Precalculus Examples

$f (x) = \sqrt{2 x - 5}$ from $x_{1} = 3$ to $x_{2} = 5$

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.

$\frac{f (5) - f (3)}{(5) - (3)}$

Step 1.2

Substitute the equation $y = \sqrt{2 x - 5}$ for $f (5)$ and $f (3)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Multiply $2$ by $5$ .

$\frac{\sqrt{10 - 5} - \sqrt{2 (3) - 5}}{5 - (3)}$

Step 2.1.2

Subtract $5$ from $10$ .

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

Step 2.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{5} - \sqrt{6 - 5}}{5 - (3)}$

Step 2.1.4

Subtract $5$ from $6$ .

$\frac{\sqrt{5} - \sqrt{1}}{5 - (3)}$

Step 2.1.5

Any root of $1$ is $1$ .

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

Step 2.1.6

Multiply $- 1$ by $1$ .

$\frac{\sqrt{5} - 1}{5 - (3)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $3$ .

$\frac{\sqrt{5} - 1}{5 - 3}$

Step 2.2.2

Subtract $3$ from $5$ .

$\frac{\sqrt{5} - 1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5} - 1}{2}$

Decimal Form:

$0.61803398 \dots$