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.