2.4 Absolute Value Functions
Quiz

Find the Domain and Range of f(x) = |x|

Answer:
Domain: (-∞, ∞)
Range: [ 0 , ∞)
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

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
Basic Properties of Absolute Values
|ab| = |a| * |b|
 |a/b| = |a| / |b|
 |a| = |-a|
 |a| + |b| ≥ |a + b| (triangle inequality)

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?
Absolute Value of Functions
f(x)
|f(x)|
y
y
x
x
Absolute Value of Functions

Given the graph of f(x) below, sketch the
graph of |f(x)|
y
x
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
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|
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| < 5
|x - 5| + 2 ≥ 6
Discussion

Solve
|3x + 2| = -2
| x -7 | > -1
| 1 – 2x| < -5
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