Uploaded on

 

More in: Technology
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
1,091
On Slideshare
0
From Embeds
0
Number of Embeds
1

Actions

Shares
Downloads
52
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. Functions
    Prepared by:
    Teresita P. Liwanag - Zapanta
  • 2. OBJECTIVES
    • distinguish functions and relations
    • 3. identify domain and range of a function/relation
    evaluate functions/relations.
    • perform operation on functions/relations
    • 4. graph functions/relations
  • DEFINITION
    Relation is referred to as any set of ordered pair.
    Conventionally, It is represented by the ordered pair
    ( x , y ). x is called the first element or x-coordinate
    while y is the second element or y-coordinate of the
    ordered pair.
  • 5. Ways of Expressing a Relation
    4. Graph
    1. Set notation
    5. Mapping
    2. Tabular form
    3. Equation
  • 6. Example: Express the relation y = 2x;x= 0,1,2,3
    in 5 ways.
    .
    1. Set notation
    (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or
    (b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 }
    2. Tabular form
  • 7. y
    5
    5
    4
    3
    2
    1
    x
    5
    -4
    -2
    1
    3
    5
    -5
    -1
    4
    -3
    2
    -5
    -1
    -2
    -3
    -4
    -5
    -5
    3. Equation: y = 2x
    5. Mapping
    4. Graph
    x
    y

    0
    0

    1
    2

    2
    4
    6
    3
  • 8. DEFINITION: Domain and Range
    All the possible values of x is called the domain and all the possible values of y is called the range. In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively.
    Example: Identify the domain and range of the following
    relations.
    1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) }
    Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}
  • 9. 2.) S = { ( x , y ) s. t. y = | x | ; x  R }
    Answer: D: all real nos.R: all real nos. > 0
    3) y = x 2 – 5
    Answer. D: all real nos. R: all real nos. > -5
    4) | y | = x
    Answer: D: all real nos. > 0 R: all real nos.
  • 10. g)
    Answer:
    D: all real nos. except -2
    R: all real nos. except 2
    5.
    Answer :
    D: all real nos. > –1
    R: all real nos. > 0
    6.
    Answer:
    D: all real nos. < 3
    R: all real nos. except 0
    7.
  • 11. Exercises: Identify the domain and range of the
    following relations.
    1. {(x,y) | y = x 2 – 4 }
    2.
    5.
    7. y = 25 – x 2
    y = | x – 7 |
    6.
    4.
    3.
    8. y = (x 2 – 3) 2
    9.
    10.
  • 12. PROBLEM SET #5-1
    FUNCTIONS
    Identify the domain and range of the following relations.
  • 13. Definition: Function
    • A function is a special relation such that every first element is paired to a unique second element.
    • 14. It is a set of ordered pairs with no two pairs having the same first element.
  • Functions
    One-to-one and many-to-one functions
    Consider the following graphs
    and
    Each value of x maps to only one value of y . . .
    Each value of x maps to only one value of y . . .
    and each y is mapped from only one x.
    BUT manyother x values map to that y.
  • 15. is an example of a many-to-one function
    is an example of a one-to-one function
    Functions
    One-to-one and many-to-one functions
    Consider the following graphs
    and
    One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function.
  • 16. Example: Identify which of the following relations are functions.
    a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) }
    b) S = { ( x , y ) s. t. y = | x | ; x  R }
    c) y = x 2 – 5
    d) | y | = x
    e)
    f)
  • 17. DEFINITION: Function Notation
    • Letters like f , g , h and the likes are used to designate functions.
    • 18. When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f .
    • 19. The notation f ( x ) is read as “ f of x”.
  • EXAMPLE: Evaluate each function value
    1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ?
    2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?
    3. If h ( x ) = x 2 + 5 , find h ( x + 1 ).
    If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 ,
    Find: a) f(g(x)) b) g(f(x))
  • 20. Piecewise Defined Function
    A piecewise defined function is defined by different formulas on different parts of its domain.
    Example:
    if x<0
    if
  • 21. Piecewise Defined Function
    EXAMPLE: Evaluate the piecewise function at the
    indicated values.
    if x<0
    f(-2), f(-1), f(0), f(1), f(2)
    if
    if
    if
    if
    f(-5), f(0), f(1), f(5)
  • 22. DEFINITION: Operations on Functions
    If f (x) and g (x) are two functions, then
    Sum and Difference
    ( f + g ) ( x ) = f(x) + g(x)
    Product
    ( f g ) ( x ) = [ f(x) ] [ g(x) ]
    Quotient
    ( f / g ) ( x ) = f(x) / g(x)
    d) Composite
    ( f ◦ g ) ( x ) = f (g(x))
  • 23. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10
    evaluate each ofthe following functions
    f(-5)
    g(2)
    (f g)(5)
    (f - g)(4)
    f(7)+g(x)
    g(-1) – f(-4)
    (f ○ g)(x)
    (g ○ f)(x)
    (g ○ f)(2)
    (f○ g)
  • 24.  
  • 25.  
  • 26. DEFINITION: Graph of a Function
    • If f(x) is a function, then its graph is the set of all points
    (x,y) in the two-dimensional plane for which (x,y) is an
    ordered pair in f(x)
    • One way to graph a function is by point plotting.
    • 27. We can also find the domain and range from the
    graph of a function.
  • 28. Example: Graph each of the following functions.
    6.
  • 29. Graph of piecewise defined function
    The graph of a piecewise function consists of separate functions.
    Example: Graph each piecewise function.
    if
    if
    if
    if
    if
  • 30. Plot the points in the coordinate plane
    y
    x
    -2
    1
  • 31. Plot the points in the coordinate plane
    y
    x
    -2
    1
  • 32. Graph of absolute value function.
    Recall that
    if
    if
    Using the same method that we used in graphing
    piecewise function, we note that the graph of f
    coincides with the line y=x to the right of the y axis
    and coincides with the line y= -x the left of the y-axis.
  • 33. Example: Graph each of the follow functions.
    y = | x – 7 |
    y = x-| x - 2 |
    1.
    4.
  • 34. Plot the points in the coordinate plane
    y
    x
    -2
    1
  • 35.  
  • 36. Definition: Greatest integer function.
    The greatest integer function is defined by
    greatest integer less than or equal to x
    Example:
    1
    3
    0
    -4
    1
    0
    -1
    1
    0
    2
    0
    2
    1
  • 37. Definition: Least integer function.
    The least integer function is defined by
    least integer greater than or equal to x
    Example:
    2
    4
    0
    -3
    2
    1
    0
    2
    1
    2
    1
    3
    1
  • 38. Graph of greatest integer function.
    Sketch the graph of
  • 39. Plot the points in the coordinate plane
    y
    x
    -2
    1
  • 40. Graph of least integer function.
    Sketch the graph of
  • 41. y
    Plot the points in the coordinate plane
    x
    1
    -2