MATHS

BOOKS - KUMAR PRAKASHAN KENDRA MATHS (GUJRATI

ENGLISH)

RELATIONS AND FUNCTIONS

Exercise 1 1

1. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A = {1, 2, 3, .... .13, 14} defined as

R = {(x, y) : 3x − y = 0}

Watch Video Solution

2. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R is the set N of natural numbers defined as R = {(x,y): y = x +5

and x < 4}.

Watch Video Solution

3. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A = {1,2,3,4,5,6} as R = {(x,y):y is divisible by x}.

Watch Video Solution

4. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set Z of all integers defined as ={(x, y) : x − y is an

integers }
 Watch Video Solution

5. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A of human beings in a town at a particular time

given by

(a) r = {(x, y) : x and y works at the same place }

Watch Video Solution

6. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A of human beings in a town at a particular time

given by

R = {(x,y)} : x and y live in the same locality }

Watch Video Solution

7. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A of human beings in a town at a particular time

given by

R = {(x,y)} : x is exactly 7 cm taller than y }

Watch Video Solution

8. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A of human beings in a town at a particular time

given by

R = {(x, y) : x is wife of y }

Watch Video Solution

9. Determine whether each of the following relations are reflexive ,

symmetric and transitive :

Relation R in the set A of human beings in a town at a particular time

given by

R = {(x, y) : x is father of y }

Watch Video Solution

10. Show that the relation R in the set R of real number , defined as

R = {(a, b) : a ≤ b }
2
is neither reflexive nor symmetric nor transitive.

Watch Video Solution

11. Check whether the relation R defined in the set {1,2,3,4,5,6} as R = {(a,b) :

b = a +1 } is reflexive , symmetric or transitive.

Watch Video Solution

12. Show that the relation R is R defined as R = {(a, b) : a ≤ b} is

reflexive and transitive but not symmetric.

 Watch Video Solution

13. Check whether the relation R defined by 3

R = {(a, b) : a ≤ b } is

reflexive , symmetric or transitive.

Watch Video Solution

14. Show that the relation R in the set {1, 2, 3} given by

R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Watch Video Solution

15. Show that the relation R in the set A of all the books in a library of a

college , given by R = {(x, y) : x and y have same number of pages} is an

equivalence relation.

Watch Video Solution

16. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by

R{(a, b) : |a − b| is even } , is an equivalence relation . Show that all the

elements of {1,3,5} are related to each other all the elements of {2,4} are

related to each other . But no element of {1,3,5} is related to any element

of {2,4} .

Watch Video Solution

17. Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}

given by

R = {(a, b) : |a − b| is multiple of 4} is in equivelance.

Watch Video Solution

18. Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}

given by

R = {(a, b) : a = b} is an equivalence relation . Find the set of all

elements related to 1 each case.

 Watch Video Solution

19. Give an example of relation . Which is

Symmetric but neither reflexive nor transitive.

Watch Video Solution

20. Give an example of relation . Which is

Transitive but neither reflexive nor symmetric .

Watch Video Solution

21. Give an example of relation . Which is

Reflexive and symmetric but not transitive .

Watch Video Solution

22. Give an example of relation . Which is

Reflexive and transitive but not symmetric.

Watch Video Solution

23. Give an example of relation . Which is

Symmetric and transitive but not reflexive.

Watch Video Solution

24. Show that the relation R in the set A of points in a plane give by R =

{(P,Q) : distance of the point P from the origin is same as the distance of

the point Q from the origin} , is an equivalence relation. Further , show

that the set equivalence relation . Further , show that the set of all points

related to a point P ≠ (0, 0) is the circle passing through P with origin

as centre.

Watch Video Solution

25. Show that the relation R defined in the set A of all triangles as

R = {(T1 , T2 }T1 is similar to T2 } , is equivalence relation . Consider

three right angle triangles T1 with sides 3,4,5 , T with sides 5,12, 13 and
2

T3 with sides 6, 8, 10 . Which triangles among T 1

, T2 and T3 are related ?

Watch Video Solution

26. Show that the relation R defined in the set A of all polygons as

R = {(P1 , P2 ) : P1 and P2 have same number of sides} , is an

equivalence relation . What is the set of all elements in A related to the

right angle triangle T with sides 3,4 and 5 ?

