Precalculus Examples

$f (x) = \sqrt{x}$ from $x_{1} = 4$ to $x_{2} = 9$

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.

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

Step 1.2

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

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Multiply $- 1$ by $2$ .

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

Step 2.1.6

Subtract $2$ from $3$ .

$\frac{1}{9 - (4)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $4$ .

$\frac{1}{9 - 4}$

Step 2.2.2

Subtract $4$ from $9$ .

$\frac{1}{5}$