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1 | 518-521 | 3 Types of Functions
The notion of a function along with some special functions like identity function, constant
function, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been
studied As the concept of function is of paramount importance in mathematics and
among other disciplines as well, we would like to extend our study about function from
where we finished earlier |
1 | 519-522 | along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been
studied As the concept of function is of paramount importance in mathematics and
among other disciplines as well, we would like to extend our study about function from
where we finished earlier In this section, we would like to study different types of
functions |
1 | 520-523 | Addition, subtraction, multiplication and division of two functions have also been
studied As the concept of function is of paramount importance in mathematics and
among other disciplines as well, we would like to extend our study about function from
where we finished earlier In this section, we would like to study different types of
functions Consider the functions f1, f2, f3 and f4 given by the following diagrams |
1 | 521-524 | As the concept of function is of paramount importance in mathematics and
among other disciplines as well, we would like to extend our study about function from
where we finished earlier In this section, we would like to study different types of
functions Consider the functions f1, f2, f3 and f4 given by the following diagrams In Fig 1 |
1 | 522-525 | In this section, we would like to study different types of
functions Consider the functions f1, f2, f3 and f4 given by the following diagrams In Fig 1 2, we observe that the images of distinct elements of X1 under the function
f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,
namely b |
1 | 523-526 | Consider the functions f1, f2, f3 and f4 given by the following diagrams In Fig 1 2, we observe that the images of distinct elements of X1 under the function
f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,
namely b Further, there are some elements like e and f in X2 which are not images of
any element of X1 under f1, while all elements of X3 are images of some elements of X1
under f3 |
1 | 524-527 | In Fig 1 2, we observe that the images of distinct elements of X1 under the function
f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,
namely b Further, there are some elements like e and f in X2 which are not images of
any element of X1 under f1, while all elements of X3 are images of some elements of X1
under f3 The above observations lead to the following definitions:
Definition 5 A function f : X β Y is defined to be one-one (or injective), if the images
of distinct elements of X under f are distinct, i |
1 | 525-528 | 2, we observe that the images of distinct elements of X1 under the function
f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same,
namely b Further, there are some elements like e and f in X2 which are not images of
any element of X1 under f1, while all elements of X3 are images of some elements of X1
under f3 The above observations lead to the following definitions:
Definition 5 A function f : X β Y is defined to be one-one (or injective), if the images
of distinct elements of X under f are distinct, i e |
1 | 526-529 | Further, there are some elements like e and f in X2 which are not images of
any element of X1 under f1, while all elements of X3 are images of some elements of X1
under f3 The above observations lead to the following definitions:
Definition 5 A function f : X β Y is defined to be one-one (or injective), if the images
of distinct elements of X under f are distinct, i e , for every x1, x2 β X, f(x1) = f(x2)
implies x1 = x2 |
1 | 527-530 | The above observations lead to the following definitions:
Definition 5 A function f : X β Y is defined to be one-one (or injective), if the images
