Precalculus Examples

f (x) = 2 x

$f (x) = 2 x$ ,

[3, 5]

$[3, 5]$

Step 1

Write

f (x) = 2 x

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

y = 2 x

$y = 2 x$

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

$y$ values of the two points divided by the change in

x

$x$ values of the two points.

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

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

Step 2.2

Substitute the equation

y = 2 x

$y = 2 x$ for

f (5)

$f (5)$ and

f (3)

$f (3)$ , replacing

x

$x$ in the function with the corresponding

x

$x$ value.

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

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

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

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

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

$2$ by

5

$5$ .

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

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

Step 3.1.2

Multiply

- 1 \cdot 2 \cdot 3

$- 1 \cdot 2 \cdot 3$ .

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Step 3.1.2.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 3.1.2.2

Multiply

- 2

$- 2$ by

3

$3$ .

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

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

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

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

Step 3.1.3

Subtract

6

$6$ from

10

$10$ .

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

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

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

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply

- 1

$- 1$ by

3

$3$ .

\frac{4}{5 - 3}

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

Step 3.2.2

Subtract

3

$3$ from

5

$5$ .

\frac{4}{2}

$\frac{4}{2}$

\frac{4}{2}

$\frac{4}{2}$

Step 3.3

Divide

4

$4$ by

2

$2$ .

2

$2$

2

$2$

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

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

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