Precalculus Examples

$g (x) = \sqrt[3]{x} + 2$ ; $- 3 \leq x \leq 0$

Step 1

Write $g (x) = \sqrt[3]{x} + 2$ as an equation.

$y = \sqrt[3]{x} + 2$

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

$\frac{f (0) - f (- 3)}{(0) - (- 3)}$

Step 2.2

Substitute the equation $y = \sqrt[3]{x} + 2$ for $f (0)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt[3]{0} + 2) - (\sqrt[3]{- 3} + 2)}{(0) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $0$ as $0^{3}$ .

$\frac{\sqrt[3]{0^{3}} + 2 - (\sqrt[3]{- 3} + 2)}{0 - (- 3)}$

Step 3.1.2

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

$\frac{0 + 2 - (\sqrt[3]{- 3} + 2)}{0 - (- 3)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

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Step 3.1.3.1.1

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{0 + 2 - (\sqrt[3]{- 1 (3)} + 2)}{0 - (- 3)}$

Step 3.1.3.1.2

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

$\frac{0 + 2 - (\sqrt[3]{{(- 1)}^{3} \cdot 3} + 2)}{0 - (- 3)}$

Step 3.1.3.2

Pull terms out from under the radical.

$\frac{0 + 2 - (- 1 \sqrt[3]{3} + 2)}{0 - (- 3)}$

Step 3.1.3.3

Rewrite $- 1 \sqrt[3]{3}$ as $- \sqrt[3]{3}$ .

$\frac{0 + 2 - (- \sqrt[3]{3} + 2)}{0 - (- 3)}$

Step 3.1.4

Apply the distributive property.

$\frac{0 + 2 - - \sqrt[3]{3} - 1 \cdot 2}{0 - (- 3)}$

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{0 + 2 + 1 \sqrt[3]{3} - 1 \cdot 2}{0 - (- 3)}$

Step 3.1.5.2

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

$\frac{0 + 2 + \sqrt[3]{3} - 1 \cdot 2}{0 - (- 3)}$

Step 3.1.6

Multiply $- 1$ by $2$ .

$\frac{0 + 2 + \sqrt[3]{3} - 2}{0 - (- 3)}$

Step 3.1.7

Add $0$ and $2$ .

$\frac{2 + \sqrt[3]{3} - 2}{0 - (- 3)}$

Step 3.1.8

Subtract $2$ from $2$ .

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

Step 3.1.9

Add $0$ and $\sqrt[3]{3}$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

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

Step 3.2.2

Add $0$ and $3$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.48074985 \dots$