Precalculus Examples

$f (x) = \sqrt{x + 6}$ on $[4, 9]$

Step 1

Write $f (x) = \sqrt{x + 6}$ as an equation.

$y = \sqrt{x + 6}$

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.

$\frac{f (9) - f (4)}{(9) - (4)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 6}$ for $f (9)$ and $f (4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{(9) + 6}) - (\sqrt{(4) + 6})}{(9) - (4)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $9$ and $6$ .

$\frac{\sqrt{15} - \sqrt{4 + 6}}{9 - (4)}$

Step 3.1.2

Add $4$ and $6$ .

$\frac{\sqrt{15} - \sqrt{10}}{9 - (4)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $4$ .

$\frac{\sqrt{15} - \sqrt{10}}{9 - 4}$

Step 3.2.2

Subtract $4$ from $9$ .

$\frac{\sqrt{15} - \sqrt{10}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15} - \sqrt{10}}{5}$

Decimal Form:

$0.14214113 \dots$