Even and Odd Functions

Functions can be categorised into one of three types - odd, even and neither.
Odd functions have the property that  
\[f(x)=-f(-x)\]
. Each minus sign indicate reflection so that  
\[f(-x)\]
  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  
\[g(x)=f(x)+f(-x)\]
.