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.