Odd and Even Functions

Odd Functions

Definition

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 $\sin x$.

Even Functions

Definition

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 $\cos x$.

Properties

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.

Periodic Functions

Definition

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.

Workbooks

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