Precalculus Examples

$f (x) = \sqrt{2 x - 7}$ from $x_{1} = 4$ to $x_{2} = 6$

Step 1

Substitute using the average rate of change formula.

Tap for more steps...

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.

$\frac{f (6) - f (4)}{(6) - (4)}$

Step 1.2

Substitute the equation $y = \sqrt{2 x - 7}$ for $f (6)$ and $f (4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{2 (6) - 7}) - (\sqrt{2 (4) - 7})}{(6) - (4)}$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Simplify the numerator.

Tap for more steps...

Step 2.1.1

Multiply $2$ by $6$ .

$\frac{\sqrt{12 - 7} - \sqrt{2 (4) - 7}}{6 - (4)}$

Step 2.1.2

Subtract $7$ from $12$ .

$\frac{\sqrt{5} - \sqrt{2 (4) - 7}}{6 - (4)}$

Step 2.1.3

Multiply $2$ by $4$ .

$\frac{\sqrt{5} - \sqrt{8 - 7}}{6 - (4)}$

Step 2.1.4

Subtract $7$ from $8$ .

$\frac{\sqrt{5} - \sqrt{1}}{6 - (4)}$

Step 2.1.5

Any root of $1$ is $1$ .

$\frac{\sqrt{5} - 1 \cdot 1}{6 - (4)}$

Step 2.1.6

Multiply $- 1$ by $1$ .

$\frac{\sqrt{5} - 1}{6 - (4)}$

Step 2.2

Simplify the denominator.

Tap for more steps...

Step 2.2.1

Multiply $- 1$ by $4$ .

$\frac{\sqrt{5} - 1}{6 - 4}$

Step 2.2.2

Subtract $4$ from $6$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.61803398 \dots$