Functions and Their Graphs

1
Functions and Their Graphs

• Chapter 2

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2
Functions

• Section 2.1

3
Relations

• Relation A correspondence between two sets.
• x corresponds to y or y depends on x if a
  relation exists between x and y
• Denote by x ! y in this case.

4
Relations

• Example.

Person
Salary
Melissa John Jennifer Patrick
45,000 40,000 50,000
5
Relations

• Example.

Number
Number
0 1 1 2 2
0 1 4
6
Functions

• Function special kind of relation
• Each input corresponds to precisely one output
• If X and Y are nonempty sets, a function from X
  into Y is a relation that associates with each
  element of X exactly one element of Y

7
Functions

• Example.
• Problem Does this relation represent a function?
• Answer

Person
Salary
Melissa John Jennifer Patrick
45,000 40,000 50,000
8
Functions

• Example.
• Problem Does this relation represent a function?
• Answer

Number
Number
0 1 1 2 2
0 1 4
9
Domain and Range

• Function from X to Y
• Domain of the function the set X.
• If x in X
• The image of x or the value of the function at x
  The element y corresponding to x
• Range of the function the set of all values of
  the function

10
Domain and Range

• Example.
• Problem What is the range of this function?
• Answer

X
Y
3 2 1 0 1 2 3
0 1 4 9
11
Domain and Range

• Example. Determine whether the relation
  represents a function. If it is a function, state
  the domain and range.
• Problem
• Relation f(2,5), (6,3), (8,2), (4,3)g
• Answer

12
Domain and Range

• Example. Determine whether the relation
  represents a function. If it is a function, state
  the domain and range.
• Problem
• Relation f(1,7), (0, 3), (2,4), (1,8)g
• Answer

13
Equations as Functions

• To determine whether an equation is a function
• Solve the equation for y.
• If any value of x in the domain corresponds to
  more than one y, the equation doesnt define a
  function
• Otherwise, it does define a function.

14
Equations as Functions

• Example.
• Problem Determine if the equation
• x y2 9
• defines y as a function of x.
• Answer

15
Function as a Machine

• Accepts numbers from domain as input.
• Exactly one output for each input.

16
Finding Values of a Function

• Example. Evaluate each of the following for the
  function
• f(x) 3x2 2x
• (a) Problem f(3)
• Answer
• (b) Problem f(x) f(3)
• Answer
• (c) Problem f(x)
• Answer
• (d) Problem f(x)
• Answer
• (e) Problem f(x3)
• Answer

17
Finding Values of a Function

• Example. Evaluate the difference quotient of the
  function
• Problem f(x) 3x2 2x.
• Answer

18
Implicit Form of a Function

• A function given in terms of x and y is given
  implicitly.
• If we can solve an equation for y in terms of x,
  the function is given explicitly

19
Implicit Form of a Function

• Example. Find the explicit form of the implicit
  function.
• (a) Problem 3x y 5
• Answer
• (b) Problem xy x 1
• Answer

20
Important Facts

• For each x in the domain of f, there is exactly
  one image f(x) in the range
• An element in the range can result from more than
  one x in the domain
• We usually call x the independent variable
• y is the dependent variable

21
Finding the Domain

• If the domain isnt specified, it will always be
  the largest set of real numbers for which f(x) is
  a real number
• We cant take square roots of negative numbers
  (yet) or divide by zero

22
Finding the Domain

• Example. Find the domain of each of the following
  functions.
• (a) Problem f(x) x2 9
• Answer
• (b) Problem
• Answer
• (c) Problem
• Answer

23
Finding the Domain

• Example. A rectangular garden has a perimeter of
  100 feet.
• (a) Problem Express the area A of the garden as
  a function of the width w.
• Answer
• (b) Problem Find the domain of A(w)
• Answer

24
Operations on Functions

• Arithmetic on functions f and g
• Sum of functions
• (f g)(x) f(x) g(x)
• Difference of functions
• (f g)(x) f(x) g(x)
• Domains Set of all real numbers in the domains
  of both f and g.
• For both sum and difference

25
Operations on Functions

• Arithmetic on functions f and g
• Product of functions f and g is
• (f g)(x) f(x) g(x)
• The quotient of functions f and g is
• Domain of product Set of all real numbers in the
  domains of both f and g
• Domain of quotient Set of all real numbers in
  the domains of both f and g with g(x) ? 0

