Finite Math Examples

$f (x) = - \frac{1}{x + 1}$ , $(1, 4)$

Step 1

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

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

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.

$\frac{f (4) - f (1)}{(4) - (1)}$

Step 2.2

Substitute the equation $y = - \frac{1}{x + 1}$ for $f (4)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- \frac{1}{(4) + 1}) - (- \frac{1}{(1) + 1})}{(4) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $4 + 1$ .

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Step 3.1.1

Multiply $\frac{- \frac{1}{4 + 1} - - \frac{1}{1 + 1}}{4 - (1)}$ by $\frac{4 + 1}{4 + 1}$ .

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

Step 3.1.2

Combine.

$\frac{(4 + 1) (- \frac{1}{4 + 1} - - \frac{1}{1 + 1})}{(4 + 1) (4 - (1))}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by adding numbers.

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Step 3.3.1

Add $4$ and $1$ .

$\frac{(4 + 1) (- \frac{1}{5}) + (4 + 1) (- - \frac{1}{1 + 1})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.3.2

Add $1$ and $1$ .

$\frac{(4 + 1) (- \frac{1}{5}) + (4 + 1) (- - \frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $4$ and $1$ .

$\frac{5 (- \frac{1}{5}) + (4 + 1) (- - \frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.2

Cancel the common factor of $5$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{5 (\frac{- 1}{5}) + (4 + 1) (- - \frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{5 (\frac{- 1}{5}) + (4 + 1) (- - \frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.2.3

Rewrite the expression.

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

Step 3.4.3

Add $4$ and $1$ .

$\frac{- 1 + 5 (- - \frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.4

Multiply $- - \frac{1}{2}$ .

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Step 3.4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 1 + 5 (1 (\frac{1}{2}))}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.4.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{- 1 + 5 (\frac{1}{2})}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.5

Combine $5$ and $\frac{1}{2}$ .

$\frac{- 1 + \frac{5}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 1 \cdot \frac{2}{2} + \frac{5}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.7

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{\frac{- 1 \cdot 2}{2} + \frac{5}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.8

Combine the numerators over the common denominator.

$\frac{\frac{- 1 \cdot 2 + 5}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $2$ .

$\frac{\frac{- 2 + 5}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.4.9.2

Add $- 2$ and $5$ .

$\frac{\frac{3}{2}}{(4 + 1) \cdot 4 + (4 + 1) (- (1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Add $4$ and $1$ .

$\frac{\frac{3}{2}}{5 \cdot 4 + (4 + 1) (- (1))}$

Step 3.5.2

Multiply $5$ by $4$ .

$\frac{\frac{3}{2}}{20 + (4 + 1) (- (1))}$

Step 3.5.3

Add $4$ and $1$ .

$\frac{\frac{3}{2}}{20 + 5 (- (1))}$

Step 3.5.4

Multiply $5 (- (1))$ .

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Step 3.5.4.1

Multiply $- 1$ by $1$ .

$\frac{\frac{3}{2}}{20 + 5 \cdot - 1}$

Step 3.5.4.2

Multiply $5$ by $- 1$ .

$\frac{\frac{3}{2}}{20 - 5}$

Step 3.5.5

Subtract $5$ from $20$ .

$\frac{\frac{3}{2}}{15}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} \cdot \frac{1}{15}$

Step 3.7

Cancel the common factor of $3$ .

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Step 3.7.1

Factor $3$ out of $15$ .

$\frac{3}{2} \cdot \frac{1}{3 (5)}$

Step 3.7.2

Cancel the common factor.

$\frac{3}{2} \cdot \frac{1}{3 \cdot 5}$

Step 3.7.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{1}{5}$

Step 3.8

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$\frac{1}{2 \cdot 5}$

Step 3.9

Multiply $2$ by $5$ .

$\frac{1}{10}$