Precalculus Examples

$f (x) = \sqrt{x + 8}$ ; $[4, 10]$

Step 1

Write $f (x) = \sqrt{x + 8}$ as an equation.

$y = \sqrt{x + 8}$

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.

$\frac{f (10) - f (4)}{(10) - (4)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 8}$ for $f (10)$ and $f (4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{(10) + 8}) - (\sqrt{(4) + 8})}{(10) - (4)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $10$ and $8$ .

$\frac{\sqrt{18} - \sqrt{4 + 8}}{10 - (4)}$

Step 3.1.2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.1.2.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} - \sqrt{4 + 8}}{10 - (4)}$

Step 3.1.2.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2} - \sqrt{4 + 8}}{10 - (4)}$

Step 3.1.3

Pull terms out from under the radical.

$\frac{3 \sqrt{2} - \sqrt{4 + 8}}{10 - (4)}$

Step 3.1.4

Add $4$ and $8$ .

$\frac{3 \sqrt{2} - \sqrt{12}}{10 - (4)}$

Step 3.1.5

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.1.5.1

Factor $4$ out of $12$ .

$\frac{3 \sqrt{2} - \sqrt{4 (3)}}{10 - (4)}$

Step 3.1.5.2

Rewrite $4$ as $2^{2}$ .

$\frac{3 \sqrt{2} - \sqrt{2^{2} \cdot 3}}{10 - (4)}$

Step 3.1.6

Pull terms out from under the radical.

$\frac{3 \sqrt{2} - (2 \sqrt{3})}{10 - (4)}$

Step 3.1.7

Multiply $2$ by $- 1$ .

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{10 - (4)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $4$ .

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{10 - 4}$

Step 3.2.2

Subtract $4$ from $10$ .

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{6}$

Decimal Form:

$0.12975651 \dots$