## Even and Odd Functions

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 \[g(x)=f(x)+f(-x)\]

.