Finite Math Examples

$f (x) = \frac{1}{r}$ , $(2, 3)$

Step 1

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

$y = \frac{1}{r}$

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 $r$ values of the two points.

$\frac{f (3) - f (2)}{(3) - (2)}$

Step 2.2

Substitute the equation $y = \frac{1}{r}$ for $f (3)$ and $f (2)$ , replacing $r$ in the function with the corresponding $r$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{3} - \frac{1}{2}}{3 - (2)}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\frac{1}{3} - \frac{1}{2}}{3 - (2)}$

Step 3.1.2

Combine.

$\frac{6 (\frac{1}{3} - \frac{1}{2})}{6 (3 - (2))}$

Step 3.2

Apply the distributive property.

$\frac{6 (\frac{1}{3}) + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- (2))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $3$ .

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Step 3.3.1.1

Factor $3$ out of $6$ .

$\frac{3 (2) \frac{1}{3} + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{3 \cdot 2 \frac{1}{3} + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{2 + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{2 + 6 (\frac{- 1}{2})}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.2.2

Factor $2$ out of $6$ .

$\frac{2 + 2 (3) \frac{- 1}{2}}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{2 + 2 \cdot 3 \frac{- 1}{2}}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{2 + 3 \cdot - 1}{6 \cdot 3 + 6 (- (2))}$

Step 3.3.3

Multiply $3$ by $- 1$ .

$\frac{2 - 3}{6 \cdot 3 + 6 (- (2))}$

Step 3.4

Subtract $3$ from $2$ .

$\frac{- 1}{6 \cdot 3 + 6 (- (2))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $6$ by $3$ .

$\frac{- 1}{18 + 6 (- (2))}$

Step 3.5.2

Multiply $6 (- (2))$ .

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Step 3.5.2.1

Multiply $- 1$ by $2$ .

$\frac{- 1}{18 + 6 \cdot - 2}$

Step 3.5.2.2

Multiply $6$ by $- 2$ .

$\frac{- 1}{18 - 12}$

Step 3.5.3

Subtract $12$ from $18$ .

$\frac{- 1}{6}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{1}{6}$