2.1 Represent Relations and Functions by chenmeixiu


									2.1: Represent Relations and Functions

1.   To determine if a relation is a function
2.   To find the domain and range of a relation
     or function
3.   To classify and evaluate functions
4.   To distinguish between discrete and
     continuous functions
As a group, define      Relation      Function
 each of these
 without your book.     Input         Output
 Give an example of
 each word and          Domain        Range
 leave a bit of space
                        Independent   Dependent
 for additions and      Variable      Variable
A mathematical relation is the pairing up
  (mapping) of inputs and outputs.
A mathematical relation is the pairing up
  (mapping) of inputs and outputs.

• Domain: the set of all input values
• Range: the set of all output values
A mathematical relation is the pairing up
  (mapping) of inputs and outputs.

  What’s the domain and range of each relation?
Exercise 1
Consider the relation given by the ordered
  pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3).
1. Identify the domain and range
2. Represent the relation as a graph and as
   a mapping diagram
Calvin and Hobbes!

A toaster is an example of a function. You
  put in bread, the toaster performs a
  toasting function, and out pops toasted
Calvin and Hobbes!

“What comes out of a toaster?”
“It depends on what you put in.”
  – You can’t input bread and expect a waffle!
What’s Your Function?
A function is a relation
  in which each input
  has exactly one
• A function is a          Functions
  dependent relation
• Output depends on
  the input
What’s Your Function?
A function is a relation
  in which each input
  has exactly one
• Each output does not     Functions
  necessarily have only
  one input
How Many Girlfriends?
If you think of the
   inputs as boys and
   the output as girls,
   then a function
   occurs when each
   boy has only one
   girlfriend. Otherwise
   the boy gets in BIG
What’s a Function Look Like?
What’s a Function Look Like?
What’s a Function Look Like?
What’s a Function Look Like?
Exercise 2a
Tell whether or not each table represents a
 function. Give the domain and range of
 each relationship.
Exercise 2b
The size of a set is called its cardinality.
 What must be true about the cardinalities
 of the domain and range of any function?
Exercise 3
Which sets of ordered pairs represent
1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}
4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}
Exercise 4
Which of the
 What is an
 easy way to
 tell that each
 input has only
 one output?
Vertical Line Test
A relation is a function iff no vertical line
  intersects the graph of the relation at more
  than one point
If it does,
    then an
    input has
    more than
                 Function       Not a Function
Domain and Range: Graphs
• Domain: All x-
  values (L → R)
   – {x: -∞ < x < ∞}
                                                  than or
• Range: All y-                                   to -4
  values (D ↑ U)
   – {y: y ≥ -4}       Domain: All real numbers
Exercise 5
Determine the domain and range of each
Domain and Range: Equations
• Domain: What you are allowed to plug in
  for x.
   – Easier to ask what you can’t plug in for x.

• Range: What you can get out for y using
  the domain.
   – Easier to ask what you can’t get for y.
Exercise 6
Determine the domain of each function.
1. y = x2 + 2

2. y  x  2

3. y 
Protip: Domains of Equations
When you have to find the domain of a
  function given its equation there’s really
  only two limiting factors:
1. The denominator of any fractions can’t be
  – Set denominators ≠ zero and solve
2. Square roots can’t be negative
  – Set square roots ≥ zero and solve
Dependent Quantities
Functions can also be thought of as
  dependent relationships. In a function, the
  value of the output depends on the value
  of the input.
• Independent quantity: Input values, x-
  values, domain
• Dependent quantity: Output value, which
  depends on the input value, y-values,
Exercise 7
The number of pretzels, p, that can be
 packaged in a box with a volume of V cubic
 units is given by the equation p = 45V + 10.
 In this relationship, which is the dependent
Function Notation
In an equation, the dependent variable is
  usually represented as f (x).
• Read “f of x”
  –   f = name of function; x = independent variable
  –   Takes place of y
  –   f (x) does NOT mean multiplication!
  –   f (3) means “the function evaluated at 3”
      where you plug 3 in for x.
Exercise 8
Evaluate each function when x = −3.
1. f (x) = −2x3 + 5

2. g (x) = 12 – 8x
Flavors of Functions
Functions come in a variety of flavors. You
 will need to be able to distinguish a linear
 from a nonlinear function.
     Linear Function      Nonlinear Function
 f (x) = 3x             f (x) = x2 – 2x + 5
 g (x) = ½ x – 5        g (x) = 1/x
 h (x) = 15 – 5x        h (x) = |x| + 2
Analog vs. Digital
• Analog: A signal created by some physical
  – Sound, temperature, etc.
  – Contain an infinite amount of data
• Digital: A numerical representation of an
  analog signal created by samples
  – Not continuous = set of points
  – Contain a finite amount of data
Digital Signal Processing
                      Original Analog Signal

    Digital Samples
Digital Signal Processing
Digital signal processing is about converting
  an analog signal into digital information,
  doing something to it, and usually
  converting it back into an analog signal.
Continuous vs. Discrete
• Continuous Function      • Discrete Function
  – A function whose         – A function whose
    graph consists of an       graph consists of a set
    unbroken curve             of discontinuous points
Exercise 9
Determine whether each situation describes
  a continuous or a discrete function.
1. The cheerleaders are selling candy bars for
   $1 each to pay for new pom-poms. The
   function f (x) gives the amount of money
   collected after selling x bars.
2. Kenny determined that his shower head
   releases 1.9 gallons of water per minute.
   The function V(x) gives the volume of water
   released after x minutes.
• P. 76-79: 1-3, 9-21
  odd, 24, 34-38, 41,
  45, 46, 51, 52
• P. 81: 1-9 odd
• Working with
  Domain and Range
  Worksheet: 1-3, 6-
  11, 13, 14, 15, 16,
  18, 20, 21

To top