Precalculus Examples

$f (x) = \frac{x - 3}{x}$ ; $[5, 12]$

Step 1

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

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

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

$\frac{f (12) - f (5)}{(12) - (5)}$

Step 2.2

Substitute the equation $y = \frac{x - 3}{x}$ for $f (12)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{(12) - 3}{12}) - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3

Simplify the expression.

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Step 3.1

Cancel the common factor of $(12) - 3$ and $12$ .

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Step 3.1.1

Factor $3$ out of $12$ .

$\frac{\frac{3 (4) - 3}{12} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.1.2

Factor $3$ out of $- 3$ .

$\frac{\frac{3 \cdot 4 + 3 \cdot - 1}{12} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.1.3

Factor $3$ out of $3 \cdot 4 + 3 \cdot - 1$ .

$\frac{\frac{3 \cdot (4 - 1)}{12} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $3$ out of $12$ .

$\frac{\frac{3 \cdot (4 - 1)}{3 (4)} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.1.4.2

Cancel the common factor.

$\frac{\frac{3 \cdot (4 - 1)}{3 \cdot 4} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.1.4.3

Rewrite the expression.

$\frac{\frac{4 - 1}{4} - (\frac{(5) - 3}{5})}{(12) - (5)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $20$ .

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Step 3.2.1

Multiply $\frac{\frac{4 - 1}{4} - \frac{5 - 3}{5}}{12 - (5)}$ by $\frac{20}{20}$ .

$\frac{20}{20} \cdot \frac{\frac{4 - 1}{4} - \frac{5 - 3}{5}}{12 - (5)}$

Step 3.2.2

Combine.

$\frac{20 (\frac{4 - 1}{4} - \frac{5 - 3}{5})}{20 (12 - (5))}$

Step 3.3

Apply the distributive property.

$\frac{20 \frac{4 - 1}{4} + 20 (- \frac{5 - 3}{5})}{20 \cdot 12 + 20 (- (5))}$

Step 3.4

Simplify by cancelling.

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Step 3.4.1

Cancel the common factor of $4$ .

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Step 3.4.1.1

Factor $4$ out of $20$ .

$\frac{4 (5) \frac{4 - 1}{4} + 20 (- \frac{5 - 3}{5})}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.1.2

Cancel the common factor.

$\frac{4 \cdot 5 \frac{4 - 1}{4} + 20 (- \frac{5 - 3}{5})}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.1.3

Rewrite the expression.

$\frac{5 (4 - 1) + 20 (- \frac{5 - 3}{5})}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.2

Cancel the common factor of $5$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{5 - 3}{5}$ into the numerator.

$\frac{5 (4 - 1) + 20 \frac{- (5 - 3)}{5}}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.2.2

Factor $5$ out of $20$ .

$\frac{5 (4 - 1) + 5 (4) \frac{- (5 - 3)}{5}}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.2.3

Cancel the common factor.

$\frac{5 (4 - 1) + 5 \cdot 4 \frac{- (5 - 3)}{5}}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.2.4

Rewrite the expression.

$\frac{5 (4 - 1) + 4 (- (5 - 3))}{20 \cdot 12 + 20 (- (5))}$

Step 3.4.3

Multiply $- 1$ by $4$ .

$\frac{5 (4 - 1) - 4 (5 - 3)}{20 \cdot 12 + 20 (- (5))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Subtract $1$ from $4$ .

$\frac{5 \cdot 3 - 4 (5 - 3)}{20 \cdot 12 + 20 (- (5))}$

Step 3.5.2

Multiply $5$ by $3$ .

$\frac{15 - 4 (5 - 3)}{20 \cdot 12 + 20 (- (5))}$

Step 3.5.3

Subtract $3$ from $5$ .

$\frac{15 - 4 \cdot 2}{20 \cdot 12 + 20 (- (5))}$

Step 3.5.4

Multiply $- 4$ by $2$ .

$\frac{15 - 8}{20 \cdot 12 + 20 (- (5))}$

Step 3.5.5

Subtract $8$ from $15$ .

$\frac{7}{20 \cdot 12 + 20 (- (5))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $20$ by $12$ .

$\frac{7}{240 + 20 (- (5))}$

Step 3.6.2

Multiply $20 (- (5))$ .

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Step 3.6.2.1

Multiply $- 1$ by $5$ .

$\frac{7}{240 + 20 \cdot - 5}$

Step 3.6.2.2

Multiply $20$ by $- 5$ .

$\frac{7}{240 - 100}$

Step 3.6.3

Subtract $100$ from $240$ .

$\frac{7}{140}$

Step 3.7

Cancel the common factor of $7$ and $140$ .

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Step 3.7.1

Factor $7$ out of $7$ .

$\frac{7 (1)}{140}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $7$ out of $140$ .

$\frac{7 \cdot 1}{7 \cdot 20}$

Step 3.7.2.2

Cancel the common factor.

$\frac{7 \cdot 1}{7 \cdot 20}$

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{20}$