Precalculus Examples

$f (x) = \frac{1}{x}$ on $[1, 3]$

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.

$\frac{f (3) - f (1)}{(3) - (1)}$

Step 2.2

Substitute the equation $y = \frac{1}{x}$ for $f (3)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Cancel the common factor of $1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

$\frac{\frac{1}{3} - 1 \cdot 1}{(3) - (1)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.2.1

Multiply $\frac{\frac{1}{3} - 1 \cdot 1}{3 - (1)}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\frac{1}{3} - 1 \cdot 1}{3 - (1)}$

Step 3.2.2

Combine.

$\frac{3 (\frac{1}{3} - 1 \cdot 1)}{3 (3 - (1))}$

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3 (- 1 \cdot 1)$ .

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Step 3.5.1.1

Multiply $- 1$ by $1$ .

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

Step 3.5.1.2

Multiply $3$ by $- 1$ .

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

Step 3.5.2

Subtract $3$ from $1$ .

$\frac{- 2}{3 \cdot 3 + 3 (- (1))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $3$ by $3$ .

$\frac{- 2}{9 + 3 (- (1))}$

Step 3.6.2

Multiply $3 (- (1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $1$ .

$\frac{- 2}{9 + 3 \cdot - 1}$

Step 3.6.2.2

Multiply $3$ by $- 1$ .

$\frac{- 2}{9 - 3}$

Step 3.6.3

Subtract $3$ from $9$ .

$\frac{- 2}{6}$

Step 3.7

Reduce the expression by cancelling the common factors.

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Step 3.7.1

Cancel the common factor of $- 2$ and $6$ .

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Step 3.7.1.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{6}$

Step 3.7.1.2

Cancel the common factors.

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Step 3.7.1.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot - 1}{2 \cdot 3}$

Step 3.7.1.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 3}$

Step 3.7.1.2.3

Rewrite the expression.

$\frac{- 1}{3}$

Step 3.7.2

Move the negative in front of the fraction.

$- \frac{1}{3}$