Precalculus Examples

$f (x) = \frac{1}{x}$ ; $[8, 9]$

Step 1

Write $f (x) = \frac{1}{x}$ as an equation.

$y = \frac{1}{x}$

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.

$\frac{f (9) - f (8)}{(9) - (8)}$

Step 2.2

Substitute the equation $y = \frac{1}{x}$ for $f (9)$ and $f (8)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{9}) - (\frac{1}{8})}{(9) - (8)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $72$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{9} - \frac{1}{8}}{9 - (8)}$ by $\frac{72}{72}$ .

$\frac{72}{72} \cdot \frac{\frac{1}{9} - \frac{1}{8}}{9 - (8)}$

Step 3.1.2

Combine.

$\frac{72 (\frac{1}{9} - \frac{1}{8})}{72 (9 - (8))}$

Step 3.2

Apply the distributive property.

$\frac{72 (\frac{1}{9}) + 72 (- \frac{1}{8})}{72 \cdot 9 + 72 (- (8))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $9$ .

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Step 3.3.1.1

Factor $9$ out of $72$ .

$\frac{9 (8) \frac{1}{9} + 72 (- \frac{1}{8})}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{9 \cdot 8 \frac{1}{9} + 72 (- \frac{1}{8})}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{8 + 72 (- \frac{1}{8})}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.2

Cancel the common factor of $8$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$\frac{8 + 72 (\frac{- 1}{8})}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.2.2

Factor $8$ out of $72$ .

$\frac{8 + 8 (9) \frac{- 1}{8}}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{8 + 8 \cdot 9 \frac{- 1}{8}}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{8 + 9 \cdot - 1}{72 \cdot 9 + 72 (- (8))}$

Step 3.3.3

Multiply $9$ by $- 1$ .

$\frac{8 - 9}{72 \cdot 9 + 72 (- (8))}$

Step 3.4

Subtract $9$ from $8$ .

$\frac{- 1}{72 \cdot 9 + 72 (- (8))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $72$ by $9$ .

$\frac{- 1}{648 + 72 (- (8))}$

Step 3.5.2

Multiply $72 (- (8))$ .

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Step 3.5.2.1

Multiply $- 1$ by $8$ .

$\frac{- 1}{648 + 72 \cdot - 8}$

Step 3.5.2.2

Multiply $72$ by $- 8$ .

$\frac{- 1}{648 - 576}$

Step 3.5.3

Subtract $576$ from $648$ .

$\frac{- 1}{72}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{1}{72}$