Precalculus Examples

$f (x) = \sqrt{x + 5}$ ; $[- 1, 7]$

Step 1

Write $f (x) = \sqrt{x + 5}$ as an equation.

$y = \sqrt{x + 5}$

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.

$\frac{f (7) - f (- 1)}{(7) - (- 1)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 5}$ for $f (7)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{(7) + 5}) - (\sqrt{(- 1) + 5})}{(7) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $7$ and $5$ .

$\frac{\sqrt{12} - \sqrt{- 1 + 5}}{7 - (- 1)}$

Step 3.1.2

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

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Step 3.1.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} - \sqrt{- 1 + 5}}{7 - (- 1)}$

Step 3.1.2.2

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

$\frac{\sqrt{2^{2} \cdot 3} - \sqrt{- 1 + 5}}{7 - (- 1)}$

Step 3.1.3

Pull terms out from under the radical.

$\frac{2 \sqrt{3} - \sqrt{- 1 + 5}}{7 - (- 1)}$

Step 3.1.4

Add $- 1$ and $5$ .

$\frac{2 \sqrt{3} - \sqrt{4}}{7 - (- 1)}$

Step 3.1.5

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

$\frac{2 \sqrt{3} - \sqrt{2^{2}}}{7 - (- 1)}$

Step 3.1.6

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

$\frac{2 \sqrt{3} - 1 \cdot 2}{7 - (- 1)}$

Step 3.1.7

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3} - 2}{7 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{2 \sqrt{3} - 2}{7 + 1}$

Step 3.2.2

Add $7$ and $1$ .

$\frac{2 \sqrt{3} - 2}{8}$

Step 3.3

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

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Step 3.3.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 3.3.2

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{3}) + 2 (- 1)}{8}$

Step 3.3.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (- 1)$ .

$\frac{2 (\sqrt{3} - 1)}{8}$

Step 3.3.4

Cancel the common factors.

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Step 3.3.4.1

Factor $2$ out of $8$ .

$\frac{2 (\sqrt{3} - 1)}{2 (4)}$

Step 3.3.4.2

Cancel the common factor.

$\frac{2 (\sqrt{3} - 1)}{2 \cdot 4}$

Step 3.3.4.3

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3} - 1}{4}$

Decimal Form:

$0.18301270 \dots$