Precalculus Examples

$f (x) = 2 x + 3$ on $[- 1, 2]$

Step 1

Write $f (x) = 2 x + 3$ as an equation.

$y = 2 x + 3$

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.

$\frac{f (2) - f (- 1)}{(2) - (- 1)}$

Step 2.2

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

$\frac{(2 (2) + 3) - (2 (- 1) + 3)}{(2) - (- 1)}$

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 $2$ .

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

Step 3.1.2

Multiply $2$ by $- 1$ .

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

Step 3.1.3

Add $- 2$ and $3$ .

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

Step 3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.1.5

Add $4$ and $3$ .

$\frac{7 - 1}{2 - (- 1)}$

Step 3.1.6

Subtract $1$ from $7$ .

$\frac{6}{2 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{6}{2 + 1}$

Step 3.2.2

Add $2$ and $1$ .

$\frac{6}{3}$

Step 3.3

Divide $6$ by $3$ .

$2$