A function $f$ is *odd* if the following equation holds for all $x$ and $-x$ in the domain of $f$: \[-f(x) = f(-x)\] Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after a rotation of $180^{\circ}$ about the origin.

Examples of odd functions include $x$, $x^3$ and $\sin x$.

A function $f$ is *even* if the following equation holds for all $x$ and $-x$ in the domain of $f$: \[f(x) = f(-x)\] Geometrically, the graph of an even function is symmetric with respect to the $y$-axis, meaning that its graph remains unchanged after reflection about the $y$-axis.

Examples of even functions include $\vert x \vert$, $x^2$ and $\cos x$.

Some basic properties of odd and even functions are:

- The only function whose domain is all real numbers which is
**both**odd and even, is the constant function which is identically zero, $f(x)=0$. - The sum of two even functions is even, and the sum of two odd functions is odd.
- The difference of two even functions is even, and the difference of two odd functions is odd.
- The product of two even functions is even, and the product of two odd functions is even.
- The product of an even function and an odd function is an odd function.
- The quotient of two even functions is even, and the quotient of two odd functions is even.
- The quotient of an even function and an odd function is an odd function.
- The derivative of an even function is odd, and the derivative of an odd function is even.
- The composition of two even functions is even, and the composition of two odd functions is odd.
- The composition of an even function and an odd function is even.

**Note**: the sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain.

A *periodic function* is a function that repeats itself in regular intervals or *periods*. A function $f$ is said to be periodic with period $P$ if: \[f(x+P)=f(x)\] for all values of $x$ and where $P$ is a nonzero constant.

Periodic functions are used to describe oscillations and waves, and the most important periodic functions are the trigonometric functions. Any function which is not periodic is called *aperiodic*.

**Example**: The sine function is periodic with period $2\pi$ since $\sin(x+2\pi)=\sin x$ for all values of $x$. The function repeats itself on intervals of length $2\pi$ which can also be clearly seen from a graph.

These workbooks produced by HELM are good revision aids, containing key points for revision and many worked examples.

- Periodic functions
- Even and odd functions
- Characterising functions including worked examples on periodic, odd and even functions.