26
Operations on Functions

• Example. Given f(x) 2x2 3 and g(x) 4x3
  1.
• (a) Problem Find fg and its domain
• Answer
• (b) Problem Find f g and its domain
• Answer

27
Operations on Functions

• Example. Given f(x) 2x2 3 and g(x) 4x3
  1.
• (c) Problem Find fg and its domain
• Answer
• (d) Problem Find f/g and its domain
• Answer

28
Key Points

• Relations
• Functions
• Domain and Range
• Equations as Functions
• Function as a Machine
• Finding Values of a Function
• Implicit Form of a Function
• Important Facts
• Finding the Domain

29
Key Points (cont.)

• Operations on Functions

30
The Graph of a Function

• Section 2.2

31
Vertical-line Test

• Theorem. Vertical-Line TestA set of points in
  the xy-plane is the graph of a function if and
  only if every vertical line intersects the graphs
  in at most one point.

32
Vertical-line Test

• Example.
• Problem Is the graph that of a function?
• Answer

33
Vertical-line Test

• Example.
• Problem Is the graph that of a function?
• Answer

34
Finding Information From Graphs

• Example. Answer the questions about the graph.
• (a) Problem Find f(0)
• Answer
• (b) Problem Find f(2)
• Answer
• (c) Problem Find the domain
• Answer
• (d) Problem Find the range
• Answer

35
Finding Information From Graphs

• Example. Answer the questions about the graph.
• (e) Problem Find the x-intercepts
• Answer
• (f) Problem Find the y-intercepts
• Answer

36
Finding Information From Graphs

• Example. Answer the questions about the graph.
• (g) Problem How often does the line y 3
  intersect the graph?
• Answer
• (h) Problem For what values of x does f(x) 2?
• Answer
• (i) Problem For what values of x is f(x) gt 0?
• Answer

37
Finding Information From Formulas

• Example. Answer the following questions for the
  function
• f(x) 2x2 5
• (a) Problem Is the point (2,3) on the graph of
  y f(x)?
• Answer
• (b) Problem If x 1, what is f(x)? What is the
  corresponding point on the graph?
• Answer
• (c) Problem If f(x) 1, what is x? What is
  (are) the corresponding point(s) on the graph?
• Answer

38
Key Points

• Vertical-line Test
• Finding Information From Graphs
• Finding Information From Formulas

39
Properties of Functions

• Section 2.3

40
Even and Odd Functions

• Even function
• For every number x in its domain, the number x
  is also in the domain
• f(x) f(x)
• Odd function
• For every number x in its domain, the number x
  is also in the domain
• f(x) f(x)

41
Description of Even and Odd Functions

• Even functions
• If (x, y) is on the graph, so is (x, y)
• Odd functions
• If (x, y) is on the graph, so is (x, y)

42
Description of Even and Odd Functions

• Theorem. A function is even if and only if its
  graph is symmetric with respect to the y-axis.A
  function is odd if and only if its graph is
  symmetric with respect to the origin.

43
Description of Even and Odd Functions

• Example.
• Problem Does the graph represent a function
  which is even, odd, or neither?
• Answer

44
Description of Even and Odd Functions

• Example.
• Problem Does the graph represent a function
  which is even, odd, or neither?
• Answer

45
Description of Even and Odd Functions

• Example.
• Problem Does the graph represent a function
  which is even, odd, or neither?
• Answer

46
Identifying Even and Odd Functions from the
Equation

• Example. Determine whether the following
  functions are even, odd or neither.
• (a) Problem
• Answer
• (b) Problem g(x) 3x2 4
• Answer
• (c) Problem
• Answer

47
Increasing, Decreasing and Constant Functions

• Increasing function (on an open interval I)
• For any choice of x1 and x2 in I, with x1 lt x2,
  we have f(x1) lt f(x2)
• Decreasing function (on an open interval I)
• For any choice of x1 and x2 in I, with x1 lt x2,
  we have f(x1) gt f(x2)
• Constant function (on an open interval I)
• For all choices of x in I, the values f(x) are
  equal.

48
Increasing, Decreasing and Constant Functions
49
Increasing, Decreasing and Constant Functions

• Example. Answer the questions about the function
  shown.
• (a) Problem Where is the function increasing?
• Answer
• (b) Problem Where is the function decreasing?
• Answer
• (c) Problem Where is the function constant
• Answer

50
Increasing, Decreasing and Constant Functions

• WARNING!
• Describe the behavior of a graph in terms of its
  x-values.
• Answers for these questions should be open
  intervals.

