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:
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.
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