Precalculus Examples

$f (x) = \sqrt{x}$ from $x_{1} = 9$ to $x_{2} = 16$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

$\frac{f (16) - f (9)}{(16) - (9)}$

Step 1.2

Substitute the equation $y = \sqrt{x}$ for $f (16)$ and $f (9)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{16}) - (\sqrt{9})}{(16) - (9)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}} - \sqrt{9}}{16 - (9)}$

Step 2.1.2

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

$\frac{4 - \sqrt{9}}{16 - (9)}$

Step 2.1.3

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

$\frac{4 - \sqrt{3^{2}}}{16 - (9)}$

Step 2.1.4

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

$\frac{4 - 1 \cdot 3}{16 - (9)}$

Step 2.1.5

Multiply $- 1$ by $3$ .

$\frac{4 - 3}{16 - (9)}$

Step 2.1.6

Subtract $3$ from $4$ .

$\frac{1}{16 - (9)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $9$ .

$\frac{1}{16 - 9}$

Step 2.2.2

Subtract $9$ from $16$ .

$\frac{1}{7}$