Precalculus Examples

$f (x) = \sqrt{3 x - 2}$ from $x_{1} = 1$ to $x_{2} = 4$

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.

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

Step 1.2

Substitute the equation $y = \sqrt{3 x - 2}$ for $f (4)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Multiply $3$ by $4$ .

$\frac{\sqrt{12 - 2} - \sqrt{3 (1) - 2}}{4 - (1)}$

Step 2.1.2

Subtract $2$ from $12$ .

$\frac{\sqrt{10} - \sqrt{3 (1) - 2}}{4 - (1)}$

Step 2.1.3

Multiply $3$ by $1$ .

$\frac{\sqrt{10} - \sqrt{3 - 2}}{4 - (1)}$

Step 2.1.4

Subtract $2$ from $3$ .

$\frac{\sqrt{10} - \sqrt{1}}{4 - (1)}$

Step 2.1.5

Any root of $1$ is $1$ .

$\frac{\sqrt{10} - 1 \cdot 1}{4 - (1)}$

Step 2.1.6

Multiply $- 1$ by $1$ .

$\frac{\sqrt{10} - 1}{4 - (1)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $1$ .

$\frac{\sqrt{10} - 1}{4 - 1}$

Step 2.2.2

Subtract $1$ from $4$ .

$\frac{\sqrt{10} - 1}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{10} - 1}{3}$

Decimal Form:

$0.72075922 \dots$