Determine if Odd, Even, or Neither x/(x^2-5)

$\frac{x}{x^{2} - 5}$

Step 1

Write $\frac{x}{x^{2} - 5}$ as a function.

$f (x) = \frac{x}{x^{2} - 5}$

Step 2

Find $f (- x)$ .

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \frac{- x}{{(- x)}^{2} - 5}$

Simplify the denominator.

Apply the product rule to $- x$ .

$f (- x) = \frac{- x}{{(- 1)}^{2} x^{2} - 5}$

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{- x}{1 x^{2} - 5}$

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{- x}{x^{2} - 5}$

Move the negative in front of the fraction.

$f (- x) = - \frac{x}{x^{2} - 5}$

Step 3

A function is even if $f (- x) = f (x)$ .

Check if $f (- x) = f (x)$ .

Since $- \frac{x}{x^{2} - 5}$ $\neq$ $\frac{x}{x^{2} - 5}$ , the function is not even.

The function is not even

Step 4

A function is odd if $f (- x) = - f (x)$ .

Multiply $- 1$ by $\frac{x}{x^{2} - 5}$ .

$- f (x) = - \frac{x}{x^{2} - 5}$

Since $- \frac{x}{x^{2} - 5} = - \frac{x}{x^{2} - 5}$ , the function is odd.

The function is odd

Step 5