51
Local Extrema

• Local maximum at c
• Open interval I containing x so that, for all x
  c in I, f(x) f(c).
• f(c) is a local maximum of f.
• Local minimum at c
• Open interval I containing x so that, for all x
  c in I, f(x) f(c).
• f(c) is a local minimum of f.
• Local extrema
• Collection of local maxima and minima

52
Local Extrema

• For local maxima
• Graph is increasing to the left of c
• Graph is decreasing to the right of c.
• For local minima
• Graph is decreasing to the left of c
• Graph is increasing to the right of c.

53
Local Extrema

• Example. Answer the questions about the given
  graph of f.
• (a) Problem At which number(s) does f have a
  local maximum?
• Answer
• (b) Problem At which number(s) does f have a
  local minimum?
• Answer

54
Average Rate of Change

• Slope of a line can be interpreted as the average
  rate of change
• Average rate of change If c is in the domain of
  y f(x)
• Also called the difference quotient of f at c

55
Average Rate of Change

• Example. Find the average rates of change of
• (a) Problem From 0 to 1.
• Answer
• (b) Problem From 0 to 3.
• Answer
• (c) Problem From 1 to 3
• Answer

56
Secant Lines

• Geometric interpretation to the average rate of
  change
• Label two points (c, f(c)) and (x, f(x))
• Draw a line containing the points.
• This is the secant line.
• Theorem. Slope of the Secant LineThe average
  rate of change of a function equals the slope of
  the secant line containing two points on its graph

57
Secant Lines
58
Secant Lines

• Example.
• Problem Find an equation of the secant line to
  containing (0, f(0)) and (5, f(5))
• Answer

59
Key Points

• Even and Odd Functions
• Description of Even and Odd Functions
• Identifying Even and Odd Functions from the
  Equation
• Increasing, Decreasing and Constant Functions
• Local Extrema
• Average Rate of Change

60
Key Points (cont.)

• Secant Lines

61
Linear Functions and Models

• Section 2.4

62
Linear Functions

• Linear function
• Function of the form f(x) mx b
• Graph Line with slope m and y-intercept b.
• Theorem. Average Rate of Change of Linear
  FunctionLinear functions have a constant
  average rate of change. The constant average rate
  of change of f(x) mx b is

63
Linear Functions

• Example.
• Problem Graph the linear functionf(x) 2x 5
• Answer

64
Application Straight-Line Depreciation

• Example. Suppose that a company has just
  purchased a new machine for its manufacturing
  facility for 120,000. The company chooses to
  depreciate the machine using the straight-line
  method over 10 years.For straight-line
  depreciation, the value of the asset declines by
  a fixed amount every year.

65
Application Straight-Line Depreciation

• Example. (cont.)
• (a) Problem Write a linear function that
  expresses the book value of the machine as a
  function of its age, x
• Answer
• (b) Problem Graph the linear function
• Answer

66
Application Straight-Line Depreciation

• Example. (cont.)
• (c) Problem What is the book value of the
  machine after 4 years?
• Answer
• (d) Problem When will the machine be worth
  20,000?
• Answer

67
Scatter Diagrams

• Example. The amount of money that a lending
  institution will allow you to borrow mainly
  depends on the interest rate and your annual
  income.The following data represent the annual
  income, I, required by a bank in order to lend L
  dollars at an interest rate of 7.5 for 30 years.

68
Scatter Diagrams
Annual Income, I () Loan Amount, L ()
15,000 44,600
20,000 59,500
25,000 74,500
30,000 89,400
35,000 104,300
40,000 119,200
45,000 134,100
50,000 149,000
55,000 163,900
60,000 178,800
65,000 193,700
70,000 208,600

• Example. (cont.)

