Help with solutions to the functional equations f(x) = f(x^2 + 2012)
Hey guys, I've never solved problems of these sorts and I'm not quite sure where to begin. The problem is given the following equation
where the domain of f is defined to be the reals, how do I find the maximum of the expression $f(a) - f(b)$ for some a,b in reals?
I thought about finding a class of solutions to f(x) = f(x^2 + 2012), namely given the relation , and the function g such that , one naive equivalence class containing x, , is just the sequence of . However, because may have solutions that are not in , we have to also include those within . At this point, I'm just a bit confused by the algebra involved and what I should do next. Any/all help would be greatly appreciated.
Thanks guys for looking at this
Re: Help with solutions to the functional equations f(x) = f(x^2 + 2012)
Okay, I think I've figured it out
First: Show that each of the equivalence classes are distinct. Because g(x) is strictly increasing, then it must be the case that , then so must the sequence x, g(x), g(g(x)) be strictly increasing, meaning that , which means that
Hence, we only need to look at the values of f(x) within the domain of (0,2012). The answer is then just