Precalculus Examples

$f (x) = \frac{1}{2 x} - 5$ $- 1 \leq x \leq 1$

Step 1

Write $f (x) = \frac{1}{2 x} - 5$ as an equation.

$y = \frac{1}{2 x} - 5$

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.

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

Step 2.2

Substitute the equation $y = \frac{1}{2 x} - 5$ for $f (1)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{2 (1)} - 5) - (\frac{1}{2 (- 1)} - 5)}{(1) - (- 1)}$

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 $1$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{\frac{1}{2 \cdot 1} - 5 - (\frac{1}{2 (- 1)} - 5)}{(1) - (- 1)}$

Step 3.1.1.2

Rewrite the expression.

$\frac{\frac{1}{2} - 5 - (\frac{1}{2 (- 1)} - 5)}{(1) - (- 1)}$

Step 3.1.2

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{\frac{1}{2} - 5 - (\frac{- 1 (- 1)}{2 (- 1)} - 5)}{(1) - (- 1)}$

Step 3.1.2.2

Move the negative in front of the fraction.

$\frac{\frac{1}{2} - 5 - (- \frac{1}{2} - 5)}{(1) - (- 1)}$

Step 3.2

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

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Step 3.2.1

Multiply $\frac{\frac{1}{2} - 5 - (- \frac{1}{2} - 5)}{1 - (- 1)}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{1}{2} - 5 - (- \frac{1}{2} - 5)}{1 - (- 1)}$

Step 3.2.2

Combine.

$\frac{2 (\frac{1}{2} - 5 - (- \frac{1}{2} - 5))}{2 (1 - (- 1))}$

Step 3.3

Apply the distributive property.

$\frac{2 (\frac{1}{2}) + 2 \cdot - 5 + 2 (- (- \frac{1}{2} - 5))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) + 2 \cdot - 5 + 2 (- (- \frac{1}{2} - 5))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.4.2

Rewrite the expression.

$\frac{1 + 2 \cdot - 5 + 2 (- (- \frac{1}{2} - 5))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $- 5$ .

$\frac{1 - 10 + 2 (- (- \frac{1}{2} - 5))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1 - 10 + 2 (- (- \frac{1}{2} - 5 \cdot \frac{2}{2}))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.3

Combine $- 5$ and $\frac{2}{2}$ .

$\frac{1 - 10 + 2 (- (- \frac{1}{2} + \frac{- 5 \cdot 2}{2}))}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.4

Combine the numerators over the common denominator.

$\frac{1 - 10 + 2 (- \frac{- 1 - 5 \cdot 2}{2})}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Multiply $- 5$ by $2$ .

$\frac{1 - 10 + 2 (- \frac{- 1 - 10}{2})}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.5.2

Subtract $10$ from $- 1$ .

$\frac{1 - 10 + 2 (- \frac{- 11}{2})}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.6

Cancel the common factor of $2$ .

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Step 3.5.6.1

Move the leading negative in $- \frac{- 11}{2}$ into the numerator.

$\frac{1 - 10 + 2 (\frac{11}{2})}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.6.2

Cancel the common factor.

$\frac{1 - 10 + 2 (\frac{11}{2})}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.6.3

Rewrite the expression.

$\frac{1 - 10 + 11}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.7

Subtract $10$ from $1$ .

$\frac{- 9 + 11}{2 \cdot 1 + 2 (- (- 1))}$

Step 3.5.8

Add $- 9$ and $11$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $2$ by $1$ .

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

Step 3.6.2

Multiply $2 (- (- 1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $- 1$ .

$\frac{2}{2 + 2 \cdot 1}$

Step 3.6.2.2

Multiply $2$ by $1$ .

$\frac{2}{2 + 2}$

Step 3.6.3

Add $2$ and $2$ .

$\frac{2}{4}$

Step 3.7

Cancel the common factor of $2$ and $4$ .

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Step 3.7.1

Factor $2$ out of $2$ .

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

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $2$ out of $4$ .

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

Step 3.7.2.2

Cancel the common factor.

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

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{2}$