# 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 →→ 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)=|5x2|g(x)=|5x−2|

(ii)  and

Ans. To find:   and

(i)  and g(x)=|5x2|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) = 4y43y4y4−3y

Therefore, option (B) is correct.