Precalculus Examples

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

Step 1

Write $f (x) = 4 \sqrt{x}$ as an equation.

$y = 4 \sqrt{x}$

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{4 \sqrt{3^{2}} - 1 \cdot 4 \sqrt{4}}{9 - (4)}$

Step 3.1.2

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

$\frac{4 \cdot 3 - 1 \cdot 4 \sqrt{4}}{9 - (4)}$

Step 3.1.3

Multiply $4$ by $3$ .

$\frac{12 - 1 \cdot 4 \sqrt{4}}{9 - (4)}$

Step 3.1.4

Multiply $- 1$ by $4$ .

$\frac{12 - 4 \sqrt{4}}{9 - (4)}$

Step 3.1.5

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

$\frac{12 - 4 \sqrt{2^{2}}}{9 - (4)}$

Step 3.1.6

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

$\frac{12 - 4 \cdot 2}{9 - (4)}$

Step 3.1.7

Multiply $- 4$ by $2$ .

$\frac{12 - 8}{9 - (4)}$

Step 3.1.8

Subtract $8$ from $12$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $4$ .

$\frac{4}{9 - 4}$

Step 3.2.2

Subtract $4$ from $9$ .

$\frac{4}{5}$