Question 31: Let be a finite set. If
is a one-one function, show that
is onto also.
Answer:
In order to prove that f is onto function, we will have to show that every element in A (co-domain) has its pre-image in the domain . In other words, range of
Since is a one-one function. Therefore,
are distinct elements of set
.
But, has only
elements. Therefore,
i.e. Co-domain=Range.
Hence, is onto.
Question 32: Let be a finite set. If
is an onto function, show that
is one-one also.
Answer:
Let
In order to prove that is a one-one function, we will have to show that
are distinct elements of
Clearly, Range of
Since, is an onto function. Therefore,
Range of
But is a a finite set consisting of
elements. Therefore,
are distinct elements of
. Hence,
is one-one.
Question 33: Let be the set of real numbers. If
Then, find
Also show that
Answer:
Clearly; range of is a subset of domain of
and range of
is a subset of domain of
.
Answer:
We have, Range of
Clearly, it is a subset of domain of . Hence,
exists and
such that
Answer:
Clearly,
[CBSE 2014]
Answer:
Answer:
Question 38: If are defined respectively by
, find
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
We have,
Therefore, for any we have
Answer:
Answer:
We have,
Question 46: Let be a function such that
.Show that
is onto if and only if
is one-one. Describe
in this case.
Answer:
Let be onto. Then, we have to prove that
is one-one.
Question 47: lf be functions defined by
then find
Answer:
We have,
Answer:
We observe that
Answer:
We observe that
We find that and
have distinct domains. Also, their formulas are not same.
Answer:
Computation of We observe that: Range
Answer:
Answer:
Clearly, domain
In order to find the range of , we proceed as follows:
Let . Then,
Since takes real values. Therefore
Question 53: Let be a real function defined by
Find
Also, show that
Answer:
Answer:
Answer:
Let Then,
Question 56: Let be defined as
Find:
Answer:
(i) Let Then,
(ii) Let Then,
(iii)
Question 57: Let Determine whether the function
defined as below have inverse. Find
if it exists
Answer:
(i) Clearly, is a bijection. Hence,
is invertible and its inverse is given by
(ii) Clearly, Therefore,
is many-one and hence it is not invertible.
(iii) Clearly, is a bijection and hence invertible. The inverse of
is given by
Answer:
Clearly, is a bijection and hence invertible. The inverse of
is given by
Show that is invertible and
Answer:
Therefore,
Question 60: Prove that the function defined as
is invertible. Also, find
Answer:
In order to prove that is invertible, it is sufficient to show that
is a bijection.
Hence,
Answer:
In order to prove the invertibility of , it is sufficient to show that it is a bijection.
is one-one: For any
Now,
Answer:
Answer:
It is given that f is invertible with as its inverse.
Question 64: Let be a function defined as
,where
Show that f is invertible. Find its inverse
Answer:
In order to prove that is invertible, it is sufficient to show that it is a bijection.
is one-one: For any
, we find that
is onto : Let
be an arbitrary element of
. Then there exists
such that
Thus, for each there exists
such that
. So,
is onto.
Thus, is both one and onto. Consequently, it is invertible. Let
be the inverse of
.
Then,
Question 65: Let Consider
given by
. Show that
is invertible. Find the inverse of
.
Answer:
In order to prove that is invertible, it is sufficient to show that it is a bijection.
Question 66: Let be a function defined as
Show that
is invertible. Find the inverse of
Answer:
In order to prove that is invertible, it is sufficient to show that
is a bijection.