Exercise 1.2
Download as PDF.
NCERT Solutions class 12 Maths Relations and Functions
1. Show that the function 
defined by 
is one-one and onto, where 
is the set of all non-zero real numbers. Is the result true, if the domain is replaced by N with co-domain being same as ?
Ans. 
Part I: 
and 
If 
then 
is one-one.
is onto.
Part II: If
where 
N
is one-one.
But every real number belonging to co-domain may not have a pre-image in N.
e.g. 
N is not onto.
2. Check the injectivity and surjectivity of the following functions:
(i) 
given by 
(ii) 
given by
(iii) 
given by
(iv) given by 
(v) given by
Ans. (i) given by
If then 
is injective.
There are such numbers of co-domain which have no image in domain N.
e.g. 3
co-domain N, but there is no pre-image in domain of 
therefore is not onto. is not surjective.
(ii) given by
Since, Z = 
therefore, 
and 1 have same image. is not injective.
There are such numbers of co-domain which have no image in domain Z.
e.g. 3 co-domain, but 
domain of is not surjective.
(iii) given by
As
and 1 have same image. is not injective.
e.g. 
co-domain, but 
domain R of is not surjective.
(iv) given by
If then 
i.e., for every 
N, has a unique image in its co-domain. is injective.
There are many such members of co-domain of which do not have pre-image in its domain e.g., 2, 3, etc.
Therefore is not onto. is not surjective.
(v) given by
If then
i.e., for every Z, has a unique image in its co-domain. is injective.
There are many such members of co-domain of which do not have pre-image in its domain.
Therefore is not onto.
3. Prove that the Greatest integer Function , given by 
is neither one-one nor onto, where 
denotes the greatest integer less than or equal to 
Ans. Function , given by 
and 
is not one-one.
All the images of 
R belong to its domain have integers as the images in co-domain. But no fraction proper or improper belonging to co-domain of has any pre-image in its domain.
4. Show that the Modulus Function , given by 
is neither one-one nor onto, where is 
if 
is positive or 0 and 
is 
if is negative.
Ans. Modulus Function , given by 
Now 
contains 
Thus negative integers are not images of any element. is not one-one.
Also second set R contains some negative numbers which are not images of any real number.
is not onto.
5. Show that the Signum Function , given by 
is neither one-one nor onto.
Ans. Signum Function , given by
for 
is not one-one.
Except 
no other members of co-domain of has any pre-image its domain.
is not onto.
NCERT Solutions class 12 Maths Exercise 1.2
6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that is one-one.
Ans. A = {1, 2, 3}, B = {4, 5, 6, 7} and = {(1, 4), (2, 5), (3, 6)}
Here, 
and 
Here, also distinct elements of A have distinct images in B.
Therefore, is not one-one and is not bijective.
NCERT Solutions class 12 Maths Exercise 1.2
7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) defined by 
(ii) defined by 
Ans. (i) defined by
Now, if 
R, then 
and 
And if , then is one-one.
Again, if every element of Y (– R) is image of some element of X (R) under 
i.e., for every 
Y, there exists an element in X such that 
Now 
is onto or bijective function.
(ii) defined by
Now, if R, then 
and 
And if , then
is not one-one.
Again, if every element of Y (– R) is image of some element of X (R) under i.e., for every Y, there exists an element in X such that
Now, 
is not onto.
Therefore, is not bijective.
NCERT Solutions class 12 Maths Exercise 1.2
8. Let A and B be sets. Show that : A x B 
B x A such that 
is bijective function.
Ans. Injectivity: Let 
and 
A x B such that 
and 
= 
= for all 
A x B
So, is injective.
Surjectivity: Let 
be an arbitrary element of B x A. Then 
B and 
A.
A x B
Thus, for all 
B x A, their exists A x B such that
So, 
A x B B x A is an onto function, therefore is bijective.
9. Let be defined by 
for all 
N.
State whether the function is bijective. Justify your answer.
Ans. be defined by
(a) 
and 
The elements 1, 2, belonging to domain of have the same image 1 in its co-domain.
So, is not one-one, therefore, is not injective.
(b) Every number of co-domain has pre-image in its domain e.g., 1 has two pre-images 1 and 2.
So, is onto, therefore, is not bijective.
NCERT Solutions class 12 Maths Exercise 1.2
10. Let A = R – {3} and B = R – {1}. Consider the function : A B defined by 
Is one-one and onto? Justify your answer.
Ans. A = R – {3} and B = R – {1} and 
Let A, then 
and 
Now, for
is one-one function.
Now 
= 
Therefore, is an onto function.
NCERT Solutions class 12 Maths Exercise 1.2
11. Let be defined as 
Choose the correct answer:
(A) is one-one onto
(B) is many-one onto
(C) is one-one but not onto
(D) is neither one-one nor onto
Ans. 
and 
Let 
, then 
and 
Therefore, is not one-one function.
Now, 
and 
Therefore, is not onto function.
Therefore, option (D) is correct.
NCERT Solutions class 12 Maths Exercise 1.2
12. Let be defined as 
Choose the correct answer:
(A) is one-one onto
(B) is many-one onto
(C) is one-one but not onto
(D) is neither one-one nor onto
Ans. Let 
R such that
Therefore, is one-one function.
Now, consider R (co-domain of ) certainly 
R (domain of )
Thus for all R (co-domain of ) there exists 
R (domain of ) such that
Therefore, is onto function.
Therefore, option (A) is correct.