What are even and odd functions?

Nov 12, 2014

Even & Odd Functions

A function $f \left(x\right)$ is said to be $\left\{\begin{matrix}\text{even if "f(-x)=f(x) \\ "odd if } f \left(- x\right) = - f \left(x\right)\end{matrix}\right.$

Note that the graph of an even function is symmetric about the $y$-axis, and the graph of an odd function is symmetric about the origin.

Examples

$f \left(x\right) = {x}^{4} + 3 {x}^{2} + 5$ is an even function since

$f \left(- x\right) = {\left(- x\right)}^{4} + {\left(- x\right)}^{2} + 5 = {x}^{4} + 3 {x}^{2} + 5 = f \left(x\right)$

$g \left(x\right) = {x}^{5} - {x}^{3} + 2 x$ is an odd function since

$g \left(- x\right) = {\left(- x\right)}^{5} - {\left(- x\right)}^{3} + 2 \left(- x\right) = - {x}^{5} + {x}^{3} - 2 x = - f \left(x\right)$

I hope that this was helpful.