Precalculus Examples

$f (x) = \sqrt{x + 9}$ on $[- 4, 2]$

Step 1

Write $f (x) = \sqrt{x + 9}$ as an equation.

$y = \sqrt{x + 9}$

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.

$\frac{f (2) - f (- 4)}{(2) - (- 4)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 9}$ for $f (2)$ and $f (- 4)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $2$ and $9$ .

$\frac{\sqrt{11} - \sqrt{- 4 + 9}}{2 - (- 4)}$

Step 3.1.2

Add $- 4$ and $9$ .

$\frac{\sqrt{11} - \sqrt{5}}{2 - (- 4)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 4$ .

$\frac{\sqrt{11} - \sqrt{5}}{2 + 4}$

Step 3.2.2

Add $2$ and $4$ .

$\frac{\sqrt{11} - \sqrt{5}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{11} - \sqrt{5}}{6}$

Decimal Form:

$0.18009280 \dots$