Even and odd functions

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

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 \(\mathbb{R} \to \mathbb{R}\): in other words, each element of the group describes a function of type

meaning that it takes as input a function \(\mathbb{R} \to \mathbb{R}\) and returns as output another function \(\mathbb{R} \to \mathbb{R}\).

\(C_2\) has two elements which we’ll call \(1\) and \(-1\). \(1\) is the identity element: it acts on functions by sending a function \(f(x)\) to the same function \(f(x)\) again.\(-1\) 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 \(f(-(-x)) = f(x)\).

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:

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.}