Precalculus Examples

$f (x) = \frac{x - 4}{x}$ ; $[2, 8]$

Step 1

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

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

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.

$\frac{f (8) - f (2)}{(8) - (2)}$

Step 2.2

Substitute the equation $y = \frac{x - 4}{x}$ for $f (8)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{(8) - 4}{8}) - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Reduce the expression by cancelling the common factors.

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Step 3.1.1

Cancel the common factor of $(8) - 4$ and $8$ .

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Step 3.1.1.1

Factor $4$ out of $8$ .

$\frac{\frac{4 (2) - 4}{8} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.1.2

Factor $4$ out of $- 4$ .

$\frac{\frac{4 \cdot 2 + 4 \cdot - 1}{8} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.1.3

Factor $4$ out of $4 \cdot 2 + 4 \cdot - 1$ .

$\frac{\frac{4 \cdot (2 - 1)}{8} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.1.4

Cancel the common factors.

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Step 3.1.1.4.1

Factor $4$ out of $8$ .

$\frac{\frac{4 \cdot (2 - 1)}{4 (2)} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.1.4.2

Cancel the common factor.

$\frac{\frac{4 \cdot (2 - 1)}{4 \cdot 2} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.1.4.3

Rewrite the expression.

$\frac{\frac{2 - 1}{2} - (\frac{(2) - 4}{2})}{(8) - (2)}$

Step 3.1.2

Cancel the common factor of $(2) - 4$ and $2$ .

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Step 3.1.2.1

Factor $2$ out of $2$ .

$\frac{\frac{2 - 1}{2} - \frac{2 \cdot 1 - 4}{2}}{(8) - (2)}$

Step 3.1.2.2

Factor $2$ out of $- 4$ .

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

Step 3.1.2.3

Factor $2$ out of $2 \cdot 1 + 2 \cdot - 2$ .

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

Step 3.1.2.4

Cancel the common factors.

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Step 3.1.2.4.1

Factor $2$ out of $2$ .

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

Step 3.1.2.4.2

Cancel the common factor.

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

Step 3.1.2.4.3

Rewrite the expression.

$\frac{\frac{2 - 1}{2} - \frac{1 - 2}{1}}{(8) - (2)}$

Step 3.1.2.4.4

Divide $1 - 2$ by $1$ .

$\frac{\frac{2 - 1}{2} - (1 - 2)}{(8) - (2)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.2.1

Multiply $\frac{\frac{2 - 1}{2} - (1 - 2)}{8 - (2)}$ by $\frac{2}{2}$ .

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

Step 3.2.2

Combine.

$\frac{2 (\frac{2 - 1}{2} - (1 - 2))}{2 (8 - (2))}$

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Subtract $2$ from $1$ .

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

Step 3.5.2

Multiply $2 (- - 1)$ .

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Step 3.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2.2

Multiply $2$ by $1$ .

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

Step 3.5.3

Subtract $1$ from $2$ .

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

Step 3.5.4

Add $1$ and $2$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $2$ by $8$ .

$\frac{3}{16 + 2 (- (2))}$

Step 3.6.2

Multiply $2 (- (2))$ .

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Step 3.6.2.1

Multiply $- 1$ by $2$ .

$\frac{3}{16 + 2 \cdot - 2}$

Step 3.6.2.2

Multiply $2$ by $- 2$ .

$\frac{3}{16 - 4}$

Step 3.6.3

Subtract $4$ from $16$ .

$\frac{3}{12}$

Step 3.7

Cancel the common factor of $3$ and $12$ .

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Step 3.7.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{12}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 1}{3 \cdot 4}$

Step 3.7.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 4}$

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{4}$