of distinct elements of X under f are distinct, i e , for every x1, x2 β X, f(x1) = f(x2)
implies x1 = x2 Otherwise, f is called many-one |
1 | 528-531 | e , for every x1, x2 β X, f(x1) = f(x2)
implies x1 = x2 Otherwise, f is called many-one The function f1 and f4 in Fig 1 |
1 | 529-532 | , for every x1, x2 β X, f(x1) = f(x2)
implies x1 = x2 Otherwise, f is called many-one The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3
in Fig 1 |
1 | 530-533 | Otherwise, f is called many-one The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3
in Fig 1 2 (ii) and (iii) are many-one |
1 | 531-534 | The function f1 and f4 in Fig 1 2 (i) and (iv) are one-one and the function f2 and f3
in Fig 1 2 (ii) and (iii) are many-one Definition 6 A function f : X β Y is said to be onto (or surjective), if every element
of Y is the image of some element of X under f, i |
1 | 532-535 | 2 (i) and (iv) are one-one and the function f2 and f3
in Fig 1 2 (ii) and (iii) are many-one Definition 6 A function f : X β Y is said to be onto (or surjective), if every element
of Y is the image of some element of X under f, i e |
1 | 533-536 | 2 (ii) and (iii) are many-one Definition 6 A function f : X β Y is said to be onto (or surjective), if every element
of Y is the image of some element of X under f, i e , for every y β Y, there exists an
element x in X such that f (x) = y |
1 | 534-537 | Definition 6 A function f : X β Y is said to be onto (or surjective), if every element
of Y is the image of some element of X under f, i e , for every y β Y, there exists an
element x in X such that f (x) = y The function f3 and f4 in Fig 1 |
1 | 535-538 | e , for every y β Y, there exists an
element x in X such that f (x) = y The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1 |
1 | 536-539 | , for every y β Y, there exists an
element x in X such that f (x) = y The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is
not onto as elements e, f in X2 are not the image of any element in X1 under f1 |
1 | 537-540 | The function f3 and f4 in Fig 1 2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is
not onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24
MATHEMATICS
8
Remark f : X β Y is onto if and only if Range of f = Y |
1 | 538-541 | 2 (iii), (iv) are onto and the function f1 in Fig 1 2 (i) is
not onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24
MATHEMATICS
8
Remark f : X β Y is onto if and only if Range of f = Y Definition 7 A function f : X β Y is said to be one-one and onto (or bijective), if f is
both one-one and onto |
1 | 539-542 | 2 (i) is
not onto as elements e, f in X2 are not the image of any element in X1 under f1 Rationalised 2023-24
MATHEMATICS
8
Remark f : X β Y is onto if and only if Range of f = Y Definition 7 A function f : X β Y is said to be one-one and onto (or bijective), if f is
both one-one and onto The function f4 in Fig 1 |
1 | 540-543 | Rationalised 2023-24
MATHEMATICS
8
Remark f : X β Y is onto if and only if Range of f = Y Definition 7 A function f : X β Y is said to be one-one and onto (or bijective), if f is
both one-one and onto The function f4 in Fig 1 2 (iv) is one-one and onto |
1 | 541-544 | Definition 7 A function f : X β Y is said to be one-one and onto (or bijective), if f is
both one-one and onto The function f4 in Fig 1 2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school |
1 | 542-545 | The function f4 in Fig 1 2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school Let f : A β N be
function defined by f (x) = roll number of the student x |
1 | 543-546 | 2 (iv) is one-one and onto Example 7 Let A be the set of all 50 students of Class X in a school Let f : A β N be
function defined by f (x) = roll number of the student x Show that f is one-one
but not onto |
1 | 544-547 | Example 7 Let A be the set of all 50 students of Class X in a school Let f : A β N be
function defined by f (x) = roll number of the student x Show that f is one-one
but not onto Solution No two different students of the class can have same roll number |
