Finite Math Examples

$f (x) = 4 x - \frac{1}{2}$ , $(1, 3)$

Step 1

Write $f (x) = 4 x - \frac{1}{2}$ as an equation.

$y = 4 x - \frac{1}{2}$

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 $y = 4 x - \frac{1}{2}$ for $f (3)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

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

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Step 3.1.1

Multiply $\frac{4 (3) - \frac{1}{2} - (4 (1) - \frac{1}{2})}{3 - (1)}$ by $\frac{2}{2}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $2 \cdot 4 \cdot 3$ .

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Step 3.4.1.1

Multiply $2$ by $4$ .

$\frac{8 \cdot 3 - 1 + 2 (- (4 (1) - \frac{1}{2}))}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.1.2

Multiply $8$ by $3$ .

$\frac{24 - 1 + 2 (- (4 (1) - \frac{1}{2}))}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.2

Multiply $4$ by $1$ .

$\frac{24 - 1 + 2 (- (4 - \frac{1}{2}))}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{24 - 1 + 2 (- (4 \cdot \frac{2}{2} - \frac{1}{2}))}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.4

Combine $4$ and $\frac{2}{2}$ .

$\frac{24 - 1 + 2 (- (\frac{4 \cdot 2}{2} - \frac{1}{2}))}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.5

Combine the numerators over the common denominator.

$\frac{24 - 1 + 2 (- \frac{4 \cdot 2 - 1}{2})}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Multiply $4$ by $2$ .

$\frac{24 - 1 + 2 (- \frac{8 - 1}{2})}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.6.2

Subtract $1$ from $8$ .

$\frac{24 - 1 + 2 (- \frac{7}{2})}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.7

Cancel the common factor of $2$ .

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Step 3.4.7.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$\frac{24 - 1 + 2 (\frac{- 7}{2})}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.7.2

Cancel the common factor.

$\frac{24 - 1 + 2 (\frac{- 7}{2})}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.7.3

Rewrite the expression.

$\frac{24 - 1 - 7}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.8

Subtract $1$ from $24$ .

$\frac{23 - 7}{2 \cdot 3 + 2 (- (1))}$

Step 3.4.9

Subtract $7$ from $23$ .

$\frac{16}{2 \cdot 3 + 2 (- (1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $2$ by $3$ .

$\frac{16}{6 + 2 (- (1))}$

Step 3.5.2

Multiply $2 (- (1))$ .

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Step 3.5.2.1

Multiply $- 1$ by $1$ .

$\frac{16}{6 + 2 \cdot - 1}$

Step 3.5.2.2

Multiply $2$ by $- 1$ .

$\frac{16}{6 - 2}$

Step 3.5.3

Subtract $2$ from $6$ .

$\frac{16}{4}$

Step 3.6

Divide $16$ by $4$ .

$4$