Precalculus Examples

$f (x) = \frac{x}{x + 3}$ ; $[- 1, 5]$

Step 1

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

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

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.

$\frac{f (5) - f (- 1)}{(5) - (- 1)}$

Step 2.2

Substitute the equation $y = \frac{x}{x + 3}$ for $f (5)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{5}{(5) + 3}) - (\frac{- 1}{(- 1) + 3})}{(5) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(5 + 3) (- 1 + 3)$ .

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Step 3.1.1

Multiply $\frac{\frac{5}{5 + 3} - \frac{- 1}{- 1 + 3}}{5 - (- 1)}$ by $\frac{(5 + 3) (- 1 + 3)}{(5 + 3) (- 1 + 3)}$ .

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

Step 3.1.2

Combine.

$\frac{(5 + 3) (- 1 + 3) (\frac{5}{5 + 3} - \frac{- 1}{- 1 + 3})}{(5 + 3) (- 1 + 3) (5 - (- 1))}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $5 + 3$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

Cancel the common factor of $- 1 + 3$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{- 1}{- 1 + 3}$ into the numerator.

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

Step 3.3.2.2

Factor $- 1 + 3$ out of $(5 + 3) (- 1 + 3)$ .

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

Step 3.3.2.3

Cancel the common factor.

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

Step 3.3.2.4

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $- 1$ and $3$ .

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

Step 3.4.2

Multiply $2$ by $5$ .

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

Step 3.4.3

Add $10$ and $5$ .

$\frac{15 + 3}{(5 + 3) (- 1 + 3) \cdot 5 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.4.4

Add $15$ and $3$ .

$\frac{18}{(5 + 3) (- 1 + 3) \cdot 5 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Add $5$ and $3$ .

$\frac{18}{8 (- 1 + 3) \cdot 5 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.5.2

Add $- 1$ and $3$ .

$\frac{18}{8 \cdot 2 \cdot 5 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.5.3

Multiply $8 \cdot 2 \cdot 5$ .

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Step 3.5.3.1

Multiply $8$ by $2$ .

$\frac{18}{16 \cdot 5 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.5.3.2

Multiply $16$ by $5$ .

$\frac{18}{80 + (5 + 3) (- 1 + 3) (- (- 1))}$

Step 3.5.4

Add $5$ and $3$ .

$\frac{18}{80 + 8 (- 1 + 3) (- (- 1))}$

Step 3.5.5

Add $- 1$ and $3$ .

$\frac{18}{80 + 8 \cdot 2 (- (- 1))}$

Step 3.5.6

Multiply $8 \cdot 2 (- (- 1))$ .

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Step 3.5.6.1

Multiply $8$ by $2$ .

$\frac{18}{80 + 16 (- (- 1))}$

Step 3.5.6.2

Multiply $- 1$ by $- 1$ .

$\frac{18}{80 + 16 \cdot 1}$

Step 3.5.6.3

Multiply $16$ by $1$ .

$\frac{18}{80 + 16}$

Step 3.5.7

Add $80$ and $16$ .

$\frac{18}{96}$

Step 3.6

Cancel the common factor of $18$ and $96$ .

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Step 3.6.1

Factor $6$ out of $18$ .

$\frac{6 (3)}{96}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $6$ out of $96$ .

$\frac{6 \cdot 3}{6 \cdot 16}$

Step 3.6.2.2

Cancel the common factor.

$\frac{6 \cdot 3}{6 \cdot 16}$

Step 3.6.2.3

Rewrite the expression.

$\frac{3}{16}$