1. Absolute Value as Piecewise Functions
Lesson2.5

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

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:
-x, if x < 0
x, if x ≥ 0
f (x) =

4. 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.

5. 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))

6. 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))

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

8. EOCT Practice A

9. EOCT Practice C

10. Writing Abs. Value as Piecewise
Using a graph

11. Writing Abs. Value as Piecewise
Try this one...