1 | 545-548 | Let f : A β N be
function defined by f (x) = roll number of the student x Show that f is one-one
but not onto Solution No two different students of the class can have same roll number Therefore,
f must be one-one |
1 | 546-549 | Show that f is one-one
but not onto Solution No two different students of the class can have same roll number Therefore,
f must be one-one We can assume without any loss of generality that roll numbers of
students are from 1 to 50 |
1 | 547-550 | Solution No two different students of the class can have same roll number Therefore,
f must be one-one We can assume without any loss of generality that roll numbers of
students are from 1 to 50 This implies that 51 in N is not roll number of any student of
the class, so that 51 can not be image of any element of X under f |
1 | 548-551 | Therefore,
f must be one-one We can assume without any loss of generality that roll numbers of
students are from 1 to 50 This implies that 51 in N is not roll number of any student of
the class, so that 51 can not be image of any element of X under f Hence, f is not onto |
1 | 549-552 | We can assume without any loss of generality that roll numbers of
students are from 1 to 50 This implies that 51 in N is not roll number of any student of
the class, so that 51 can not be image of any element of X under f Hence, f is not onto Example 8 Show that the function f : N β N, given by f(x) = 2x, is one-one but not
onto |
1 | 550-553 | This implies that 51 in N is not roll number of any student of
the class, so that 51 can not be image of any element of X under f Hence, f is not onto Example 8 Show that the function f : N β N, given by f(x) = 2x, is one-one but not
onto Solution The function f is one-one, for f (x1) = f(x2) β 2x1 = 2x2 β x1 = x2 |
1 | 551-554 | Hence, f is not onto Example 8 Show that the function f : N β N, given by f(x) = 2x, is one-one but not
onto Solution The function f is one-one, for f (x1) = f(x2) β 2x1 = 2x2 β x1 = x2 Further,
f is not onto, as for 1 β N, there does not exist any x in N such that f(x) = 2x = 1 |
1 | 552-555 | Example 8 Show that the function f : N β N, given by f(x) = 2x, is one-one but not
onto Solution The function f is one-one, for f (x1) = f(x2) β 2x1 = 2x2 β x1 = x2 Further,
f is not onto, as for 1 β N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1 |
1 | 553-556 | Solution The function f is one-one, for f (x1) = f(x2) β 2x1 = 2x2 β x1 = x2 Further,
f is not onto, as for 1 β N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1 2 (i) to (iv)
Rationalised 2023-24
RELATIONS AND FUNCTIONS
9
Example 9 Prove that the function f : R β R, given by f (x) = 2x, is one-one and onto |
1 | 554-557 | Further,
f is not onto, as for 1 β N, there does not exist any x in N such that f(x) = 2x = 1 Fig 1 2 (i) to (iv)
Rationalised 2023-24
RELATIONS AND FUNCTIONS
9
Example 9 Prove that the function f : R β R, given by f (x) = 2x, is one-one and onto Solution f is one-one, as f (x1) = f (x2) β 2x1 = 2x2 β x1 = x2 |
1 | 555-558 | Fig 1 2 (i) to (iv)
Rationalised 2023-24
RELATIONS AND FUNCTIONS
9
Example 9 Prove that the function f : R β R, given by f (x) = 2x, is one-one and onto Solution f is one-one, as f (x1) = f (x2) β 2x1 = 2x2 β x1 = x2 Also, given any real
number y in R, there exists 2
y in R such that f ( 2
y ) = 2 |
1 | 556-559 | 2 (i) to (iv)
Rationalised 2023-24
RELATIONS AND FUNCTIONS
9
Example 9 Prove that the function f : R β R, given by f (x) = 2x, is one-one and onto Solution f is one-one, as f (x1) = f (x2) β 2x1 = 2x2 β x1 = x2 Also, given any real
number y in R, there exists 2
y in R such that f ( 2
y ) = 2 ( 2
y ) = y |
1 | 557-560 | Solution f is one-one, as f (x1) = f (x2) β 2x1 = 2x2 β x1 = x2 Also, given any real
number y in R, there exists 2
y in R such that f ( 2
y ) = 2 ( 2
y ) = y Hence, f is onto |
1 | 558-561 | Also, given any real
number y in R, there exists 2
y in R such that f ( 2
y ) = 2 ( 2
y ) = y Hence, f is onto Fig 1 |
