Precalculus Examples

$f (x) = \sqrt{2 x - 1}$ ; $[5, 25]$

Step 1

Write $f (x) = \sqrt{2 x - 1}$ as an equation.

$y = \sqrt{2 x - 1}$

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.

$\frac{f (25) - f (5)}{(25) - (5)}$

Step 2.2

Substitute the equation $y = \sqrt{2 x - 1}$ for $f (25)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{2 (25) - 1}) - (\sqrt{2 (5) - 1})}{(25) - (5)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $25$ .

$\frac{\sqrt{50 - 1} - \sqrt{2 (5) - 1}}{25 - (5)}$

Step 3.1.2

Subtract $1$ from $50$ .

$\frac{\sqrt{49} - \sqrt{2 (5) - 1}}{25 - (5)}$

Step 3.1.3

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2}} - \sqrt{2 (5) - 1}}{25 - (5)}$

Step 3.1.4

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

$\frac{7 - \sqrt{2 (5) - 1}}{25 - (5)}$

Step 3.1.5

Multiply $2$ by $5$ .

$\frac{7 - \sqrt{10 - 1}}{25 - (5)}$

Step 3.1.6

Subtract $1$ from $10$ .

$\frac{7 - \sqrt{9}}{25 - (5)}$

Step 3.1.7

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

$\frac{7 - \sqrt{3^{2}}}{25 - (5)}$

Step 3.1.8

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

$\frac{7 - 1 \cdot 3}{25 - (5)}$

Step 3.1.9

Multiply $- 1$ by $3$ .

$\frac{7 - 3}{25 - (5)}$

Step 3.1.10

Subtract $3$ from $7$ .

$\frac{4}{25 - (5)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $5$ .

$\frac{4}{25 - 5}$

Step 3.2.2

Subtract $5$ from $25$ .

$\frac{4}{20}$

Step 3.3

Cancel the common factor of $4$ and $20$ .

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Step 3.3.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{20}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $4$ out of $20$ .

$\frac{4 \cdot 1}{4 \cdot 5}$

Step 3.3.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 5}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{5}$