Precalculus Examples

$f (x) = 2 x$ , $[3, 5]$

Step 1

Write $f (x) = 2 x$ as an equation.

$y = 2 x$

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.

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

Step 2.2

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

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

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 $5$ .

$\frac{10 - 1 \cdot 2 \cdot 3}{5 - (3)}$

Step 3.1.2

Multiply $- 1 \cdot 2 \cdot 3$ .

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Step 3.1.2.1

Multiply $- 1$ by $2$ .

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

Step 3.1.2.2

Multiply $- 2$ by $3$ .

$\frac{10 - 6}{5 - (3)}$

Step 3.1.3

Subtract $6$ from $10$ .

$\frac{4}{5 - (3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $3$ .

$\frac{4}{5 - 3}$

Step 3.2.2

Subtract $3$ from $5$ .

$\frac{4}{2}$

Step 3.3

Divide $4$ by $2$ .

$2$