1 | 559-562 | ( 2
y ) = y Hence, f is onto Fig 1 3
Example 10 Show that the function f : N β N, given by f (1) = f (2) = 1 and f(x) = x β 1,
for every x > 2, is onto but not one-one |
1 | 560-563 | Hence, f is onto Fig 1 3
Example 10 Show that the function f : N β N, given by f (1) = f (2) = 1 and f(x) = x β 1,
for every x > 2, is onto but not one-one Solution f is not one-one, as f (1) = f (2) = 1 |
1 | 561-564 | Fig 1 3
Example 10 Show that the function f : N β N, given by f (1) = f (2) = 1 and f(x) = x β 1,
for every x > 2, is onto but not one-one Solution f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y β N, y β 1,
we can choose x as y + 1 such that f (y + 1) = y + 1 β 1 = y |
1 | 562-565 | 3
Example 10 Show that the function f : N β N, given by f (1) = f (2) = 1 and f(x) = x β 1,
for every x > 2, is onto but not one-one Solution f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y β N, y β 1,
we can choose x as y + 1 such that f (y + 1) = y + 1 β 1 = y Also for 1 β N, we
have f (1) = 1 |
1 | 563-566 | Solution f is not one-one, as f (1) = f (2) = 1 But f is onto, as given any y β N, y β 1,
we can choose x as y + 1 such that f (y + 1) = y + 1 β 1 = y Also for 1 β N, we
have f (1) = 1 Example 11 Show that the function f : R β R,
defined as f (x) = x2, is neither one-one nor onto |
1 | 564-567 | But f is onto, as given any y β N, y β 1,
we can choose x as y + 1 such that f (y + 1) = y + 1 β 1 = y Also for 1 β N, we
have f (1) = 1 Example 11 Show that the function f : R β R,
defined as f (x) = x2, is neither one-one nor onto Solution Since f (β 1) = 1 = f (1), f is not one-
one |
1 | 565-568 | Also for 1 β N, we
have f (1) = 1 Example 11 Show that the function f : R β R,
defined as f (x) = x2, is neither one-one nor onto Solution Since f (β 1) = 1 = f (1), f is not one-
one Also, the element β 2 in the co-domain R is
not image of any element x in the domain R
(Why |
1 | 566-569 | Example 11 Show that the function f : R β R,
defined as f (x) = x2, is neither one-one nor onto Solution Since f (β 1) = 1 = f (1), f is not one-
one Also, the element β 2 in the co-domain R is
not image of any element x in the domain R
(Why ) |
1 | 567-570 | Solution Since f (β 1) = 1 = f (1), f is not one-
one Also, the element β 2 in the co-domain R is
not image of any element x in the domain R
(Why ) Therefore f is not onto |
1 | 568-571 | Also, the element β 2 in the co-domain R is
not image of any element x in the domain R
(Why ) Therefore f is not onto Example 12 Show that f : N β N, given by
1,if
is odd,
( )
1,if
is even
x
x
f x
x
x
+
=
β
is both one-one and onto |
1 | 569-572 | ) Therefore f is not onto Example 12 Show that f : N β N, given by
1,if
is odd,
( )
1,if
is even
x
x
f x
x
x
+
=
β
is both one-one and onto Fig 1 |
1 | 570-573 | Therefore f is not onto Example 12 Show that f : N β N, given by
1,if
is odd,
( )
1,if
is even
x
x
f x
x
x
+
=
β
is both one-one and onto Fig 1 4
Rationalised 2023-24
MATHEMATICS
10
Solution Suppose f (x1) = f (x2) |
1 | 571-574 | Example 12 Show that f : N β N, given by
1,if
is odd,
( )
1,if
is even
x
x
f x
x
x
+
=
β
is both one-one and onto Fig 1 4
Rationalised 2023-24
MATHEMATICS
10
Solution Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have
x1 + 1 = x2 β 1, i |
1 | 572-575 | Fig 1 4
Rationalised 2023-24
MATHEMATICS
10
Solution Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have
x1 + 1 = x2 β 1, i e |
1 | 573-576 | 4
Rationalised 2023-24
MATHEMATICS
10
Solution Suppose f (x1) = f (x2) Note that if x1 is odd and x2 is even, then we will have
x1 + 1 = x2 β 1, i e , x2 β x1 = 2 which is impossible |
1 | 574-577 | Note that if x1 is odd and x2 is even, then we will have
x1 + 1 = x2 β 1, i e , x2 β x1 = 2 which is impossible Similarly, the possibility of x1 being
even and x2 being odd can also be ruled out, using the similar argument |
1 | 575-578 | e , x2 β x1 = 2 which is impossible Similarly, the possibility of x1 being
even and x2 being odd can also be ruled out, using the similar argument Therefore,