69
Scatter Diagrams

• Example. (cont.)
• Problem Use a graphing utility to draw a scatter
  diagram of the data.
• Answer

70
Linear and Nonlinear Relationships
Linear
Linear
Nonlinear
Nonlinear
Linear
Nonlinear
71
Line of Best Fit

• For linearly related scatter diagram
• Line is line of best fit.
• Use graphing calculator to find
• Example.
• (a) Problem Use a graphing utility to find the
  line of best fit to the data in the last example.
• Answer

72
Line of Best Fit

• Example. (cont.)
• (b) Problem Graph the line of best fit from the
  last example on the scatter diagram.
• Answer

73
Line of Best Fit

• Example. (cont.)
• (c) Problem Determine the loan amount that an
  individual would qualify for if her income is
  42,000.
• Answer

74
Direct Variation

• Variation or proportionality.
• y varies directly with x, or is directly
  proportional to x
• There is a nonzero number such that y kx.
• k is the constant of proportionality.

75
Direct Variation

• Example. Suppose y varies directly with x.
  Suppose as well that y 15 when x 3.
• (a) Problem Find the constant of
  proportionality.
• Answer
• (b) Problem Find x when y 124.53.
• Answer

76
Key Points

• Linear Functions
• Application Straight-Line Depreciation
• Scatter Diagrams
• Linear and Nonlinear Relationships
• Line of Best Fit
• Direct Variation

77
Library of FunctionsPiecewise-defined Functions

• Section 2.5

78
Linear Functions

• f(x) mxb, m and b a real number
• Domain (1, 1)
• Range (1, 1) unless m 0
• Increasing on (1, 1) (if m gt 0)
• Decreasing on (1, 1) (if m lt 0)
• Constant on (1, 1) (if m 0)

79
Constant Function

• f(x) b, b a real number
• Special linear functions
• Domain (1, 1)
• Range fbg
• Even/odd/neither Even (also odd if b 0)
• Constant on (1, 1)
• x-intercepts None (unless b 0)
• y-intercept y b.

80
Identity Function

• f(x) x
• Special linear function
• Domain (1, 1)
• Range (1, 1)
• Even/odd/neither Odd
• Increasing on (1, 1)
• x-intercepts x 0
• y-intercept y 0.

81
Square Function

• f(x) x2
• Domain (1, 1)
• Range 0, 1)
• Even/odd/neither Even
• Increasing on (0, 1)
• Decreasing on (1, 0)
• x-intercepts x 0
• y-intercept y 0.

82
Cube Function

• f(x) x3
• Domain (1, 1)
• Range (1, 1)
• Even/odd/neither Odd
• Increasing on (1, 1)
• x-intercepts x 0
• y-intercept y 0.

83
Square Root Function

• Domain 0, 1)
• Range 0, 1)
• Even/odd/neither Neither
• Increasing on (0, 1)
• x-intercepts x 0
• y-intercept y 0

84
Cube Root Function

• Domain (1, 1)
• Range (1, 1)
• Even/odd/neither Odd
• Increasing on (1, 1)
• x-intercepts x 0
• y-intercept y 0

85
Reciprocal Function

• Domain x ? 0
• Range x ? 0
• Even/odd/neither Odd
• Decreasing on (1, 0) (0, 1)
• x-intercepts None
• y-intercept None

86
Absolute Value Function

• f(x) jxj
• Domain (1, 1)
• Range 0, 1)
• Even/odd/neither Even
• Increasing on (0, 1)
• Decreasing on (1, 0)
• x-intercepts x 0
• y-intercept y 0

87
Absolute Value Function

• Can also write the absolute value function as
• This is a piecewise-defined function.

88
Greatest Integer Function

• f(x) int(x)
• greatest integer less than or equal to x
• Domain (1, 1)
• Range Integers (Z)
• Even/odd/neither Neither
• y-intercept y 0
• Called a step function

89
Greatest Integer Function
90
Piecewise-defined Functions

• Example. We can define a function differently on
  different parts of its domain.
• (a) Problem Find f(2)
• Answer
• (b) Problem Find f(1)
• Answer
• (c) Problem Find f(2)
• Answer
• (d) Problem Find f(3)
• Answer

91
Key Points

• Linear Functions
• Constant Function
• Identity Function
• Square Function
• Cube Function
• Square Root Function
• Cube Root Function
• Reciprocal Function
• Absolute Value Function

92
Key Points (cont.)

• Greatest Integer Function
• Piecewise-defined Functions

93
Graphing Techniques Transformations

• Section 2.6

94
Transformations

• Use basic library of functions and
  transformations to plot many other functions.
• Plot graphs that look almost like one of the
  basic functions.

