Section 1'4 Functions

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Section 1.4 Functions

• Functions

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What you should learn

• How to determine whether relations between two
  variables are functions.
• How to use function notation and evaluate two
  functions.
• How to find the domains of functions.
• How to use functions to model and solve real-life
  problems.

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Relation

• A relation is a set of ordered pairs of real
  numbers.
• F (3, 2) (4, 1) (2, 4) (1, 3)
• If I say (2, __ ) , can you fill in the blank?
• G (3, 3) (4, 1) (2, 1) (1, 3)
• If I say (4, __ ) , can you fill in the blank?

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DomainF (3, 2) (4, 1) (2, 4) (1, 3)

• In a relation the set of all of the values of the
  independent variable is called the domain.
• What is the domain of F?
• 3, 4, 2, 1
• Does G (3, 3) (4, 1) (2, 1) (1, 3) have the
  same domain?

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Range G (3, 3) (4, 1) (2, 1) (1, 3)

• In a relation the set of all of the values of the
  dependent variable is called the range.
• What is the range of G?
• 3, 1
• Does F (3, 2) (4, 1) (2, 4) (1, 3) have the
  same range?

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(Domain, Range)

• Notice the alphabetical characteristic of Domain
  and Range.
• (x, y)
• (a, b)
• (abscissa, ordinate)
• Unfortunately (independent, dependent) breaks the
  rule.

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Function

• A function is a relation in which , for each
  value of the first component there is exactly one
  value of the second component.
• H (3, 2) (4, 1) (3, 4) (1, 3)
• K (2, 3) (4, 1) (3, 2) (1, 3)
• H is not a function,but K is a function.

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Definition of a Function (page 27)

• A function from set A to set B is a relation that
  assigns to each element x in the set A exactly
  one element y in the set B.
• The set A is the domain (or set of inputs) of the
  function f.
• The set B contains the range (or the set of
  outputs)

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Function Expressed as a Mapping
Domain
Range

• F
• (A,1)
• (C, 2)
• (B, 3)

A
1
C
2
3
B
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Function Expressed as a Mapping
Domain
Range

• G
• (A,1)
• (C, 2)
• (B, 3)
• (A, 4)

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A
1
C
2
3
B
Since A goes to two ranges G is not a function.
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Characteristics of a function from Set A to Set B
(page 40)

• Each element in A must be matched with an element
  in B.
• Some elements in B may not be matched with any
  element in A. (leftovers)
• Two or more elements in A may be matched with the
  same element in B.
• An element in A (the domain) cannot be matched
  with two different elements in B.

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Four Ways to Represent a Function

• Verbally by a sentence that describes how the
  input variable is related to the output variable.
• Numerically by a table or a list of ordered pairs
  that matches input values with output values
• Graphically by points on a graph in a coordinate
  plane in which the inputs are represented on the
  horizontal axis and the output values are
  represented by the vertical axis.
• Algebraically by an equation in two variables.

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Testing for Functions Example 1a

• Determine whether the relation represents y as a
  function of x.
• The input value x is the number of
  representatives from a state, and the output
  value y is the number of senators.
• (x, 2)
• This is a constant function.

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Testing for Functions Example 1b

• Determine whether the relation represents y as a
  function of x.
• Since x 2 has two outputs the table does not
  describe a function.

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Testing for Functions Represented Algebraically
Example 2a

• Solve for y
• For each value of x there is only one value for
  y.
• So y is a function of x.

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Testing for Functions Represented Algebraically
Example 2b

• Solve for y
• For each value of x there are two values for y.
• So y is not a function of x.

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Functional Notation

• y F(x)
• F(x) read F of x
• It does not mean F x (multiplication)

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Functional Notation

• Consider y 2x 5
• Suppose that you wanted to tell someone to
  substitute in x 3 into an equation.
• With functional notation y 2x 5 becomes f(x)
  2x 5.
• And f(3) means substitute in 3 everyplace you see
  an x.

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Example 3a Evaluating a FunctionFind g(2)
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Example 3b Evaluating a FunctionFind g(t)
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Example 3c Evaluating a FunctionFind g(x2)
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Example 4 A Piecewise-Defined Function
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The Domain of a Function

• The implied domain is the set of all real numbers
  for which the expression is defined.
• For what values of x is f(x) undefined?

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The Domain of a Function

• The implied domain is the set of all real numbers
  for which the expression is defined.
• For what values of x is g(x) undefined?

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Example 7 Baseball

• A baseball is hit at a point 3 feet above the
  ground at a velocity of 100 feet per second and
  an angle of 45. The path of the ball is given by
  the function
• Will the baseball clear a10-foot fence located
  300 feet from home plate?

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Example 9 Evaluating a Difference Quotient

• For
• find