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