Determine if Odd, Even, or Neither 1/(9x^3)

$\frac{1}{9 x^{3}}$

Step 1

Write $\frac{1}{9 x^{3}}$ as a function.

$f (x) = \frac{1}{9 x^{3}}$

Step 2

Find $f (- x)$ .

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

$f (- x) = \frac{1}{9 {(- x)}^{3}}$

Simplify the denominator.

Apply the product rule to $- x$ .

$f (- x) = \frac{1}{9 {(- 1)}^{3} x^{3}}$

Raise $- 1$ to the power of $3$ .

$f (- x) = \frac{1}{9 \cdot (- 1 x^{3})}$

Combine exponents.

Factor out negative.

$f (- x) = \frac{1}{- (9 x^{3})}$

Multiply $9$ by $- 1$ .

$f (- x) = \frac{1}{- 9 x^{3}}$

Move the negative in front of the fraction.

$f (- x) = - \frac{1}{9 x^{3}}$

Step 3

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

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

Since $- \frac{1}{9 x^{3}}$ $\neq$ $\frac{1}{9 x^{3}}$ , the function is not even.

The function is not even

Step 4

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

Multiply $- 1$ by $\frac{1}{9 x^{3}}$ .

$- f (x) = - \frac{1}{9 x^{3}}$

Since $- \frac{1}{9 x^{3}} = - \frac{1}{9 x^{3}}$ , the function is odd.

The function is odd

Step 5