2.4 Absolute Value Functions

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

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|

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| < 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