95
Shifts

• Example.
• Problem Plot f(x) x3, g(x) x3 1 and h(x)
  x3 2 on the same axes
• Answer

96
Shifts

• Vertical shift
• A real number k is added to the right side of a
  function y f(x),
• New function y f(x) k
• Graph of new function
• Graph of f shifted vertically up k units (if k gt
  0)
• Down jkj units (if k lt 0)

97
Shifts

• Example.
• Problem Use the graph of f(x) jxj to obtain
  the graph of g(x) jxj 2
• Answer

98
Shifts

• Example.
• Problem Plot f(x) x3, g(x) (x 1)3 and h(x)
  (x 2)3 on the same axes
• Answer

99
Shifts

• Horizontal shift
• Argument x of a function f is replaced by x h,
• New function y f(x h)
• Graph of new function
• Graph of f shifted horizontally right h units (if
  h gt 0)
• Left jhj units (if h lt 0)
• Also y f(x h) in latter case

100
Shifts

• Example.
• Problem Use the graph of f(x) jxj to obtain
  the graph of g(x) jx2j
• Answer

101
Shifts

• Example.
• Problem The graph of a function y f(x) is
  given. Use it to plot g(x) f(x 3) 2
• Answer

102
Compressions and Stretches

• Example.
• Problem Plot f(x) x3, g(x) 2x3 and
  on the same axes
• Answer

103
Compressions and Stretches

• Vertical compression/stretch
• Right side of function y f(x) is multiplied by
  a positive number a,
• New function y af(x)
• Graph of new function
• Multiply each y-coordinate on the graph of y
  f(x) by a.
• Vertically compressed (if 0 lt a lt 1)
• Vertically stretched (if a gt 1)

104
Compressions and Stretches

• Example.
• Problem Use the graph of f(x) x2 to obtain the
  graph of g(x) 3x2
• Answer

105
Compressions and Stretches

• Example.
• Problem Plot f(x) x3, g(x) (2x)3 and
  on the same axes
• Answer

106
Compressions and Stretches

• Horizontal compression/stretch
• Argument x of a function y f(x) is multiplied
  by a positive number a
• New function y f(ax)
• Graph of new function
• Divide each x-coordinate on the graph of y
  f(x) by a.
• Horizontally compressed (if a gt 1)
• Horizontally stretched (if 0 lt a lt 1)

107
Compressions and Stretches

• Example.
• Problem Use the graph of f(x) x2 to obtain the
  graph of g(x) (3x)2
• Answer

108
Compressions and Stretches

• Example.
• Problem The graph of a function y f(x) is
  given. Use it to plot g(x) 3f(2x)
• Answer

109
Reflections

• Example.
• Problem f(x) x3 1 and g(x) (x3 1) on
  the same axes
• Answer

110
Reflections

• Reflections about x-axis
• Right side of the function y f(x) is
  multiplied by 1,
• New function y f(x)
• Graph of new function
• Reflection about the x-axis of the graph of the
  function y f(x).

111
Reflections

• Example.
• Problem f(x) x3 1 and g(x) (x)3 1 on
  the same axes
• Answer

112
Reflections

• Reflections about y-axis
• Argument of the function y f(x) is multiplied
  by 1,
• New function y f(x)
• Graph of new function
• Reflection about the y-axis of the graph of the
  function y f(x).

113
Summary of Transformations
114
Summary of Transformations
115
Summary of Transformations
116
Summary of Transformations

• Example.
• Problem Use transformations to graph the
  function
• Answer

117
Key Points

• Transformations
• Shifts
• Compressions and Stretches
• Reflections
• Summary of Transformations

118
Mathematical Models Constructing Functions

• Section 2.7

119
Mathematical Models

• Example.
• Problem The volume V of a right circular
  cylinder is V ¼r2h. If the height is three
  times the radius, express the volume V as a
  function of r.
• Answer

120
Mathematical Models

• Example. Anne has 5000 feet of fencing available
  to enclose a rectangular field. One side of the
  field lies along a river, so only three sides
  require fencing.
• (a) Problem Express the area A of the rectangle
  as a function of x, where x is the length of the
  side parallel to the river.
• Answer

121
Mathematical Models

• Example (cont.)
• (b) Problem Graph A A(x) and find what value
  of x makes the area largest.
• Answer
• (c) Problem What value of x makes the area
  largest?
• Answer

122
Key Points

• Mathematical Models