Precalculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Multiply $\frac{1}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$f (x) = \frac{1}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Step 1.2

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.1

Multiply $\frac{1}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 1.2.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 1.2.3

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

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 1.2.4

Add $1$ and $2$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

Step 1.2.5

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

Tap for more steps...

Step 1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

Step 1.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 1.2.5.3

Combine $\frac{1}{3}$ and $3$ .

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 1.2.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.5.4.1

Cancel the common factor.

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 1.2.5.4.2

Rewrite the expression.

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{x}$

Step 1.2.5.5

Simplify.

$f (x) = \frac{{\sqrt[3]{x}}^{2}}{x}$

Step 1.3

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f (x) = \frac{\sqrt[3]{x^{2}}}{x}$

Step 2

Find $f (- x)$ .

Tap for more steps...

Step 2.1

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

$f (- x) = \frac{\sqrt[3]{{(- x)}^{2}}}{- x}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{\sqrt[3]{{(- 1)}^{2} x^{2}}}{- x}$

Step 2.2.2

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{\sqrt[3]{1 x^{2}}}{- x}$

Step 2.2.3

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{\sqrt[3]{x^{2}}}{- x}$

Step 2.3

Move the negative in front of the fraction.

$f (- x) = - \frac{\sqrt[3]{x^{2}}}{x}$

Step 3

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

Tap for more steps...

Step 3.1

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

Step 3.2

Since $- \frac{\sqrt[3]{x^{2}}}{x}$ $\neq$ $\frac{\sqrt[3]{x^{2}}}{x}$ , the function is not even.

The function is not even

Step 4

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

Tap for more steps...

Step 4.1

Multiply $- 1$ by $\frac{\sqrt[3]{x^{2}}}{x}$ .

$- f (x) = - \frac{\sqrt[3]{x^{2}}}{x}$

Step 4.2

Since $- \frac{\sqrt[3]{x^{2}}}{x} = - \frac{\sqrt[3]{x^{2}}}{x}$ , the function is odd.

The function is odd

Step 5