Finite Math Examples

$f (x) = 7 x - \frac{3}{4}$ , $(0, 100)$

Step 1

Write $f (x) = 7 x - \frac{3}{4}$ as an equation.

$y = 7 x - \frac{3}{4}$

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.

$\frac{f (100) - f (0)}{(100) - (0)}$

Step 2.2

Substitute the equation $y = 7 x - \frac{3}{4}$ for $f (100)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(7 (100) - \frac{3}{4}) - (7 (0) - \frac{3}{4})}{(100) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

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

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Step 3.1.1

Multiply $\frac{7 (100) - \frac{3}{4} - (7 (0) - \frac{3}{4})}{100 - (0)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{7 (100) - \frac{3}{4} - (7 (0) - \frac{3}{4})}{100 - (0)}$

Step 3.1.2

Combine.

$\frac{4 (7 (100) - \frac{3}{4} - (7 (0) - \frac{3}{4}))}{4 (100 - (0))}$

Step 3.2

Apply the distributive property.

$\frac{4 (7 (100)) + 4 (- \frac{3}{4}) + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{4 (7 (100)) + 4 (\frac{- 3}{4}) + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.3.2

Cancel the common factor.

$\frac{4 (7 (100)) + 4 (\frac{- 3}{4}) + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.3.3

Rewrite the expression.

$\frac{4 (7 (100)) - 3 + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $4 \cdot 7 \cdot 100$ .

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Step 3.4.1.1

Multiply $4$ by $7$ .

$\frac{28 \cdot 100 - 3 + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.1.2

Multiply $28$ by $100$ .

$\frac{2800 - 3 + 4 (- (7 (0) - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.2

Multiply $7$ by $0$ .

$\frac{2800 - 3 + 4 (- (0 - \frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.3

Subtract $\frac{3}{4}$ from $0$ .

$\frac{2800 - 3 + 4 (- - \frac{3}{4})}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.4

Multiply $- - \frac{3}{4}$ .

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Step 3.4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2800 - 3 + 4 (1 (\frac{3}{4}))}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.4.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{2800 - 3 + 4 (\frac{3}{4})}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.5

Cancel the common factor of $4$ .

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Step 3.4.5.1

Cancel the common factor.

$\frac{2800 - 3 + 4 (\frac{3}{4})}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.5.2

Rewrite the expression.

$\frac{2800 - 3 + 3}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.6

Subtract $3$ from $2800$ .

$\frac{2797 + 3}{4 \cdot 100 + 4 (- (0))}$

Step 3.4.7

Add $2797$ and $3$ .

$\frac{2800}{4 \cdot 100 + 4 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $4$ by $100$ .

$\frac{2800}{400 + 4 (- (0))}$

Step 3.5.2

Multiply $4 (- (0))$ .

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Step 3.5.2.1

Multiply $- 1$ by $0$ .

$\frac{2800}{400 + 4 \cdot 0}$

Step 3.5.2.2

Multiply $4$ by $0$ .

$\frac{2800}{400 + 0}$

Step 3.5.3

Add $400$ and $0$ .

$\frac{2800}{400}$

Step 3.6

Divide $2800$ by $400$ .

$7$