Precalculus Examples

$\sqrt[3]{x}$

Step 1

Write $\sqrt[3]{x}$ as a function.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Remove parentheses.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

Pull terms out from under the radical.

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

Step 2.5

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

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

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 $- \sqrt[3]{x}$ $\neq$ $\sqrt[3]{x}$ , 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 $- 1$ by $\sqrt[3]{x}$ .

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

Step 4.2

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

The function is odd

Step 5