# How to prove Big-O when $F(N)$ is even or odd

If I'm given a function $$f(n)$$ which is for example $$4n+1$$ when even and $$3n^2+2$$ when odd and I have to prove or disprove $$f(n)$$ is $$\mathcal O( n^2 )$$. Do I have to do $$f(n) < c n^2$$ for all $$n > n0$$ for both expressions or do I just select the $$3n^2+2$$ expression?

you have to do for both expressions. But in this case, one expression is strictly smaller than the other one, so you can do it just for the $$3n^2+2$$ and say $$4n+1<3n^2+2$$ and therefore also $$4n+1.