Watch Video Solution

27. Let L be the set of all lines in XY plane and R be the relation in L

defined as R = {(L1 , L2 } : L1 is parallel to L2 } . Show that R is an

equivalence relation . Find the set of all lines related to the line y = 2x + 4.
 Watch Video Solution

28. Let R be the relation in the set {(1, 2, 3, 4} given by

R = {(1, 2), (2, 2), (1, 1)(4, 4), (1, 3), (3, 3), (3, 2)}. Choose the

correct answer.

A. R is reflexive and symmetric but not transitive .

B. R is reflexive and transitive but not symmetric.

C. R is symmetric and transitive but not reflexive.

D. R is an equivalence relation.

Answer: B

Watch Video Solution

29. Let R be the relation on the set N given by

R = {(a, b) : a = b − 2, b > 6} . Choose the correct answer.

A. (2, 4) ∈ R
 B. (3, 8) ∈ R

C. (6, 8) ∈ R

D. (8, 7) ∈ R

Answer: C

Watch Video Solution

Exercise 1 2

1
1. Show that the function f : R → R , defined by f(x) = is one - one
x

and onto , where R is the set of all non - zero real number . is the result

true, if the domain R is replaced by N with co-domain being same as R ?

Watch Video Solution

2. Check the injectiveity and surjectivity of the following functions :

f:N → N given by f(x) = x

2
 Watch Video Solution

3. Check the injectiveity and surjectivity of the following functions :

f:Z → Z given by f(x) = x

2

Watch Video Solution

4. Check the injectiveity and surjectivity of the following functions :

f:R → R given by f(x) = x

2

Watch Video Solution

5. Check the injectiveity and surjectivity of the following functions :

f:N → N given by f(x) = x

3

Watch Video Solution

6. Check the injectiveity and surjectivity of the following functions :

f:Z → Z given by f(x) = x

3

Watch Video Solution

7. Prove that the Greatest Integer Function f : R → R , given by f(x) = [x] ,

is neither one - one nor onto , where [x] denotes the greatest integer less

than or equal to x.

Watch Video Solution

8. Show that the Modulus Function f:R → R , given by f(x) = |x| , is

neither oneone nor onto , where |x| is x, if x is positive or 0 and |x| is - x , if

x is negative.

Watch Video Solution

9. Show that the Signum Function f:R → R given by

⎧ 1 if x > 0
⎪

f(x) = ⎨ 0 if x = 0 is neither one - one nor onto.

⎩
⎪
−1 if x < 0

Watch Video Solution

10. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)}

be a function from A to B. Show that f is one - one.

Watch Video Solution

11. In each of the following cases , state whether the function is one - one ,

onto or bijective. Justify your answer.

f:R → R defined by f(x) = 3 − 4x.

Watch Video Solution

12. In each of the following cases , state whether the function is one - one

, onto or bijective. Justify your answer.

f:R → R defined by f(x) = 1 + x 2

Watch Video Solution

13. Let A and B be sets. Show that f:A × B → B × A such that f(a,b) =

(b,a) is bijecive function.

Watch Video Solution

n+1
if n is odd
14. Let : f:N → N be defined by f(n) = {
n
2
for all
if n is even
2

n ∈ N .

State whether the function f is bijective . Justify your answer.

Watch Video Solution

15. Let A = R - {3} and B = R - {1}. Consider the function f:A → B defined
x − 2
by , f(x) = ( ) is f one - one and onto ?
x − 3

Watch Video Solution

16. Let f : R → R be defined as f(x) = x

4
. Choose the correct answer.

A. f is one - one onto

B. f is many - one onto

C. f is one - one but not onto

D. f is neither one - one nor onto

Answer: D

Watch Video Solution

17. Let f : R → R be defined as f(x) = 3x . Choose the correct answer.

 A. f is one - one onto

B. f is many - one onto

C. f is one - one but not onto

D. f is neither one - one nor onto

Answer: A

Watch Video Solution

Exercise 1 3

1. Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by

f = {(1, 2), (3, 5), (4, 1) and g{(1, 3), (2, 3), (5, 1)} . Write down gof.

Watch Video Solution

2. Let f , g and h be functions from R to R . Show that , (f +g) oh = foh +

goh
(f.g) oh = (foh)+ (goh)

Watch Video Solution

3. Find gof and fog , if

f(x) = |x| and g(x) = |5x − 2| .

