## MA101.5 Odd and even functions

A function is said to be an even function if,

$\forall x \in D(f), f(x) = f(-x)$

and to be an odd function if,

$\forall x \in D(f), f(x) = -f(x)$.

Note that these definitions require D(f) to lie symmetrically about x=0 (ie $x \in D(f) \Rightarrow -x \in D(f)$ ). Clearly even functions have graphs that are symmetrical about the y-axis, whilst for odd functions the origin is a centre of rotational symmetry.

Examples are $x^2, \; |x|$ for even, and $x^3, \; sin(x)$ for odd. Most functions are neither odd nor even. However every function (subject to having a symmetrical domain) may be expressed as the sum of an odd and even function since $f(x) = \frac{1}{2}\{f(x) +f(-x)\} + \frac{1}{2}\{f(x) - f(-x)\}$, which is even + odd.

Example

\begin{aligned} f(x) &= e^x sin(x) \\ &= \frac{1}{2}\{e^x sin(x) + e^{-x} sin(-x)\} + \frac{1}{2}\{e^x sin(x) - e^{-x} sin(-x)\} \\&= sin(x) \frac{e^x-e^{-x}}{2} + sin(x) \frac {e^x + e^{-x}}{2} \\&= sin(x)sinh(x) + sin(x)cosh(x) \end{aligned}

ie. even + odd.