# 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

$$(\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R})$$

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:

$$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd}}.$$

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.

1That 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.

Parents:

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