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TOPIC :        FUNCTIONS                                             Absolute Value Functions
                                                                                                                                      Eg: f (x) = x − 2
                                                                                             Function Notations                       1) Find the possible values of x
                                                                              • f :x → x+2                                            if f(x) = 3
 CONCEPTS MAP                                                                   (means the function f maps x onto ” x + 2             Solution
                                                                                                                                        x-2 =3 , x=5
                                                                              • f ( x) = x + 2 is read as ’ f of x is x + 2
                                                                                                                                      - (x - 2) = 3 , x = -1
                              RELATIONS                                       • x is the object , f (x) is the image
                                                                                                                                      2) Sketch the Graph of
                                                                              Examples:
                                                                               a) If 5 is the object , find the image                   f (x) = x − 2 for the
                                                                                  Solution : f ( 5) = 5 + 2 = 7,                        domain -1 ≤ x ≤ 6
     How to represent              Types Of Relations                                                                                 Solution:
        Relations                                                             b)    Given f (3 y) = 11 , find y.                      x = -1 , f(x) = 3
                                 1. One-to-one relation                             Solution: f ( 3y) = 3y + 2 = 11,                  x = 5 , f(x) = 3
    1) Arrow diagram                                                                                 3y=9 ,                           f(x) = x-3 = 0, x = 3
                                                                                                      y=3                                    f (x)
          a•             •1
                                       2
                                       4
                                                 3
                                                 5
                                                                F
                                                                                                                                                     3
          b•
                                       6         7              U                          Composite Functions
                         •2
                                                                N              • If a function f is followed by a function g ,                                                 x
          c•             •3                                                   we obtain the composite function g f .                      -1 0       3               6
                                 2. One-to-many relation        C                                                                     3) Corresponding
                                                                                             f                 g
          d•             •4                                     T         •                                                              Range:
                                                                                       x             f (x)             gf                 0 ≤ f(x) ≤ 3
    2) Ordered Pairs                   1         4              I
                                                                                                     gf
      (a,1) (b,2), (c,2),              3         5              O                                                                         Inverse functions (f-1)
      (d,3)                                      6                        •     In general   gf ≠ fg .
                                                                N                                                                     • Concept: f(x) = y ,
                                                                                                                                          Then, f –1 (y) = x
3 3) Graph                                                      S         •   How to determine composite function:                    • Eg:
                                 3. Many-to-one relation                      Example : Given f : x  x +1                             Given f : x  2x + 1. Find f –1
2                    ×                                                                           g : x  2x                           Solution:

1              × ×                     7         6                                      Determine i.) f g ii ) f 2                                                y −1
                                                                                                                                              y = 2x + 1 , x =                 s
                                       9                                       Solutions;                                                                           2
         ×                                       10
                                                                              i.) fg (x) = f (g (x)     (ii) f 2 (x) = f f (x)                                x −1
                                       11        14                                       = f ( 2x )                  = f (x +1)            So ; f –1 (x) =
         a     b c d                                                                                                                                            2
                                                                                          = 2x + 1                    = (x +1) +1
                                                                                                                      = x + 2         •Note: Only one – to – one functions will
Objects : a, b, c, d                                        Eg.• Given f : x  2x and                                                 give one – to – one inverse functions.
                                 4. Many-to-many relation
Domain : {a, b, c, d}                                            g f: x  3x - 1, find g.                                             •   ff-1(x) = f -1f (x) = x
                                                            Solution: g (f(x) = 3 x -1               Eg.• Given f : x  2x and
Codomain : {1, 2, 3,4}                1              5             g( 2x)      = 3x - 1                     f g: x  x + 3, find g.   f(x) = 2x+1                           x −1
                                                                                                     Solution: f (g(x) ) = x + 3                              f −1( x ) =
                                                                              y                                                                                               2
Images : 1,2,3                                       7      Lets 2x = y, x=      ,                               2 g(x) = x + 3
                                      2                                      2                                                                    ax + b                 − dx + b
Range : { 1, 2, 3 }                                                                                                      x+3           f ( x) =               f ( x) =
                                                                      y              3x                        g ( x) =                           cx + d                  cx − a
                                                            g(y) = 3( ) -1,g(x) =        −1                                2
                                                                      2               2

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Concept map function

  • 1. TOPIC : FUNCTIONS Absolute Value Functions Eg: f (x) = x − 2 Function Notations 1) Find the possible values of x • f :x → x+2 if f(x) = 3 CONCEPTS MAP (means the function f maps x onto ” x + 2 Solution x-2 =3 , x=5 • f ( x) = x + 2 is read as ’ f of x is x + 2 - (x - 2) = 3 , x = -1 RELATIONS • x is the object , f (x) is the image 2) Sketch the Graph of Examples: a) If 5 is the object , find the image f (x) = x − 2 for the Solution : f ( 5) = 5 + 2 = 7, domain -1 ≤ x ≤ 6 How to represent Types Of Relations Solution: Relations b) Given f (3 y) = 11 , find y. x = -1 , f(x) = 3 1. One-to-one relation Solution: f ( 3y) = 3y + 2 = 11, x = 5 , f(x) = 3 1) Arrow diagram 3y=9 , f(x) = x-3 = 0, x = 3 y=3 f (x) a• •1 2 4 3 5 F 3 b• 6 7 U Composite Functions •2 N • If a function f is followed by a function g , x c• •3 we obtain the composite function g f . -1 0 3 6 2. One-to-many relation C 3) Corresponding f g d• •4 T • Range: x f (x) gf 0 ≤ f(x) ≤ 3 2) Ordered Pairs 1 4 I gf (a,1) (b,2), (c,2), 3 5 O Inverse functions (f-1) (d,3) 6 • In general gf ≠ fg . N • Concept: f(x) = y , Then, f –1 (y) = x 3 3) Graph S • How to determine composite function: • Eg: 3. Many-to-one relation Example : Given f : x  x +1 Given f : x  2x + 1. Find f –1 2 × g : x  2x Solution: 1 × × 7 6 Determine i.) f g ii ) f 2 y −1 y = 2x + 1 , x = s 9 Solutions; 2 × 10 i.) fg (x) = f (g (x) (ii) f 2 (x) = f f (x) x −1 11 14 = f ( 2x ) = f (x +1) So ; f –1 (x) = a b c d 2 = 2x + 1 = (x +1) +1 = x + 2 •Note: Only one – to – one functions will Objects : a, b, c, d Eg.• Given f : x  2x and give one – to – one inverse functions. 4. Many-to-many relation Domain : {a, b, c, d} g f: x  3x - 1, find g. • ff-1(x) = f -1f (x) = x Solution: g (f(x) = 3 x -1 Eg.• Given f : x  2x and Codomain : {1, 2, 3,4} 1 5 g( 2x) = 3x - 1 f g: x  x + 3, find g. f(x) = 2x+1 x −1 Solution: f (g(x) ) = x + 3 f −1( x ) = y 2 Images : 1,2,3 7 Lets 2x = y, x= , 2 g(x) = x + 3 2 2 ax + b − dx + b Range : { 1, 2, 3 } x+3 f ( x) = f ( x) = y 3x g ( x) = cx + d cx − a g(y) = 3( ) -1,g(x) = −1 2 2 2