Precalculus Examples

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

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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 1.2

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

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

Step 2

Simplify the expression.

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Step 2.1

Multiply the numerator and denominator of the fraction by $(3 (3) - 1) (3 (1) - 1)$ .

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Step 2.1.1

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

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

Step 2.1.2

Combine.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $3 \cdot 3 - 1$ .

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Step 2.3.1.1

Multiply by $1$ .

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

Step 2.3.1.2

Cancel the common factor.

Step 2.3.1.3

Rewrite the expression.

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

Step 2.3.2

Multiply $3$ by $1$ .

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

Step 2.3.3

Cancel the common factor of $3 \cdot 1 - 1$ .

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Step 2.3.3.1

Move the leading negative in $- \frac{4}{3 (1) - 1}$ into the numerator.

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

Step 2.3.3.2

Factor $3 \cdot 1 - 1$ out of $(3 \cdot 3 - 1) (3 \cdot 1 - 1)$ .

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

Step 2.3.3.3

Multiply by $1$ .

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

Step 2.3.3.4

Cancel the common factor.

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

Step 2.3.3.5

Rewrite the expression.

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

Step 2.3.4

Multiply $3$ by $3$ .

$\frac{(3 - 1) \cdot 4 + (9 - 1) \cdot - 4}{(3 \cdot 3 - 1) (3 \cdot 1 - 1) \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Subtract $1$ from $3$ .

$\frac{2 \cdot 4 + (9 - 1) \cdot - 4}{(3 \cdot 3 - 1) (3 \cdot 1 - 1) \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.4.2

Multiply $2$ by $4$ .

$\frac{8 + (9 - 1) \cdot - 4}{(3 \cdot 3 - 1) (3 \cdot 1 - 1) \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.4.3

Subtract $1$ from $9$ .

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

Step 2.4.4

Multiply $8$ by $- 4$ .

$\frac{8 - 32}{(3 \cdot 3 - 1) (3 \cdot 1 - 1) \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.4.5

Subtract $32$ from $8$ .

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

Step 2.5

Simplify the denominator.

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Step 2.5.1

Multiply $3$ by $3$ .

$\frac{- 24}{(9 - 1) (3 \cdot 1 - 1) \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.5.2

Subtract $1$ from $9$ .

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

Step 2.5.3

Multiply $3$ by $1$ .

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

Step 2.5.4

Subtract $1$ from $3$ .

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

Step 2.5.5

Multiply $8 \cdot 2 \cdot 3$ .

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Step 2.5.5.1

Multiply $8$ by $2$ .

$\frac{- 24}{16 \cdot 3 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.5.5.2

Multiply $16$ by $3$ .

$\frac{- 24}{48 + (3 \cdot 3 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.5.6

Multiply $3$ by $3$ .

$\frac{- 24}{48 + (9 - 1) (3 \cdot 1 - 1) (- (1))}$

Step 2.5.7

Subtract $1$ from $9$ .

$\frac{- 24}{48 + 8 (3 \cdot 1 - 1) (- (1))}$

Step 2.5.8

Multiply $3$ by $1$ .

$\frac{- 24}{48 + 8 (3 - 1) (- (1))}$

Step 2.5.9

Subtract $1$ from $3$ .

$\frac{- 24}{48 + 8 \cdot 2 (- (1))}$

Step 2.5.10

Multiply $8 \cdot 2 (- (1))$ .

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Step 2.5.10.1

Multiply $8$ by $2$ .

$\frac{- 24}{48 + 16 (- (1))}$

Step 2.5.10.2

Multiply $- 1$ by $1$ .

$\frac{- 24}{48 + 16 \cdot - 1}$

Step 2.5.10.3

Multiply $16$ by $- 1$ .

$\frac{- 24}{48 - 16}$

Step 2.5.11

Subtract $16$ from $48$ .

$\frac{- 24}{32}$

Step 2.6

Reduce the expression by cancelling the common factors.

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Step 2.6.1

Cancel the common factor of $- 24$ and $32$ .

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Step 2.6.1.1

Factor $8$ out of $- 24$ .

$\frac{8 (- 3)}{32}$

Step 2.6.1.2

Cancel the common factors.

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Step 2.6.1.2.1

Factor $8$ out of $32$ .

$\frac{8 \cdot - 3}{8 \cdot 4}$

Step 2.6.1.2.2

Cancel the common factor.

$\frac{8 \cdot - 3}{8 \cdot 4}$

Step 2.6.1.2.3

Rewrite the expression.

$\frac{- 3}{4}$

Step 2.6.2

Move the negative in front of the fraction.

$- \frac{3}{4}$