Precalculus Examples

g (x) = \sqrt[3]{x} + 4

$g (x) = \sqrt[3]{x} + 4$ ;

- 5 \leq x \leq 0

$- 5 \leq x \leq 0$

Step 1

Write

g (x) = \sqrt[3]{x} + 4

$g (x) = \sqrt[3]{x} + 4$ as an equation.

y = \sqrt[3]{x} + 4

$y = \sqrt[3]{x} + 4$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in

y

$y$ values of the two points divided by the change in

x

$x$ values of the two points.

\frac{f (0) - f (- 5)}{(0) - (- 5)}

$\frac{f (0) - f (- 5)}{(0) - (- 5)}$

Step 2.2

Substitute the equation

y = \sqrt[3]{x} + 4

$y = \sqrt[3]{x} + 4$ for

f (0)

$f (0)$ and

f (- 5)

$f (- 5)$ , replacing

x

$x$ in the function with the corresponding

x

$x$ value.

\frac{(\sqrt[3]{0} + 4) - (\sqrt[3]{- 5} + 4)}{(0) - (- 5)}

$\frac{(\sqrt[3]{0} + 4) - (\sqrt[3]{- 5} + 4)}{(0) - (- 5)}$

\frac{(\sqrt[3]{0} + 4) - (\sqrt[3]{- 5} + 4)}{(0) - (- 5)}

$\frac{(\sqrt[3]{0} + 4) - (\sqrt[3]{- 5} + 4)}{(0) - (- 5)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite

0

$0$ as

0^{3}

$0^{3}$ .

\frac{\sqrt[3]{0^{3}} + 4 - (\sqrt[3]{- 5} + 4)}{0 - (- 5)}

$\frac{\sqrt[3]{0^{3}} + 4 - (\sqrt[3]{- 5} + 4)}{0 - (- 5)}$

Step 3.1.2

Pull terms out from under the radical, assuming real numbers.

\frac{0 + 4 - (\sqrt[3]{- 5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (\sqrt[3]{- 5} + 4)}{0 - (- 5)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Rewrite

- 5

$- 5$ as

{(- 1)}^{3} \cdot 5

${(- 1)}^{3} \cdot 5$ .

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Step 3.1.3.1.1

Rewrite

- 5

$- 5$ as

- 1 (5)

$- 1 (5)$ .

\frac{0 + 4 - (\sqrt[3]{- 1 (5)} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (\sqrt[3]{- 1 (5)} + 4)}{0 - (- 5)}$

Step 3.1.3.1.2

Rewrite

- 1

$- 1$ as

{(- 1)}^{3}

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

\frac{0 + 4 - (\sqrt[3]{{(- 1)}^{3} \cdot 5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (\sqrt[3]{{(- 1)}^{3} \cdot 5} + 4)}{0 - (- 5)}$

\frac{0 + 4 - (\sqrt[3]{{(- 1)}^{3} \cdot 5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (\sqrt[3]{{(- 1)}^{3} \cdot 5} + 4)}{0 - (- 5)}$

Step 3.1.3.2

Pull terms out from under the radical.

\frac{0 + 4 - (- 1 \sqrt[3]{5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (- 1 \sqrt[3]{5} + 4)}{0 - (- 5)}$

Step 3.1.3.3

Rewrite

- 1 \sqrt[3]{5}

$- 1 \sqrt[3]{5}$ as

- \sqrt[3]{5}

$- \sqrt[3]{5}$ .

\frac{0 + 4 - (- \sqrt[3]{5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (- \sqrt[3]{5} + 4)}{0 - (- 5)}$

\frac{0 + 4 - (- \sqrt[3]{5} + 4)}{0 - (- 5)}

$\frac{0 + 4 - (- \sqrt[3]{5} + 4)}{0 - (- 5)}$

Step 3.1.4

Apply the distributive property.

\frac{0 + 4 - - \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}

$\frac{0 + 4 - - \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}$

Step 3.1.5

Multiply

- - \sqrt[3]{5}

$- - \sqrt[3]{5}$ .

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Step 3.1.5.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{0 + 4 + 1 \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}

$\frac{0 + 4 + 1 \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}$

Step 3.1.5.2

Multiply

\sqrt[3]{5}

$\sqrt[3]{5}$ by

1

$1$ .

\frac{0 + 4 + \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}

$\frac{0 + 4 + \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}$

\frac{0 + 4 + \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}

$\frac{0 + 4 + \sqrt[3]{5} - 1 \cdot 4}{0 - (- 5)}$

Step 3.1.6

Multiply

- 1

$- 1$ by

4

$4$ .

\frac{0 + 4 + \sqrt[3]{5} - 4}{0 - (- 5)}

$\frac{0 + 4 + \sqrt[3]{5} - 4}{0 - (- 5)}$

Step 3.1.7

Add

0

$0$ and

4

$4$ .

\frac{4 + \sqrt[3]{5} - 4}{0 - (- 5)}

$\frac{4 + \sqrt[3]{5} - 4}{0 - (- 5)}$

Step 3.1.8

Subtract

4

$4$ from

4

$4$ .

\frac{0 + \sqrt[3]{5}}{0 - (- 5)}

$\frac{0 + \sqrt[3]{5}}{0 - (- 5)}$

Step 3.1.9

Add

0

$0$ and

\sqrt[3]{5}

$\sqrt[3]{5}$ .

\frac{\sqrt[3]{5}}{0 - (- 5)}

$\frac{\sqrt[3]{5}}{0 - (- 5)}$

\frac{\sqrt[3]{5}}{0 - (- 5)}

$\frac{\sqrt[3]{5}}{0 - (- 5)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply

- 1

$- 1$ by

- 5

$- 5$ .

\frac{\sqrt[3]{5}}{0 + 5}

$\frac{\sqrt[3]{5}}{0 + 5}$

Step 3.2.2

Add

0

$0$ and

5

$5$ .

\frac{\sqrt[3]{5}}{5}

$\frac{\sqrt[3]{5}}{5}$

\frac{\sqrt[3]{5}}{5}

$\frac{\sqrt[3]{5}}{5}$

\frac{\sqrt[3]{5}}{5}

$\frac{\sqrt[3]{5}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

\frac{\sqrt[3]{5}}{5}

$\frac{\sqrt[3]{5}}{5}$

Decimal Form:

0.34199518 \dots

$0.34199518 \dots$