Guided Example #1 - Even and Odd Functions

Is $f (x) = \frac{x}{x^{2} - 1}$ Even or Odd or neither?

a. Even
b. Odd
c. Neither

The correct answer is B. To see why, view each of the steps below.

Let's substitute x with $- x$ :
Put in $-x : f (- x) = \frac{(- x)}{{(- x)}^{2} - 1}$

Simplify:
$\frac{- x}{x^{2} - 1} = - f (x)$

So $f (- x) = - f (x)$ and hence $f (x)$ is an Odd Function.

Let's substitute x with $- x$ ::

Put in $-x : f (- x) = \frac{(- x)}{{(- x)}^{2} - 1}$

Simplify:

$\frac{- x}{x^{2} - 1} = - f (x)$

So $f (- x) = - f (x)$ and hence $f (x)$ is an Odd Function.