both x1 and x2 must be either odd or even |
1 | 576-579 | , x2 β x1 = 2 which is impossible Similarly, the possibility of x1 being
even and x2 being odd can also be ruled out, using the similar argument Therefore,
both x1 and x2 must be either odd or even Suppose both x1 and x2 are odd |
1 | 577-580 | Similarly, the possibility of x1 being
even and x2 being odd can also be ruled out, using the similar argument Therefore,
both x1 and x2 must be either odd or even Suppose both x1 and x2 are odd Then
f (x1) = f (x2) β x1 + 1 = x2 + 1 β x1 = x2 |
1 | 578-581 | Therefore,
both x1 and x2 must be either odd or even Suppose both x1 and x2 are odd Then
f (x1) = f (x2) β x1 + 1 = x2 + 1 β x1 = x2 Similarly, if both x1 and x2 are even, then also
f (x1) = f (x2) β x1 β 1 = x2 β 1 β x1 = x2 |
1 | 579-582 | Suppose both x1 and x2 are odd Then
f (x1) = f (x2) β x1 + 1 = x2 + 1 β x1 = x2 Similarly, if both x1 and x2 are even, then also
f (x1) = f (x2) β x1 β 1 = x2 β 1 β x1 = x2 Thus, f is one-one |
1 | 580-583 | Then
f (x1) = f (x2) β x1 + 1 = x2 + 1 β x1 = x2 Similarly, if both x1 and x2 are even, then also
f (x1) = f (x2) β x1 β 1 = x2 β 1 β x1 = x2 Thus, f is one-one Also, any odd number
2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number
2r in the co-domain N is the image of 2r β 1 in the domain N |
1 | 581-584 | Similarly, if both x1 and x2 are even, then also
f (x1) = f (x2) β x1 β 1 = x2 β 1 β x1 = x2 Thus, f is one-one Also, any odd number
2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number
2r in the co-domain N is the image of 2r β 1 in the domain N Thus, f is onto |
1 | 582-585 | Thus, f is one-one Also, any odd number
2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number
2r in the co-domain N is the image of 2r β 1 in the domain N Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} β {1, 2, 3} is always one-one |
1 | 583-586 | Also, any odd number
2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and any even number
2r in the co-domain N is the image of 2r β 1 in the domain N Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} β {1, 2, 3} is always one-one Solution Suppose f is not one-one |
1 | 584-587 | Thus, f is onto Example 13 Show that an onto function f : {1, 2, 3} β {1, 2, 3} is always one-one Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the
domain whose image in the co-domain is same |
1 | 585-588 | Example 13 Show that an onto function f : {1, 2, 3} β {1, 2, 3} is always one-one Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the
domain whose image in the co-domain is same Also, the image of 3 under f can be
only one element |
1 | 586-589 | Solution Suppose f is not one-one Then there exists two elements, say 1 and 2 in the
domain whose image in the co-domain is same Also, the image of 3 under f can be
only one element Therefore, the range set can have at the most two elements of the
co-domain {1, 2, 3}, showing that f is not onto, a contradiction |
1 | 587-590 | Then there exists two elements, say 1 and 2 in the
domain whose image in the co-domain is same Also, the image of 3 under f can be
only one element Therefore, the range set can have at the most two elements of the
co-domain {1, 2, 3}, showing that f is not onto, a contradiction Hence, f must be one-one |
1 | 588-591 | Also, the image of 3 under f can be
only one element Therefore, the range set can have at the most two elements of the
co-domain {1, 2, 3}, showing that f is not onto, a contradiction Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} β {1, 2, 3} must be onto |
1 | 589-592 | Therefore, the range set can have at the most two elements of the
co-domain {1, 2, 3}, showing that f is not onto, a contradiction Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} β {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different
elements of the co-domain {1, 2, 3} under f |
1 | 590-593 | Hence, f must be one-one Example 14 Show that a one-one function f : {1, 2, 3} β {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different
