Finite Math Examples

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

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Divide $- 1$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Combine.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2 (- - 1)$ .

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Step 3.5.1.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.1.2

Multiply $2$ by $1$ .

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

Step 3.5.2

Add $- 1$ and $2$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $2$ by $2$ .

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

Step 3.6.2

Multiply $2 (- (1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $1$ .

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

Step 3.6.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{4 - 2}$

Step 3.6.3

Subtract $2$ from $4$ .

$\frac{1}{2}$