Precalculus Examples

$f (x) = \frac{x - 6}{x + 4}$ on $[1, 7]$

Step 1

Write $f (x) = \frac{x - 6}{x + 4}$ as an equation.

$y = \frac{x - 6}{x + 4}$

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.

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

Step 2.2

Substitute the equation $y = \frac{x - 6}{x + 4}$ for $f (7)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{(7) - 6}{(7) + 4}) - (\frac{(1) - 6}{(1) + 4})}{(7) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(7 + 4) (1 + 4)$ .

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Step 3.1.1

Multiply $\frac{\frac{7 - 6}{7 + 4} - \frac{1 - 6}{1 + 4}}{7 - (1)}$ by $\frac{(7 + 4) (1 + 4)}{(7 + 4) (1 + 4)}$ .

$\frac{(7 + 4) (1 + 4)}{(7 + 4) (1 + 4)} \cdot \frac{\frac{7 - 6}{7 + 4} - \frac{1 - 6}{1 + 4}}{7 - (1)}$

Step 3.1.2

Combine.

$\frac{(7 + 4) (1 + 4) (\frac{7 - 6}{7 + 4} - \frac{1 - 6}{1 + 4})}{(7 + 4) (1 + 4) (7 - (1))}$

Step 3.2

Apply the distributive property.

$\frac{(7 + 4) (1 + 4) \frac{7 - 6}{7 + 4} + (7 + 4) (1 + 4) (- \frac{1 - 6}{1 + 4})}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $7 + 4$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{(7 + 4) (1 + 4) \frac{7 - 6}{7 + 4} + (7 + 4) (1 + 4) (- \frac{1 - 6}{1 + 4})}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{(1 + 4) (7 - 6) + (7 + 4) (1 + 4) (- \frac{1 - 6}{1 + 4})}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3.2

Cancel the common factor of $1 + 4$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1 - 6}{1 + 4}$ into the numerator.

$\frac{(1 + 4) (7 - 6) + (7 + 4) (1 + 4) \frac{- (1 - 6)}{1 + 4}}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3.2.2

Factor $1 + 4$ out of $(7 + 4) (1 + 4)$ .

$\frac{(1 + 4) (7 - 6) + (1 + 4) (7 + 4) \frac{- (1 - 6)}{1 + 4}}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{(1 + 4) (7 - 6) + (1 + 4) (7 + 4) \frac{- (1 - 6)}{1 + 4}}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{(1 + 4) (7 - 6) + (7 + 4) (- (1 - 6))}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $1$ and $4$ .

$\frac{5 (7 - 6) + (7 + 4) (- (1 - 6))}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.2

Subtract $6$ from $7$ .

$\frac{5 \cdot 1 + (7 + 4) (- (1 - 6))}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.3

Multiply $5$ by $1$ .

$\frac{5 + (7 + 4) (- (1 - 6))}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.4

Add $7$ and $4$ .

$\frac{5 + 11 (- (1 - 6))}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.5

Subtract $6$ from $1$ .

$\frac{5 + 11 (- - 5)}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.6

Multiply $11 (- - 5)$ .

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Step 3.4.6.1

Multiply $- 1$ by $- 5$ .

$\frac{5 + 11 \cdot 5}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.6.2

Multiply $11$ by $5$ .

$\frac{5 + 55}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.4.7

Add $5$ and $55$ .

$\frac{60}{(7 + 4) (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Add $7$ and $4$ .

$\frac{60}{11 (1 + 4) \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.5.2

Add $1$ and $4$ .

$\frac{60}{11 \cdot 5 \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.5.3

Multiply $11 \cdot 5 \cdot 7$ .

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Step 3.5.3.1

Multiply $11$ by $5$ .

$\frac{60}{55 \cdot 7 + (7 + 4) (1 + 4) (- (1))}$

Step 3.5.3.2

Multiply $55$ by $7$ .

$\frac{60}{385 + (7 + 4) (1 + 4) (- (1))}$

Step 3.5.4

Add $7$ and $4$ .

$\frac{60}{385 + 11 (1 + 4) (- (1))}$

Step 3.5.5

Add $1$ and $4$ .

$\frac{60}{385 + 11 \cdot 5 (- (1))}$

Step 3.5.6

Multiply $11 \cdot 5 (- (1))$ .

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Step 3.5.6.1

Multiply $11$ by $5$ .

$\frac{60}{385 + 55 (- (1))}$

Step 3.5.6.2

Multiply $- 1$ by $1$ .

$\frac{60}{385 + 55 \cdot - 1}$

Step 3.5.6.3

Multiply $55$ by $- 1$ .

$\frac{60}{385 - 55}$

Step 3.5.7

Subtract $55$ from $385$ .

$\frac{60}{330}$

Step 3.6

Cancel the common factor of $60$ and $330$ .

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Step 3.6.1

Factor $30$ out of $60$ .

$\frac{30 (2)}{330}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $30$ out of $330$ .

$\frac{30 \cdot 2}{30 \cdot 11}$

Step 3.6.2.2

Cancel the common factor.

$\frac{30 \cdot 2}{30 \cdot 11}$

Step 3.6.2.3

Rewrite the expression.

$\frac{2}{11}$