## Even and Odd Functions

Functions can be categorised into one of three types - odd, even and neither.
Odd functions have the property that
$f(x)=-f(-x)$
$f(-x)$
is the reflection of
$f(x)$
in the
$y$
axis, and
$-f(x)$
is the reflection of
$f(x)$
in the
$x$
axis. The composite of these two reflections is a rotation of 180 degrees about the origin.
Examples of odd functions -
$sin x, \; x^3, \;$
.
Even functions have the property that
$f(x)=f(-x)$
meaning that the graph of
$f(x)$
has reflectional symmetry in the
$y$
axis. Every function that
$f(x)$
that can be written as a function of
$x^2$
is even e.g.
$f(x)=2-x^2$
.
Examples of even functions -
$cosx, \; sin^2 x$
.
An even function can be construction from an odd function.If
$f(x)$
is odd then
$f(x^2)$
and
$(f(x))^2$
are both even.
A function is neither odd nor even if neither of the above conditions holds for a function to be either odd or even.
Examples of functions that are neither odd nor even -
$e^x, \; (x-1)^2$
.
An odd function
$g(x)$
can be constructed from a function
$f(x)$
that is neither odd nor even -
$g(x)=f(x)-f(-x)$
, and an even function
$g(x)$
can be constructed using
$g(x)=f(x)+f(-x)$
.