24 Absolute Value Functions by hcj

VIEWS: 2 PAGES: 13

									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:
                         x    , if x ≥ 0
          f(x) = |x| =
                         -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)    , if f(x) ≥ 0
        | f(x) | =
                       - 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

								
To top