We can think about evenness and oddness in terms of group theory as follows\. There is a group called the cyclic group $~$C\_2$~$ of order $~$2$~$ acting on all functions\. It has elements which we might call $~$1$~$ and $~$-1$~$\. $~$1$~$ is the identity element: it sends a function to itself\. $~$-1$~$ sends a function $~$f(x)$~$ to the function $~$f(-x)$~$, which visually corresponds to reflecting the graph of $~$f(x)$~$ through the y\-axis\. The group multiplication is what the names of the group elements suggests, and in particular $~$(-1) \\times (-1) \= 1$~$, which corresponds to the fact that $~$f(-(-x)) \= f(x)$~$\.

Would be cool to have an image of an example graph here.