Watch Video Solution

4. Find gof and fog , if

1
3
f(x) = 8x and g(x) = x 3

Watch Video Solution

4x + 3 2 2
5. If f(x) = , x ≠ , show that fof (x) = x, for all x ≠ . What
6x − 4 3 3

is the inverse of f ?

Watch Video Solution

6. State with reason whether following functions have inverse :

f : {1, 2, 3, 4} → {10} with f : {(1, 10), (2, 10), (3, 10), (4, 10)}

Watch Video Solution

7. State with reason whether following functions have inverse :

g : {5, 6, 7, 8} → {1, 2, 3, 4} with g : {(5, 4), (6, 3), (7, 4), (8, 2)}

Watch Video Solution

8. State with reason whether following functions have inverse :

h : {2, 3, 4, 5} → {7, 9, 11, 13} with h{(2, 7), (3, 9), (4, 11), (5, 13)}

Watch Video Solution

x
9. Show that f : [-1,1] → R , given by f(x) = is one - one . Find the
x + 2

inverse of the function f : [ − 1, 1] → Range f.

 x
(Hint: For y ∈ Range f, y = f(x) = , for some x in [-1,1] , i.e.,
x + 2

2y
x = ).
1 − y

Watch Video Solution

10. Consider f : R → R given by f(x) = 4x +3. Show that f is invertible. Find

inverse of f .

Watch Video Solution

11. Consider f:R

+
→ [4, ∞] given by f(x) = x
2
+ 4 show that f is f

invertible with the inverse f −1

of given by f −1
(y) = √y − 4 where R
+

is set of all non - negative real numbers .

Watch Video Solution

12. Consider f : R +
→ [ − 5, ∞) given by f(x) = 9x 2
+ 6x − 5 . Show that

√y + 6 − 1
f is invertible with f −1
(y) = ( )
3
 Watch Video Solution

13. Let f:X → Y be an invertible function . Show that f has unique

inverse .

(Hint : Suppose g1 and g2 are two inverse of f. Then for all

y ∈ Y , (fog1 )(y) = IY (y) = (fog2 )(y). Use one - one ness of f).

Watch Video Solution

14. Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) and f(3) = c . Find

and show that (f .

−1
−1 −1
f ) = f

Watch Video Solution

15. Let f : X → Y be an invertible function . Show that the inverse of f −1

is f . i.e., (f .
−1
−1
) = f

Watch Video Solution

 1

16. If f : R → R be given by f(x) = (3 − x )

3 3
then fof (x) is

A. x
1

B. x 3

C. x

D. (3 − x 2
)

Answer: C

Watch Video Solution

4 4x
17. Let f:R − { − } → R be a function defined as f(x) = .
3 3x + 4

4
The inverse of f is the map g : Range f → R − { − } given by
3

3y
A. g(y) =
3 − 4y

4y
B. g(y) =
4 − 3y

4y
C. g(y) =
3 − 4y
 3y
D. g(y) =
4 − 3y

Answer: B

Watch Video Solution

18. Determine whether or not each of the definition of ∗ given below

gives a binary operation. In the even that ∗ is not a binary operation ,

give justification for this .

On Z +
, define ∗ by a * b = a − b

Watch Video Solution

19. Determine whether or not each of the definition of ∗ given below

gives a binary operation. In the even that ∗ is not a binary operation ,

give justification for this .

On Z +
, define ∗ by a * b = ab

Watch Video Solution

20. Determine whether or not each of the definition of ∗ given below

gives a binary operation. In the even that ∗ is not a binary operation ,

give justification for this .

On R , define ∗ by a * b = ab
2

Watch Video Solution

21. Determine whether or not each of the definition of ∗ given below

gives a binary operation. In the even that ∗ is not a binary operation ,

give justification for this .

On Z +
, define ∗ by a * b = |a − b|

Watch Video Solution

22. Determine whether or not each of the definition of ∗ given below

gives a binary operation. In the even that ∗ is not a binary operation ,

give justification for this .

On Z +
, define ∗ by a * b = a

Watch Video Solution

23. For each opertion ∗ difined below, determine whether ∗ isw binary,

commutative or associative.

