Absolute Value as Piecewise Functions

Lesson2.5Lesson2.5

Example

f (x) =

x + 1, if x < 1

2, if 1 ≤ x ≤ 3

(x-3)2 + 2, if x > 3

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:

f (x) = -x, if x < 0

x, if x ≥ 0

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.

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 – 18 (when x ≥ 4)

For (x-4) < 0

2[-(x-4)] – 10 = 2(-x + 4) – 10 = -2x + 8 – 10

= -2x – 2 (when x < 4))

More Examples:

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

For (x-4) ≥ 0

2(x – 4) – 10 = 2x – 8 – 10 = 2x – 18 (when x ≥ 4)

For (x-4) < 0

2[-(x-4)] – 10 = 2(-x + 4) – 10 = -2x + 8 – 10

= -2x – 2 (when x < 4))

Graphs of Both

y=2x-18y=-2x-2

EOCT Practice

A

EOCT Practice

C

Writing Abs. Value as Piecewise

Using a graph

Writing Abs. Value as Piecewise

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