# 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:

• Group theory

What kinds of sym­me­try can an ob­ject have?

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