Even and odd functions

Recall that a function $f:R\to R$ is even if $f(-x)=f\left(x\right)$, and odd if $f(-x)=-f\left(x\right)$. A typical example of an even function is $f\left(x\right)=x^2$ or $f\left(x\right)=\cos x$, while a typical example of an odd function is $f\left(x\right)=x^3$ or $f\left(x\right)=\sin x$.

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 the set of functions $R\to R$: in other words, each element of the group describes a function of type

$$(R\to R)\to (R\to R)$$

meaning that it takes as input a function $R\to R$ and returns as output another function $R\to R$.

$C_2$ has two elements which we'll call $1$ and $-1$. $1$ is the identity element: it acts on functions by sending a function $f\left(x\right)$ to the same function $f\left(x\right)$ again. $-1$ sends a function $f\left(x\right)$ to the function $f(-x)$, which visually corresponds to reflecting the graph of $f\left(x\right)$ 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 $f(-(-x\left)\right)=f\left(x\right)$.

Any time a group $G$ acts on a set $X$, it's interesting to ask what elements are invariant under that group action. Here the invariants of functions under the action of $C_2$ above are the even functions, and they form a Subspace of the vector space of all functions.

It turns out that every function is uniquely the sum of an even and an odd function, as follows:

$$f\left(x\right)=\underset{\text{even}}{\underset{}{\frac{f\left(x\right)+f(-x)}{2}}}+\underset{\text{odd}}{\underset{}{\frac{f\left(x\right)-f(-x)}{2}}}.$$

This is a special case of various more general facts in representation theory, and in particular can be thought of as the simplest case of the discrete Fourier transform, which in turn is a toy model of the theory of Fourier series and the Fourier transform.

It's also interesting to observe that the cyclic group $C_2$ shows up in lots of other places in mathematics as well. For example, it is also the group describing how even and odd numbers add¹ (where even corresponds to $1$ and odd corresponds to $-1$); this is the simplest case of modular arithmetic.

¹_{That is: an even plus an even make an even, an odd plus an odd make an even, and an even plus an odd make an odd.}