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2.4 Absolute Value Functions

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

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2.4 Absolute Value Functions

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  1. 2.4 Absolute Value Functions

  2. Quiz • Find the Domain and Range of f(x) = |x| • Answer: Domain: (-∞, ∞) Range: [ 0 , ∞)

  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 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. 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

  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

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