elements of the co-domain {1, 2, 3} under f Hence, f has to be onto |
1 | 591-594 | Example 14 Show that a one-one function f : {1, 2, 3} β {1, 2, 3} must be onto Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different
elements of the co-domain {1, 2, 3} under f Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary
finite set X, i |
1 | 592-595 | Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different
elements of the co-domain {1, 2, 3} under f Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary
finite set X, i e |
1 | 593-596 | Hence, f has to be onto Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary
finite set X, i e , a one-one function f : X β X is necessarily onto and an onto map
f : X β X is necessarily one-one, for every finite set X |
1 | 594-597 | Remark The results mentioned in Examples 13 and 14 are also true for an arbitrary
finite set X, i e , a one-one function f : X β X is necessarily onto and an onto map
f : X β X is necessarily one-one, for every finite set X In contrast to this, Examples 8
and 10 show that for an infinite set, this may not be true |
1 | 595-598 | e , a one-one function f : X β X is necessarily onto and an onto map
f : X β X is necessarily one-one, for every finite set X In contrast to this, Examples 8
and 10 show that for an infinite set, this may not be true In fact, this is a characteristic
difference between a finite and an infinite set |
1 | 596-599 | , a one-one function f : X β X is necessarily onto and an onto map
f : X β X is necessarily one-one, for every finite set X In contrast to this, Examples 8
and 10 show that for an infinite set, this may not be true In fact, this is a characteristic
difference between a finite and an infinite set EXERCISE 1 |
1 | 597-600 | In contrast to this, Examples 8
and 10 show that for an infinite set, this may not be true In fact, this is a characteristic
difference between a finite and an infinite set EXERCISE 1 2
1 |
1 | 598-601 | In fact, this is a characteristic
difference between a finite and an infinite set EXERCISE 1 2
1 Show that the function f : Rβββββ β Rβββββ defined by f (x) = 1
x is one-one and onto,
where Rβββββ is the set of all non-zero real numbers |
1 | 599-602 | EXERCISE 1 2
1 Show that the function f : Rβββββ β Rβββββ defined by f (x) = 1
x is one-one and onto,
where Rβββββ is the set of all non-zero real numbers Is the result true, if the domain
Rβββββ is replaced by N with co-domain being same as Rβββββ |
1 | 600-603 | 2
1 Show that the function f : Rβββββ β Rβββββ defined by f (x) = 1
x is one-one and onto,
where Rβββββ is the set of all non-zero real numbers Is the result true, if the domain
Rβββββ is replaced by N with co-domain being same as Rβββββ 2 |
1 | 601-604 | Show that the function f : Rβββββ β Rβββββ defined by f (x) = 1
x is one-one and onto,
where Rβββββ is the set of all non-zero real numbers Is the result true, if the domain
Rβββββ is replaced by N with co-domain being same as Rβββββ 2 Check the injectivity and surjectivity of the following functions:
(i) f : N β N given by f(x) = x2
(ii) f : Z β Z given by f(x) = x2
(iii) f : R β R given by f(x) = x2
(iv) f : N β N given by f(x) = x3
(v) f : Z β Z given by f(x) = x3
3 |
1 | 602-605 | Is the result true, if the domain
Rβββββ is replaced by N with co-domain being same as Rβββββ 2 Check the injectivity and surjectivity of the following functions:
(i) f : N β N given by f(x) = x2
(ii) f : Z β Z given by f(x) = x2
(iii) f : R β R given by f(x) = x2
(iv) f : N β N given by f(x) = x3
(v) f : Z β Z given by f(x) = x3
3 Prove that the Greatest Integer Function f : R β R, given by f(x) = [x], is neither
one-one nor onto, where [x] denotes the greatest integer less than or equal to x |
1 | 603-606 | 2 Check the injectivity and surjectivity of the following functions:
(i) f : N β N given by f(x) = x2
(ii) f : Z β Z given by f(x) = x2
(iii) f : R β R given by f(x) = x2
(iv) f : N β N given by f(x) = x3
