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 →→ 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 g(x)=|5x−2|g(x)=|5x−2| 

(ii)  and  

Ans. To find:   and  

(i)  and g(x)=|5x−2|g(x)=|5x−2|

 and  =  = 

(ii)  and 

 

and  = 

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  [Hint: f−1(y)=y−34f−1(y)=y−34]

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 

 

 

 

 

 f^-1(y) = g(y) = 4y4−3y4y4−3y

Therefore, option (B) is correct.