Precalculus Examples

$g (x) = \frac{x + 2}{x + 3}$ , $[0, 1]$

Step 1

Write $g (x) = \frac{x + 2}{x + 3}$ as an equation.

$y = \frac{x + 2}{x + 3}$

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.

$\frac{f (1) - f (0)}{(1) - (0)}$

Step 2.2

Substitute the equation $y = \frac{x + 2}{x + 3}$ for $f (1)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{(1) + 2}{(1) + 3}) - (\frac{(0) + 2}{(0) + 3})}{(1) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(1 + 3) (0 + 3)$ .

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Step 3.1.1

Multiply $\frac{\frac{1 + 2}{1 + 3} - \frac{0 + 2}{0 + 3}}{1 - (0)}$ by $\frac{(1 + 3) (0 + 3)}{(1 + 3) (0 + 3)}$ .

$\frac{(1 + 3) (0 + 3)}{(1 + 3) (0 + 3)} \cdot \frac{\frac{1 + 2}{1 + 3} - \frac{0 + 2}{0 + 3}}{1 - (0)}$

Step 3.1.2

Combine.

$\frac{(1 + 3) (0 + 3) (\frac{1 + 2}{1 + 3} - \frac{0 + 2}{0 + 3})}{(1 + 3) (0 + 3) (1 - (0))}$

Step 3.2

Apply the distributive property.

$\frac{(1 + 3) (0 + 3) \frac{1 + 2}{1 + 3} + (1 + 3) (0 + 3) (- \frac{0 + 2}{0 + 3})}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $1 + 3$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{(1 + 3) (0 + 3) \frac{1 + 2}{1 + 3} + (1 + 3) (0 + 3) (- \frac{0 + 2}{0 + 3})}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{(0 + 3) (1 + 2) + (1 + 3) (0 + 3) (- \frac{0 + 2}{0 + 3})}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3.2

Cancel the common factor of $0 + 3$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{0 + 2}{0 + 3}$ into the numerator.

$\frac{(0 + 3) (1 + 2) + (1 + 3) (0 + 3) \frac{- (0 + 2)}{0 + 3}}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3.2.2

Factor $0 + 3$ out of $(1 + 3) (0 + 3)$ .

$\frac{(0 + 3) (1 + 2) + (0 + 3) (1 + 3) \frac{- (0 + 2)}{0 + 3}}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{(0 + 3) (1 + 2) + (0 + 3) (1 + 3) \frac{- (0 + 2)}{0 + 3}}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{(0 + 3) (1 + 2) + (1 + 3) (- (0 + 2))}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $0$ and $3$ .

$\frac{3 (1 + 2) + (1 + 3) (- (0 + 2))}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.2

Add $1$ and $2$ .

$\frac{3 \cdot 3 + (1 + 3) (- (0 + 2))}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.3

Multiply $3$ by $3$ .

$\frac{9 + (1 + 3) (- (0 + 2))}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.4

Add $1$ and $3$ .

$\frac{9 + 4 (- (0 + 2))}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.5

Add $0$ and $2$ .

$\frac{9 + 4 (- 1 \cdot 2)}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.6

Multiply $4 (- 1 \cdot 2)$ .

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Step 3.4.6.1

Multiply $- 1$ by $2$ .

$\frac{9 + 4 \cdot - 2}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.6.2

Multiply $4$ by $- 2$ .

$\frac{9 - 8}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.4.7

Subtract $8$ from $9$ .

$\frac{1}{(1 + 3) (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Add $1$ and $3$ .

$\frac{1}{4 (0 + 3) \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.5.2

Add $0$ and $3$ .

$\frac{1}{4 \cdot 3 \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.5.3

Multiply $4 \cdot 3 \cdot 1$ .

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Step 3.5.3.1

Multiply $4$ by $3$ .

$\frac{1}{12 \cdot 1 + (1 + 3) (0 + 3) (- (0))}$

Step 3.5.3.2

Multiply $12$ by $1$ .

$\frac{1}{12 + (1 + 3) (0 + 3) (- (0))}$

Step 3.5.4

Add $1$ and $3$ .

$\frac{1}{12 + 4 (0 + 3) (- (0))}$

Step 3.5.5

Add $0$ and $3$ .

$\frac{1}{12 + 4 \cdot 3 (- (0))}$

Step 3.5.6

Multiply $4 \cdot 3 (- (0))$ .

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Step 3.5.6.1

Multiply $4$ by $3$ .

$\frac{1}{12 + 12 (- (0))}$

Step 3.5.6.2

Multiply $- 1$ by $0$ .

$\frac{1}{12 + 12 \cdot 0}$

Step 3.5.6.3

Multiply $12$ by $0$ .

$\frac{1}{12 + 0}$

Step 3.5.7

Add $12$ and $0$ .

$\frac{1}{12}$