FUNCTION

1
FUNCTION GRAPH

• CHAPTER 6
• DCT1043

2
CONTENT

• 6.1 Introduction to Functions
• 6.2 Operation on Functions
• 6.3 The Graph of a Function
• 6.4 Quadratic, Cubic Rational
• Function
• 6.5 Exponential Logarithmic Functions
• 6.6 Trigonometric Hyperbolic Functions
• 6.7 Piecewise-defined Function

3
6.1 Introduction to Function

• CHAPTER 6
• DCT1043

4
OBJECTIVES

• By the end of this topic, you should be able to
• Determine whether a relation represents a
  function
• Find the value of a function
• Find the domain and range of a function

5
Relation

• A relation is correspondence between 2 sets.

A B C D
a b c
x
y
a b c d
A B C
x corresponds to y y depends on x (x, y)
6
Functions

• A function f from X to Y is a relation that
  associates with each element of X exactly one
  element of Y.
• y f (x) is the value of f at the number x

f
f
x (input) Dependent variable
x1 x2
y
y f (x) Output Independent variable
Domain Range (input)
(output)
7
Example 1 Functions
a b c
a b c
1 2 3 4
1 2 3 4
2. A function 3. Not a
function
8
EXERCISE 1 Finding values for a function

• For the function f defined by
  evaluate

9
Domain Range of a Function

• If f is a function from A to B, we say that A is
  the domain of f and B is the codomain of f.
• If , f (a) b we say that b is the image of a
  and a is a pre-image of b.
• The range of f is the set of all images of
  elements of A.

A
B
f
f
b f (a) Image of a
a
b
a Pre-image of b
domain codomain / range
f maps A to B
10
Example 2

• What are the domain, codomain and range of the
  function that assigns grades to students as
  follows?
• Domain of f Adam, Bob, Chu, Deen, Emy
• Codomain of f A, B, C, D, E
• Range of f A, B, C, D because each grade
  except E as assigned to some
  student.

Adam Bob Chu Deen Emy
A B C D E
11
EXERCISE 2 relation function

• Determine whether each relation
• represents a function. If it a function, state
• the domain and range
• (1,4), (2,5), (3,6), (4,7)
• (1,4), (2,4), (3,5), (6,10)
• (-3,9), (-2,4), (0,0), (1,1), (-3,8)

12
Example 3

• Let f be the function from Z to Z that assigns
  the square of an integer to this integer.
• What are the domain, codomain and range of the
  function?
• Domain of f the set of all integers
• Codomain of f the set of all integers
• Range of f the set of all nonnegative
  integers that are perfect squares,namely
  , 0, 1, 4, 9, .

13
EXERCISE 3

• Find the domain and range of each of the
  following functions

14
6.2 Operation on Function

• CHAPTER 6
• DCT1043

15
OBJECTIVES

• By the end of this topic, you should be able to
• Form the Sum, Difference, Product and Quotient of
  Two Functions

16
Operations on Functions
The domain consists of the number x that are in
the domains of both f and g.
17
EXERCISE 4 Operations on Functions

• Let f and g be two functions defined as
  
• Find the following and determine the domain
• in each case

18
6.3 The Graph of a Function

• CHAPTER 6
• DCT1043

19
Objectives

• By the end of this topic, you should be able to
• Identify the graph of a function
• Obtain information from or about the graph of a
  function

20
Identifying the graph of a function

• A set of points in the xy-plane is the graph of a
  function if and only if every vertical line
  intersects the graph in at most one point.
• If (x, y) is a point on the graph of a function
  f, then y is the value of f at x, that is y f
  (x)

21
EXERCISE 5 graph of a function

• Which of the graphs below are the graph of a
  function?

22
EXERCISE 6 graph of a function

• Which of the graphs below are the graph of a
  function?

23
EXERCISE 7 Obtaining information from the graph
of a function
Consider the function
a. Is the point (1, ½) on the graph of f ? b.
If x -1, what is f (x) ? What points is on the
graph of f ? c. If x 2, what is x ? What
points is on the graph of f ?
24
Summary of Graphic Techniques
25
6.4 Quadratic, Cubic Rational
Functions

• CHAPTER 6
• DCT1043

26
Objectives

• By the end of this topic you should be able to
• Graph a quadratic function
• Graph a cubic function
• Graph a rational function

27
Quadratic Function
28
Quadratic Function
Compressing or Stretching y a f (x),
a gt 0
29
Quadratic Function
Reflection about the x-axis y - f (x)
30
Quadratic Function
Horizontal Shifts
y f (x h) , h gt 0
y f (x - h) , h gt 0
31
Quadratic Function
Vertical shifts
y f (x) k, k gt 0
y f (x) - k, k gt 0
32
EXERCISE 8 Quadratic function

• Graph the following quadratic functions

33
Cubic Function
34
EXERCISE 9 Cubic function

• Graph the following cubic functions

35
Rational Function
36
6.5 Exponential Logarithmic Functions

• CHAPTER 6
• DCT1043

37
Objectives

• By the end of this topic you should be able to
• Graph an exponential function
• Graph a logarithmic function

38
Exponential Function
39
EXERCISE 10 Exponential function

• Graph the following exponential functions

40
Logarithmic Function
41
EXERCISE 11 Logarithmic function

• Graph the following logarithmic functions

42
6.6 Trigonometric Hyperbolic Functions

• CHAPTER 6
• DCT1043

43
Objectives

• By the end of this topic you should be able to
• Graph a trigonometric function
• Graph a hyperbolic function

44
Trigonometric Function
45
Hyperbolic Function
46
6.7 Piecewise-defined Function

• CHAPTER 6
• DCT1043

47
Objectives

• By the end of this topic you should be able to
• Graph a piecewise defined function

48
Piecewise-defined Function

• Sometimes a function is defined differently on
  different part of its domain
• Example
• When functions are defined by more than one
  equation, they are called Piecewise-defined
  Function

49
EXERCISE 12 Piecewise-defined function

• Graph the following function

50
EXERCISE 13 Piecewise-defined function

• The function f is defined as
• (a) Find f (0), f (1), and f (2)
• (b) Determine the domain of f
• (c) Graph f
• (d) Use the graph to find the range of f

51
FUNCTION GRAPH

• THE END
• THANK YOU