Determine if Odd, Even, or Neither 1/(x+9)

$\frac{1}{x + 9}$

Step 1

Write $\frac{1}{x + 9}$ as a function.

$f (x) = \frac{1}{x + 9}$

Step 2

Find $f (- x)$ .

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

$f (- x) = \frac{1}{(- x) + 9}$

Factor $- 1$ out of $- x$ .

$f (- x) = \frac{1}{- (x) + 9}$

Rewrite $9$ as $- 1 (- 9)$ .

$f (- x) = \frac{1}{- (x) - 1 \cdot - 9}$

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$f (- x) = \frac{1}{- (x - 9)}$

Rewrite negatives.

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$f (- x) = \frac{1}{- 1 (x - 9)}$

Move the negative in front of the fraction.

$f (- x) = - \frac{1}{x - 9}$

Step 3

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

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

$- \frac{1}{x - 9}$ $\neq$ $\frac{1}{x + 9}$

Since $- \frac{1}{x - 9}$ $\neq$ $\frac{1}{x + 9}$ , the function is not even.

The function is not even

Step 4

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

Multiply $- 1$ by $\frac{1}{x + 9}$ .

$- f (x) = - \frac{1}{x + 9}$

Since $- \frac{1}{x - 9}$ $\neq$ $- \frac{1}{x + 9}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6