Symmetry "decomposition" of functions

First, sorry for the incorrect terminology, I'm not sure what else to call this.

I know it is possible to use the group $\displaystyle C_2$ to show that a function f(x) breaks down into the sum of an even and an odd function. Since I can't find my notes or reference on the subject my questions are

1) How do you do this using $\displaystyle C_2$

and

2) How do you generalize this process? As an example, how can you do this with, say, $\displaystyle D_3$?

Thanks!

-Dan