(v) f : Z β Z given by f(x) = x3
3 Prove that the Greatest Integer Function f : R β R, given by f(x) = [x], is neither
one-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24
RELATIONS AND FUNCTIONS
11
4 |
1 | 604-607 | Check the injectivity and surjectivity of the following functions:
(i) f : N β N given by f(x) = x2
(ii) f : Z β Z given by f(x) = x2
(iii) f : R β R given by f(x) = x2
(iv) f : N β N given by f(x) = x3
(v) f : Z β Z given by f(x) = x3
3 Prove that the Greatest Integer Function f : R β R, given by f(x) = [x], is neither
one-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24
RELATIONS AND FUNCTIONS
11
4 Show that the Modulus Function f : R β R, given by f (x) = | x|, is neither one-
one nor onto, where | x | is x, if x is positive or 0 and |x | is β x, if x is negative |
1 | 605-608 | Prove that the Greatest Integer Function f : R β R, given by f(x) = [x], is neither
one-one nor onto, where [x] denotes the greatest integer less than or equal to x Rationalised 2023-24
RELATIONS AND FUNCTIONS
11
4 Show that the Modulus Function f : R β R, given by f (x) = | x|, is neither one-
one nor onto, where | x | is x, if x is positive or 0 and |x | is β x, if x is negative 5 |
1 | 606-609 | Rationalised 2023-24
RELATIONS AND FUNCTIONS
11
4 Show that the Modulus Function f : R β R, given by f (x) = | x|, is neither one-
one nor onto, where | x | is x, if x is positive or 0 and |x | is β x, if x is negative 5 Show that the Signum Function f : R β R, given by
f x
x
x
x
( )
,
,
οΏ½ ,
=
>
=
<
ο£±


1
0
0
0
1
0
if
if
if
is neither one-one nor onto |
1 | 607-610 | Show that the Modulus Function f : R β R, given by f (x) = | x|, is neither one-
one nor onto, where | x | is x, if x is positive or 0 and |x | is β x, if x is negative 5 Show that the Signum Function f : R β R, given by
f x
x
x
x
( )
,
,
οΏ½ ,
=
>
=
<
ο£±


1
0
0
0
1
0
if
if
if
is neither one-one nor onto 6 |
1 | 608-611 | 5 Show that the Signum Function f : R β R, given by
f x
x
x
x
( )
,
,
οΏ½ ,
=
>
=
<
ο£±


1
0
0
0
1
0
if
if
if
is neither one-one nor onto 6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function
from A to B |
1 | 609-612 | Show that the Signum Function f : R β R, given by
f x
x
x
x
( )
,
,
οΏ½ ,
=
>
=
<
ο£±


1
0
0
0
1
0
if
if
if
is neither one-one nor onto 6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function
from A to B Show that f is one-one |
1 | 610-613 | 6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function
from A to B Show that f is one-one 7 |
1 | 611-614 | Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function
from A to B Show that f is one-one 7 In each of the following cases, state whether the function is one-one, onto or
bijective |
1 | 612-615 | Show that f is one-one 7 In each of the following cases, state whether the function is one-one, onto or
bijective Justify your answer |
1 | 613-616 | 7 In each of the following cases, state whether the function is one-one, onto or
bijective Justify your answer (i) f : R β R defined by f (x) = 3 β 4x
(ii) f : R β R defined by f (x) = 1 + x2
8 |
1 | 614-617 | In each of the following cases, state whether the function is one-one, onto or
bijective Justify your answer (i) f : R β R defined by f (x) = 3 β 4x
(ii) f : R β R defined by f (x) = 1 + x2
8 Let A and B be sets |
1 | 615-618 | Justify your answer (i) f : R β R defined by f (x) = 3 β 4x
(ii) f : R β R defined by f (x) = 1 + x2
8 Let A and B be sets Show that f : A Γ B β B Γ A such that f (a, b) = (b, a) is
bijective function |
1 | 616-619 | (i) f : R β R defined by f (x) = 3 β 4x
(ii) f : R β R defined by f (x) = 1 + x2
8 Let A and B be sets Show that f : A Γ B β B Γ A such that f (a, b) = (b, a) is
bijective function 9 |
1 | 617-620 | Let A and B be sets Show that f : A Γ B β B Γ A such that f (a, b) = (b, a) is
bijective function 9 Let f : N β N be defined by f (n) =
n
n
n
n
+
ο£±
ο£²



1
2
2
,
,
if
is odd
if
is even
for all n β N |
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