## Even and Odd Functions

Functions can be categorised into one of three types - odd, even and neither.
Odd functions have the property that  {jatex options:inline}f(x)=-f(-x){/jatex}. Each minus sign indicate reflection so that  {jatex options:inline}f(-x){/jatex}  is the reflection of  {jatex options:inline}f(x){/jatex}  in the  {jatex options:inline}y{/jatex}  axis, and  {jatex options:inline}-f(x){/jatex}  is the reflection of  {jatex options:inline}f(x){/jatex}  in the  {jatex options:inline}x{/jatex}  axis. The composite of these two reflections is a rotation of 180 degrees about the origin.
Examples of odd functions -  {jatex options:inline}sin x, \; x^3, \; {/jatex}.
Even functions have the property that  {jatex options:inline}f(x)=f(-x){/jatex}  meaning that the graph of  {jatex options:inline}f(x){/jatex}  has reflectional symmetry in the  {jatex options:inline}y{/jatex}  axis. Every function that  {jatex options:inline}f(x){/jatex}  that can be written as a function of  {jatex options:inline}x^2{/jatex}  is even e.g.  {jatex options:inline}f(x)=2-x^2{/jatex}.
Examples of even functions -  {jatex options:inline}cosx, \; sin^2 x{/jatex}.
An even function can be construction from an odd function.If  {jatex options:inline}f(x){/jatex}  is odd then  {jatex options:inline}f(x^2){/jatex}  and  {jatex options:inline}(f(x))^2{/jatex}  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 -  {jatex options:inline}e^x, \; (x-1)^2{/jatex}.
An odd function  {jatex options:inline}g(x){/jatex}  can be constructed from a function  {jatex options:inline}f(x){/jatex}  that is neither odd nor even -  {jatex options:inline}g(x)=f(x)-f(-x){/jatex}, and an even function  {jatex options:inline}g(x){/jatex}  can be constructed using  {jatex options:inline}g(x)=f(x)+f(-x){/jatex}.