Exercise 1.3
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NCERT Solutions Class 12 Maths Relations and Functions
1. Let 
: {1, 3, 4} 
{1, 2, 5} and 
: {1, 2, 5} {1, 3} be given by = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}. Write down 
Ans. = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}
Now, 
and 
and 
Hence, 
{(1, 3), (3, 1), (4, 3)}
2. Let 
and 
be functions from R to R. Show that:
Ans. (a) To prove:
L. H. S. = 
= 
= 
= R. H. S.
(b) To prove:
L. H. S. = 
= 
= 
= R. H. S.
3. Find 
and 
, if:
(i) 
and 
(ii) 
and 
Ans. To find: and
(i) and
and 
= 
= 
(ii) and
and 
= 
NCERT Solutions class 12 Maths Exercise 1.3
4. If 
show that 
for all 
What is the inverse of 
Ans. Given: 
L.H.S. = 
= 
= 
= 
= 
= R.H.S.
Now, 
Hence inverse of 
5. State with reason whether following functions have inverse:
(i) : {1, 2, 3, 4} {10} with = {(1, 10), (2, 10), (3, 10), (4, 10)}
(ii) : {5, 6, 7, 8} {1, 2, 3, 4} with = {(5, 4), (6, 3), (7, 4), (8, 2)}
(iii) : {2, 3, 4, 5} {7, 9, 11, 13} with = {(2, 7), (3, 9), (4, 11), (5, 13)}
Ans. (i) = {(1, 10), (2, 10), (3, 10), (4, 10)}
It is many-one function, therefore has no inverse.
(ii) = {(5, 4), (6, 3), (7, 4), (8, 2)}
It is many-one function, therefore has no inverse.
(iii) = {(2, 7), (3, 9), (4, 11), (5, 13)}
is one-one onto function, therefore, has an inverse.
6. Show that 
R given by 
is one-one. Find the inverse of the function Range 
Ans. Part I: R given by 
Let 
, then 
and 
When 
then 
is one-one.
Part II: Let 
Range of
for some 
in 
As 
is onto.
Therefore, 
7. Consider : R R given by 
Show that is invertible. Find the inverse of
Ans. Consider : R R given by 
Let 
R, then 
and 
Now, for , then 
is one-one.
Let Range of
is onto.
Therefore, is invertible and hence, 
.
8. Consider 
given by 
Show that is invertible with the inverse 
of given by 
where 
is the set of all non-negative real numbers.
Ans. Consider and
Let R 
, then 
and 
is one-one.
Now 
as 
is onto.
Therefore, 
is invertible and 
.
9. Consider 
given by 
Show that is invertible with 
Ans. Consider and
Let R 
, then 
and 
Now, then 
is one-one.
Now, again 
= 
= 
= 
= 
= 
is onto.
Therefore, is invertible and 
.
10. Let 
be an invertible function. Show that has unique inverse.
(Hint: Suppose 
and 
are two inverses of Then for all 
Use one-one ness of ).
Ans. Given: be an invertible function.
Thus is 1 – 1 and onto and therefore exists.
Let and be two inverses of Then for all Y,
The inverse is unique and hence has a unique inverse.
11. Consider : {1, 2, 3} 
given by 
and 
Find and show that 
Ans. 
, then it is clear that is 1 – 1 and onto and therefore exists.
Also, 
and 
Hence, 
12. Let be an invertible function. Show that the inverse of is , i.e.,
Ans. Let be an invertible function.
Then is one-one and onto
X where is also one-one and onto such that
and 
Now, 
and 
13. If : R R given by 
then 
is:
(A) 
(B) 
(C)
(D) 
Ans. : R R and 
= 
= 
= 
=
Therefore, option (C) is correct.
14. Let : R – 
R be a function defined as 
The inverse of is the map : Range of 
given by:
(A) 
(B) 
(C) 
(D) 
Ans. Given: : R – R and 
Now, Range of 
Let 
Therefore, option (B) is correct.