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**Absolute Value as Piecewise Functions**

Lesson2.5

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**Example x + 1, if x < 1 2, if 1 ≤ x ≤ 3 (x-3)2 + 2, if x > 3**

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**Absolute Value as Piecewise**

We usually write an absolute value function as f (x)= x , but since absolute value is a measure of distance and distance is always positive, it also can be written as follows: -x, if x < 0 x, if x ≥ 0 f (x) =

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**Writing Abs. Value as Piecewise**

To identify the number in the domain, set x – h = 0 and solve for x. For I x - h I ≥ 0, simplify the equation given by distributing and combining like terms. For I x - h I < 0, substitute –(x - h) in place of I x - h I. Then, simplify the equation given by distributing and combining like terms.

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**Example: Write y = 2 Ix – 4I – 10 as a piecewise function.**

Use 4 in your domain. For (x-4) ≥ 0 2(x – 4) – 10 = 2x – 8 – 10 = 2x – (when x ≥ 4) For (x-4) < 0 2[-(x-4)] – 10 = 2(-x + 4) – 10 = -2x + 8 – 10 = -2x – 2 (when x < 4))

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**More Examples: Write y = 2 Ix – 4I – 10 as a piecewise function.**

For (x-4) ≥ 0 2(x – 4) – 10 = 2x – 8 – 10 = 2x – (when x ≥ 4) For (x-4) < 0 2[-(x-4)] – 10 = 2(-x + 4) – 10 = -2x + 8 – 10 = -2x – 2 (when x < 4))

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Graphs of Both y=-2x-2 y=2x-18

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EOCT Practice A

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EOCT Practice C

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**Writing Abs. Value as Piecewise**

Using a graph

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**Writing Abs. Value as Piecewise**

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Solving Equations. -132 = 4x – 5(6x – 10) -132 = 4x – 30x + 50 -132 = -26x + 50 -182 = -26x 7 = x.

Solving Equations. -132 = 4x – 5(6x – 10) -132 = 4x – 30x + 50 -132 = -26x + 50 -182 = -26x 7 = x.

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