags: function, geometry, algebra solved, algebra
Myth
25.12.2004 16:57
This problem was solved before (by me ). Author of topic was Moubinool.
darij grinberg
25.12.2004 17:58
CORRECTED VERSION OF THE POST! A related problem is problem 1 of the Bundeswettbewerb Mathematik 2003, 2nd round (actually this was the easiest problem of this 2nd round; I had solved it even before I did the geometry problem): Let $f:\mathbb{R}\to\mathbb{R}$ be a function whose plot has at least two centers of symmetry. Prove that the function f can be written as the sum of a linear and a periodic function. Once you have solved this problem, the Tournament of Towns problem becomes clear to you. Darij
Myth
25.12.2004 18:01
??? BTW, can you find my post about this problem?
billzhao
25.12.2004 22:55
It seems that it does disappeared...
Myth
25.12.2004 23:13
darij grinberg wrote: A related problem is problem 1 of the Bundeswettbewerb Mathematik 2003, 2nd round (actually this was the easiest problem of this 2nd round; I had solved it even before I did the geometry problem): Let $f:\mathbb{R}\to\mathbb{R}$ be a function whose plot has at least one center of symmetry. Prove that the function f can be written as the sum of a linear and a periodic function. Of course, this problem makes the idea of the Tournament of Towns problem clear. Darij But it is impossible to write $x^3$ as sum of linear and periodic function.
darij grinberg
25.12.2004 23:33
Sorry for writing nonsense. I've now corrected my post above. Darij
Myth
25.12.2004 23:34
darij grinberg wrote: CORRECTED VERSION OF THE POST! A related problem is problem 1 of the Bundeswettbewerb Mathematik 2003, 2nd round (actually this was the easiest problem of this 2nd round; I had solved it even before I did the geometry problem): Let $f:\mathbb{R}\to\mathbb{R}$ be a function whose plot has at least two centers of symmetry. Prove that the function f can be written as the sum of a linear and a periodic function. It is just obvious.