Math Club - Fun & Games

Polynomial Images - UNSOLVED

Question:

Polynomial Images

Let f and g be polynomials with rational coefficients, and let F and G denote the sets of values of f and g at rational numbers, respectively.  Prove that F = G holds if and only if f(x) = g(ax + b) for some suitable rational numbers a != 0 and b. 

Note that one direction is trivial.  The other, however, is not.