# Functional equation analysis of a continuous function

I got this question in a test of mine and it completely took me by surprise. I just can't "begin" to solve it.

Let a continuous function $f(x) : \mathbb R \rightarrow \mathbb R$ be defined such that it satisfies the relation $$f(x) + f(x + 2y) + 3xy = 2f(2y – x) + 2y^2$$ for all $x, y\in R$. Comment whether the function is odd , into , one-one , and invertible.

I know what these terms necessarily mean in general , but I'm not sure how to check for them here.

I'm always having this general trend of getting troubled by functional equations and the ways of either deriving them or solving them. Apart from this question , it would be great help if anyone could suggest me some text or video where I could learn this topic better. ( The functional equations)

• This isn't a standard question about functional equations... I believe you should solve it by "guessing" values for $x,y$ (e.g for $y=0$ you get that $2f(x)=2f(-x)$). – Yanko Dec 14 '17 at 19:06
• @yanko You mean $2 f(x) = 2 f(-x)$. $f(x) = 2 f(-x)$ would make things very easy! – Robert Israel Dec 14 '17 at 19:07
• @yanko You mean even. $x=0$ is even more useful. – Robert Israel Dec 14 '17 at 19:11
• If you put the value Robert suggested you will have a good idea what the function is. Try to put $x=0$. – Yanko Dec 14 '17 at 19:12
• Not really "random" values, rather ones that are likely to produce simpler equations (involving fewer different values of $f$). $x=0$ and $y=0$ are often good choices. – Robert Israel Dec 14 '17 at 19:15