Precalculus Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Apply the product rule to $- x$ .

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

Step 2.2.2

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

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

Step 2.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.3.1

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

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

Step 2.3.2

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

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

Step 3.2

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

The function is not even

Step 4

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

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Step 4.1

Multiply $- (- \frac{1}{x^{3}})$ .

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Step 4.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2

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

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

Step 4.2

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

The function is odd

Step 5