Precalculus Examples

$f (x) = \sqrt{x}$ , $[0, 16]$

Step 1

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

$y = \sqrt{x}$

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.

$\frac{f (16) - f (0)}{(16) - (0)}$

Step 2.2

Substitute the equation $y = \sqrt{x}$ for $f (16)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{16}) - (\sqrt{0})}{(16) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{\sqrt{4^{2}} - \sqrt{0}}{16 - (0)}$

Step 3.1.2

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

$\frac{4 - \sqrt{0}}{16 - (0)}$

Step 3.1.3

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

$\frac{4 - \sqrt{0^{2}}}{16 - (0)}$

Step 3.1.4

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

$\frac{4 - 0}{16 - (0)}$

Step 3.1.5

Multiply $- 1$ by $0$ .

$\frac{4 + 0}{16 - (0)}$

Step 3.1.6

Add $4$ and $0$ .

$\frac{4}{16 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{4}{16 + 0}$

Step 3.2.2

Add $16$ and $0$ .

$\frac{4}{16}$

Step 3.3

Cancel the common factor of $4$ and $16$ .

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Step 3.3.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{16}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 1}{4 \cdot 4}$

Step 3.3.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 4}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{4}$