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7 functions Presentation 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