Function

A relation from set A to set B is a function if every element in A has one and only one element in B.

We can represent using these symbols:-

f:A®B or A^f®B

Eg:- A= {a,b,d} ; B = {c,e,f}

f : A®B = {(a,c),(b,c),(d,e)}

1.    A = {1,2,3}

2.    B = {1,4,5,6}

a)    {(1,4), (2,4), (1,5)}

            This is not a function because x-co-ordinate repeats

b)     {1,1),(2,5),(3,1)}

            This is a funcion because every element in A has one and only one element in B.

C) {(3,1), (2,1), (3,5)}

            It is not a function since x-co-ordinate repeats.

d) {(3,4), (1,1),(2,6)}

            It is a function

e) {(1,6), (2,1), (3,5)}

            It is a function

f) {(2,6), (3,6), (1,6)}

            It is a function

2) A = {1,2} ; B = {2,4,5}

a)    {(2,1), (4,2), (5,2)}

            It is a function.

b)     {(2,1), (4,1), (5,1)}

            It is a function

c)     {(5,2), (4,1), (5,1)}

            It is not a function because x co-ordinate repeats.

d)     {(2,1), (4,1), (4,2)}

            It is not a function because x-co ordinate repeats.

f) {(5,1), (5,2), (4,2), (2,2)}

            Not a function because x-co ordinate repeats.

3) A = {1,2,3} ; B = {c,d}

Maximum number of function from A to B = B^A

Maximum no of functions from B to A = A^B

\Maximum number of functions from A to B, here

            = B^A        n(A) = 3

            = 2³         n(B) = 2

            = 8

f₁  = {(1,c), (2,c), (3,c)}

f2 = {(1,d), (2,d),(3,d)}

f3 = {(1,c), (2,c), (3,d)}

f4 = {(1,c),(2,d),(3,d)}

f5 = {(1,d), (2,d), (3,c)}

f6 = {(1,c),(2,d), (3,c)}

f7 = {(1,d), (2,c), (3,d)}

f8 = {(1,d), (2,c), 3,c)}

4)

a)    f = {(1,2), (1,3),(1,4)}

This is not a function because the x - coordinate is repeated.

b)     g = {(1,2), (2,3), (3,4)}

This can be a function

c)     h = {(1,2), (2,2), (3,2)}

This is also a function

d)     j = {(1,1), (2,2), (3,3)}

This is a function

All functions are releations. But, all relations are not functions.

Domain of a function :-

The set of all x -elements of the ordered pairs of a function is called the domain of that function

Range of function :-

The set of all y - elements of the ordered pairs of a function is called the range of that function.

The y-elements is called the image of x-element

y = f(x) ® ordered pair = (x,y)

1)

a)    False

b)     True

c)     True

d)     True

e)    False

2)

a)    {1,2), (2,3), (3,4)}

Domain  = {1,2,3}

Range = {2,3,4}

b)     {(1,1)}, (2,¹/₂), (3, ¹/₃) }

Domain = {1,2,3}

Range = {1, ¹/_2, ¹/₃}

c)     {(1,a), (2,a), (3,a)}

Domain = {1,2,3}

Range = {a}

d)     {(1,a), (2,b), (3,a)}

Domain = {1,2,3}

Range = {a,b}

3)

R = {(2,3), (5,6), (8,9), (11,12)}

It is a function

Domain of  f = {2,5,8,11}

Range of    f = {3,6,9,12}

4)

A = {2,4,6}

B = {3,5,7}

a)     {(2,3), (4,5),(2,7)}

It is not a function because x coordinate is repeated.

b)     {(2,3), (4,7),(6,3)}

It is a function

Domain :- {2,4,6}

Range :- {3,7}

c)     {(2,3), (6,5),(4,5),(2,5)}

It is not a function because x co-ordinate repeats

d)     {(2,5), (4,5),(6,5)}

It is a function

Domain :- {2,4,6}

Range :- {5}

Representaion of functions :-

1)     Roster form

Eg:- A= {1,2,3} ; B = {3,6,11,18}

f = {(1,3), (2,6), (3,11)}

2) Set builder form /Rule method.

f = {(x,y) / xÎA, yÎB, y = x²+2}

2)    Arrow diagram

A                        B

1                               3

2                       6

2                               11

                         18

4) Row and column  method:-

5) Graph

       y

   13

   12

   11                *(3,11)

   10

     9

     8

     7

     6          *(2,6)

     5

     4

     3    *(1,3)

     2

     1

     0    1    2    3    4    5                                    x

1)

a)    This is not a function because the x-coordinate(2) repeats.

b)     This is a function because every element in A has a unique image in B.

c)     This is not a function because the x coordinates (3,5) repeat.

d)     This is a function because every element in A has a  unique image in B

e)    This is a function because every element in A has a unique image in B

f)      This is not a function because x co-ordinate repeats

g)     Not a function because domain and set A are njot identical

3)

a) 

                                     y

                                  5             *(2,5)

                                  4

                                  3      *(1,3)

                                  2

                                  1 *(0,1)

                  -2    -1     0      1     2    3

                (-1,1)  *    -1

                                     y

4)

a)    A= {a, b};  B = {1, 2}

No. of  possible functions from A to B =  B ^A

                                                               = 2 ²

                                                               = 4

A                 B                                   A                     B

          f1                                                        f2

a                 1                                    a                    1

b                 2                                    b                    2

A                 B                                   A                     B

          f3                                                        f4

a                 1                                    a                    1

b                 2                                    b                    2

b)     A = {1};  B = {2,3,4}

No. of  possible functions from A to  B = B ^A

                                                                = 3 ¹

                                                                = 3