1. 2.4 Absolute Value Functions

2. Quiz Find the Domain and Range of f(x) = |x| Answer: Domain: (-∞, ∞)

3. What are we going to learn?
Nature of the graph of absolute value functions
Solve equations analytically and graphically
Solve related inequalities analytically and graphically

4. Absolute Value Definition:
- Informal: Absolute value is the magnitude of a quantity, regardless of direction; always positive; the distance from 0
- Formal:
f(x) = |x| =
x , if x ≥ 0
- x , if x < 0

5. Basic Properties of Absolute Values
|ab| = |a| * |b|
|a/b| = |a| / |b|
|a| = |-a|
|a| + |b| ≥ |a + b| (triangle inequality)

6. Absolute Value of Functions
Absolute value of any function f:
| f(x) | =
f(x) , if f(x) ≥ 0
- f(x) , if f(x) < 0
What happens to the graph of f(x) if we take its absolute value?

7. Absolute Value of Functions
f(x)
|f(x)| y y x x

8. Absolute Value of Functions
Given the graph of f(x) below, sketch the graph of |f(x)| y x

9. Solving Equations |f(x)| = K, solve the compound equation
f(x) = K or f(x) = -K
Example: |x - 3| = 7
Solve this equation analytically and graphically

10. Solving equations
Solve |2x + 7| = |6x – 1| analytically and graphically
solve for 2x + 7 = 6x – 1
and 2x + 7 = - (6x - 1)
Try: |3x + 1| = |2x - 7|

11. f(x) > M or f(x) < -M
Solving inequalities
Case 1: |f(x)| > M, solve the compound inequality
f(x) > M or f(x) < -M
Case 2: |f(x)| < M, solve the three-part inequality
-M < f(x) < M
Example: |2x + 1| < |x - 5| + 2 ≥ 6

12. Discussion
Solve
|3x + 2| = -2
| x -7 | > -1
| 1 – 2x| < -5

13. Homework
PG. 122: 3, 8 , 21, 33, 36, 49, 52, 55, 58, 61, 64, 70, 75, 80, 87, 93, 96
KEY: 52, 58, 70
Reading: 2.5 Piecewise-Defined Functions