Precalculus Examples

$f (x) = \frac{1}{x}$ ; $[9, 10]$

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.

$\frac{f (10) - f (9)}{(10) - (9)}$

Step 2.2

Substitute the equation $y = \frac{1}{x}$ for $f (10)$ and $f (9)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{10}) - (\frac{1}{9})}{(10) - (9)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $90$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{10} - \frac{1}{9}}{10 - (9)}$ by $\frac{90}{90}$ .

$\frac{90}{90} \cdot \frac{\frac{1}{10} - \frac{1}{9}}{10 - (9)}$

Step 3.1.2

Combine.

$\frac{90 (\frac{1}{10} - \frac{1}{9})}{90 (10 - (9))}$

Step 3.2

Apply the distributive property.

$\frac{90 (\frac{1}{10}) + 90 (- \frac{1}{9})}{90 \cdot 10 + 90 (- (9))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $10$ .

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Step 3.3.1.1

Factor $10$ out of $90$ .

$\frac{10 (9) \frac{1}{10} + 90 (- \frac{1}{9})}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{10 \cdot 9 \frac{1}{10} + 90 (- \frac{1}{9})}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{9 + 90 (- \frac{1}{9})}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.2

Cancel the common factor of $9$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{9 + 90 (\frac{- 1}{9})}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.2.2

Factor $9$ out of $90$ .

$\frac{9 + 9 (10) \frac{- 1}{9}}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{9 + 9 \cdot 10 \frac{- 1}{9}}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{9 + 10 \cdot - 1}{90 \cdot 10 + 90 (- (9))}$

Step 3.3.3

Multiply $10$ by $- 1$ .

$\frac{9 - 10}{90 \cdot 10 + 90 (- (9))}$

Step 3.4

Subtract $10$ from $9$ .

$\frac{- 1}{90 \cdot 10 + 90 (- (9))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $90$ by $10$ .

$\frac{- 1}{900 + 90 (- (9))}$

Step 3.5.2

Multiply $90 (- (9))$ .

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Step 3.5.2.1

Multiply $- 1$ by $9$ .

$\frac{- 1}{900 + 90 \cdot - 9}$

Step 3.5.2.2

Multiply $90$ by $- 9$ .

$\frac{- 1}{900 - 810}$

Step 3.5.3

Subtract $810$ from $900$ .

$\frac{- 1}{90}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{1}{90}$