Precalculus Examples

$f (x) = \sqrt[3]{x} - x^{3}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = \sqrt[3]{- x} - {(- x)}^{3}$

Step 1.2

Remove parentheses.

$f (- x) = \sqrt[3]{- x} - {(- x)}^{3}$

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite $- x$ as ${({(- 1)}^{3})}^{3} x$ .

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Step 1.3.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f (- x) = \sqrt[3]{{(- 1)}^{3} x} - {(- x)}^{3}$

Step 1.3.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$f (- x) = \sqrt[3]{{({(- 1)}^{3})}^{3} x} - {(- x)}^{3}$

Step 1.3.2

Pull terms out from under the radical.

$f (- x) = {(- 1)}^{3} \sqrt[3]{x} - {(- x)}^{3}$

Step 1.3.3

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

$f (- x) = - \sqrt[3]{x} - {(- x)}^{3}$

Step 1.3.4

Apply the product rule to $- x$ .

$f (- x) = - \sqrt[3]{x} - ({(- 1)}^{3} x^{3})$

Step 1.3.5

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 1.3.5.1

Move ${(- 1)}^{3}$ .

$f (- x) = - \sqrt[3]{x} + {(- 1)}^{3} \cdot (- 1 x^{3})$

Step 1.3.5.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 1.3.5.2.1

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

$f (- x) = - \sqrt[3]{x} + {(- 1)}^{3} \cdot ((- 1) x^{3})$

Step 1.3.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = - \sqrt[3]{x} + {(- 1)}^{3 + 1} x^{3}$

Step 1.3.5.3

Add $3$ and $1$ .

$f (- x) = - \sqrt[3]{x} + {(- 1)}^{4} x^{3}$

Step 1.3.6

Raise $- 1$ to the power of $4$ .

$f (- x) = - \sqrt[3]{x} + 1 x^{3}$

Step 1.3.7

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

$f (- x) = - \sqrt[3]{x} + x^{3}$

Step 2

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

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Step 2.1

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

Step 2.2

Since $- \sqrt[3]{x} + x^{3}$ $\neq$ $\sqrt[3]{x} - x^{3}$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

Find $- f (x)$ .

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Step 3.1.1

Multiply $\sqrt[3]{x} - x^{3}$ by $- 1$ .

$- f (x) = - (\sqrt[3]{x} - x^{3})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - \sqrt[3]{x} + x^{3}$

Step 3.2

Since $- \sqrt[3]{x} + x^{3} = - \sqrt[3]{x} + x^{3}$ , the function is odd.

The function is odd

Step 4