Precalculus Examples

$f (x) = \frac{| | x | | + 2}{2}$ , if $0 \leq x < 6$

Step 1

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

$y = \frac{| | x | | + 2}{2}$

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.

$\frac{f (6) - f (0)}{(6) - (0)}$

Step 2.2

Substitute the equation $y = \frac{| | x | | + 2}{2}$ for $f (6)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{| | 6 | | + 2}{2}) - (\frac{| | 0 | | + 2}{2})}{(6) - (0)}$

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 $\frac{\frac{| | 6 | | + 2}{2} - \frac{| | 0 | | + 2}{2}}{6 - (0)}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{| | 6 | | + 2}{2} - \frac{| | 0 | | + 2}{2}}{6 - (0)}$

Step 3.1.2

Combine.

$\frac{2 (\frac{| | 6 | | + 2}{2} - \frac{| | 0 | | + 2}{2})}{2 (6 - (0))}$

Step 3.2

Apply the distributive property.

$\frac{2 \frac{| | 6 | | + 2}{2} + 2 (- \frac{| | 0 | | + 2}{2})}{2 \cdot 6 + 2 (- (0))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{2 \frac{| | 6 | | + 2}{2} + 2 (- \frac{| | 0 | | + 2}{2})}{2 \cdot 6 + 2 (- (0))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{| | 6 | | + 2 + 2 (- \frac{| | 0 | | + 2}{2})}{2 \cdot 6 + 2 (- (0))}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{| | 0 | | + 2}{2}$ into the numerator.

$\frac{| | 6 | | + 2 + 2 \frac{- (| | 0 | | + 2)}{2}}{2 \cdot 6 + 2 (- (0))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{| | 6 | | + 2 + 2 \frac{- (| | 0 | | + 2)}{2}}{2 \cdot 6 + 2 (- (0))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{| | 6 | | + 2 - (| | 0 | | + 2)}{2 \cdot 6 + 2 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{| 6 | + 2 - (| | 0 | | + 2)}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.2

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{6 + 2 - (| | 0 | | + 2)}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\frac{6 + 2 - (| 0 | + 2)}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\frac{6 + 2 - (0 + 2)}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.4

Add $0$ and $2$ .

$\frac{6 + 2 - 1 \cdot 2}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.5

Multiply $- 1$ by $2$ .

$\frac{6 + 2 - 2}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.6

Add $6$ and $2$ .

$\frac{8 - 2}{2 \cdot 6 + 2 (- (0))}$

Step 3.4.7

Subtract $2$ from $8$ .

$\frac{6}{2 \cdot 6 + 2 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $2$ by $6$ .

$\frac{6}{12 + 2 (- (0))}$

Step 3.5.2

Multiply $2 (- (0))$ .

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Step 3.5.2.1

Multiply $- 1$ by $0$ .

$\frac{6}{12 + 2 \cdot 0}$

Step 3.5.2.2

Multiply $2$ by $0$ .

$\frac{6}{12 + 0}$

Step 3.5.3

Add $12$ and $0$ .

$\frac{6}{12}$

Step 3.6

Cancel the common factor of $6$ and $12$ .

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Step 3.6.1

Factor $6$ out of $6$ .

$\frac{6 (1)}{12}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $6$ out of $12$ .

$\frac{6 \cdot 1}{6 \cdot 2}$

Step 3.6.2.2

Cancel the common factor.

$\frac{6 \cdot 1}{6 \cdot 2}$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{2}$