Precalculus Examples

$f (x) = \sqrt{x} + 5$ , $0 \leq x \leq 9$

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.

$\frac{f (9) - f (0)}{(9) - (0)}$

Step 2.2

Substitute the equation $y = \sqrt{x} + 5$ for $f (9)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{9} + 5) - (\sqrt{0} + 5)}{(9) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{\sqrt{3^{2}} + 5 - (\sqrt{0} + 5)}{9 - (0)}$

Step 3.1.2

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

$\frac{3 + 5 - (\sqrt{0} + 5)}{9 - (0)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Rewrite $0$ as $0^{2}$ .

$\frac{3 + 5 - (\sqrt{0^{2}} + 5)}{9 - (0)}$

Step 3.1.3.2

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

$\frac{3 + 5 - (0 + 5)}{9 - (0)}$

Step 3.1.4

Add $0$ and $5$ .

$\frac{3 + 5 - 1 \cdot 5}{9 - (0)}$

Step 3.1.5

Multiply $- 1$ by $5$ .

$\frac{3 + 5 - 5}{9 - (0)}$

Step 3.1.6

Add $3$ and $5$ .

$\frac{8 - 5}{9 - (0)}$

Step 3.1.7

Subtract $5$ from $8$ .

$\frac{3}{9 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{3}{9 + 0}$

Step 3.2.2

Add $9$ and $0$ .

$\frac{3}{9}$

Step 3.3

Cancel the common factor of $3$ and $9$ .

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Step 3.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 1}{3 \cdot 3}$

Step 3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 3}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{3}$