Precalculus Examples

$f (x) = \frac{x^{2} + 3}{x - 2}$ ; $[4, 9]$

Step 1

Write $f (x) = \frac{x^{2} + 3}{x - 2}$ as an equation.

$y = \frac{x^{2} + 3}{x - 2}$

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.

$\frac{f (9) - f (4)}{(9) - (4)}$

Step 2.2

Substitute the equation $y = \frac{x^{2} + 3}{x - 2}$ for $f (9)$ and $f (4)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(9 - 2) (4 - 2)$ .

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Step 3.1.1

Multiply $\frac{\frac{9^{2} + 3}{9 - 2} - \frac{4^{2} + 3}{4 - 2}}{9 - (4)}$ by $\frac{(9 - 2) (4 - 2)}{(9 - 2) (4 - 2)}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $9 - 2$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

Cancel the common factor of $4 - 2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{4^{2} + 3}{4 - 2}$ into the numerator.

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

Step 3.3.2.2

Factor $4 - 2$ out of $(9 - 2) (4 - 2)$ .

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

Step 3.3.2.3

Cancel the common factor.

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

Step 3.3.2.4

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Subtract $2$ from $4$ .

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

Step 3.4.2

Raise $9$ to the power of $2$ .

$\frac{2 (81 + 3) + (9 - 2) (- (4^{2} + 3))}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.3

Add $81$ and $3$ .

$\frac{2 \cdot 84 + (9 - 2) (- (4^{2} + 3))}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.4

Multiply $2$ by $84$ .

$\frac{168 + (9 - 2) (- (4^{2} + 3))}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.5

Subtract $2$ from $9$ .

$\frac{168 + 7 (- (4^{2} + 3))}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.6

Raise $4$ to the power of $2$ .

$\frac{168 + 7 (- (16 + 3))}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.7

Add $16$ and $3$ .

$\frac{168 + 7 (- 1 \cdot 19)}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.8

Multiply $7 (- 1 \cdot 19)$ .

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Step 3.4.8.1

Multiply $- 1$ by $19$ .

$\frac{168 + 7 \cdot - 19}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.8.2

Multiply $7$ by $- 19$ .

$\frac{168 - 133}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.4.9

Subtract $133$ from $168$ .

$\frac{35}{(9 - 2) (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Subtract $2$ from $9$ .

$\frac{35}{7 (4 - 2) \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.5.2

Subtract $2$ from $4$ .

$\frac{35}{7 \cdot 2 \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.5.3

Multiply $7 \cdot 2 \cdot 9$ .

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Step 3.5.3.1

Multiply $7$ by $2$ .

$\frac{35}{14 \cdot 9 + (9 - 2) (4 - 2) (- (4))}$

Step 3.5.3.2

Multiply $14$ by $9$ .

$\frac{35}{126 + (9 - 2) (4 - 2) (- (4))}$

Step 3.5.4

Subtract $2$ from $9$ .

$\frac{35}{126 + 7 (4 - 2) (- (4))}$

Step 3.5.5

Subtract $2$ from $4$ .

$\frac{35}{126 + 7 \cdot 2 (- (4))}$

Step 3.5.6

Multiply $7 \cdot 2 (- (4))$ .

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Step 3.5.6.1

Multiply $7$ by $2$ .

$\frac{35}{126 + 14 (- (4))}$

Step 3.5.6.2

Multiply $- 1$ by $4$ .

$\frac{35}{126 + 14 \cdot - 4}$

Step 3.5.6.3

Multiply $14$ by $- 4$ .

$\frac{35}{126 - 56}$

Step 3.5.7

Subtract $56$ from $126$ .

$\frac{35}{70}$

Step 3.6

Cancel the common factor of $35$ and $70$ .

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Step 3.6.1

Factor $35$ out of $35$ .

$\frac{35 (1)}{70}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $35$ out of $70$ .

$\frac{35 \cdot 1}{35 \cdot 2}$

Step 3.6.2.2

Cancel the common factor.

$\frac{35 \cdot 1}{35 \cdot 2}$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{2}$