# How do you tell whether a function is even, odd or neither?

Sep 13, 2015

To determine this, plug $- x$ in for $x$ and see what happens.

#### Explanation:

The first step is to replace $x$ with –x. In other words, calculate $f \left(- x\right)$.

If the function doesn't change (i.e. $f \left(- x\right) = f \left(x\right)$. then it is even. For instance, $f \left(x\right) = {x}^{2}$ is even because #f(-x) = (-x)^2 = x^2.

If the function is the reverse of what it was originally (i.e. $f \left(- x\right) = - f \left(x\right)$, then it is odd. For instance, $f \left(x\right) = x$ is odd because $f \left(- x\right) = - x = - f \left(x\right)$.

If anything else happens, the function is neither even nor odd. For instance, $f \left(x\right) = {x}^{2} + x$ is neither even nor odd because $f \left(- x\right) = {\left(- x\right)}^{2} + - x = {x}^{2} - x$, and that is neither the function we started with, nor the reverse.