(i) On Z, define a ∗ b = a − b

(ii) On Q, define a ∗ b = ab + 1

ab
(iii) On Q, define a ∗ b =
2

(iv) On Z +
, define a ∗ b = 2
ab

(v) On Z +
, define a ∗ b = a
b

a
(vi) On R − { − 1}, define a ∗ b =
b + 1

Watch Video Solution

24. For each opertion ∗ difined below, determine whether ∗ isw binary,

commutative or associative.
(i) On Z, define a ∗ b = a − b

(ii) On Q, define a ∗ b = ab + 1

ab
(iii) On Q, define a ∗ b =
2

(iv) On Z +
, define a ∗ b = 2
ab

(v) On Z +
, define a ∗ b = a
b

a
(vi) On R − { − 1}, define a ∗ b =
b + 1

Watch Video Solution

25. For each opertion ∗ difined below, determine whether ∗ isw binary,

commutative or associative.

(i) On Z, define a ∗ b = a − b

(ii) On Q, define a ∗ b = ab + 1

ab
(iii) On Q, define a ∗ b =
2

(iv) On Z +
, define a ∗ b = 2
ab

(v) On Z +
, define a ∗ b = a
b

a
(vi) On R − { − 1}, define a ∗ b =
b + 1

Watch Video Solution

26. For each opertion ∗ difined below, determine whether ∗ isw binary,

commutative or associative.

(i) On Z, define a ∗ b = a − b

(ii) On Q, define a ∗ b = ab + 1

ab
(iii) On Q, define a ∗ b =
2

(iv) On Z +
, define a ∗ b = 2
ab

(v) On Z +
, define a ∗ b = a
b

a
(vi) On R − { − 1}, define a ∗ b =
b + 1

Watch Video Solution

27. For each operation ∗ defined below, determine , whether ∗ is binary

, commutative or associative.

On Z +
, define a*b = ab

Watch Video Solution

 28. For each opertion ∗ difined below, determine whether ∗ isw binary,

commutative or associative.

(i) On Z, define a ∗ b = a − b

(ii) On Q, define a ∗ b = ab + 1

ab
(iii) On Q, define a ∗ b =
2

(iv) On Z +
, define a ∗ b = 2
ab

(v) On Z +
, define a ∗ b = a
b

a
(vi) On R − { − 1}, define a ∗ b =
b + 1

Watch Video Solution

29. Consider the binary operation ∧ on the set {1,2,3,4,5} defined by a ∧ b

= min {a,b} . Write the operation table of the operation ∧ .

Watch Video Solution

Exercise 1 4
1. Consider a binary operation ∗ on the set {1,2,3,4,5} given by the

following multiplication table.

(i) Compute (2*3)*4 and 2*(3*4)

(ii) Is ∗ commutative ?

(iii) Compute (2*3)*(4*5)

(Hint: use the following table )

Watch Video Solution

2. Let ∗ be the binary operation on the set {1,2,3,4,5} defined a*b = H.C.F

of a and b Is the operation ∗ same as the operation ∗ defined in

Exercise 4 above ? Justify your answer .

Watch Video Solution

3. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

Find

5*7, 20*16

Watch Video Solution

4. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

Find

Is ∗ commutative ?

Watch Video Solution

5. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

Find

Is ∗ associative ?

Watch Video Solution

6. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

Find

Find the identity of ∗ in N.

Watch Video Solution

7. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

Find

Which elements of N are invertible for the operation ∗ ?

Watch Video Solution

8. Is ∗ defined on the set {1, 2, 3, 4, 5} by a*b = L.C.M. of a and b a binary

operation ? Justify your answer.

Watch Video Solution

9. Let ∗ be the binary operation on N defined by a*b = H.C.F of a and b . Is

∗ commutative ? Is ∗ associative ? Does there exist identity for this

binary operation on N ?

Watch Video Solution

10. Let ∗ be a binary operation on the set Q of rational numbers as

follows :

a*b = a − b

Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

11. Let ∗ be a binary operation on the set Q of rational numbers as

follows :

2 2
a*b = a + b

Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

12. Let ∗ be a binary operation on the set Q of rational numbers as

follows :

a*b = a + ab

Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

13. Let ∗ be a binary operation on the set Q of rational numbers as

follows :
 2
a*b = (a − b)

Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

14. Let ∗ be a binary operation on the set Q of rational numbers as

follows :
ab
a*b =
4

Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

15. Let ∗ be a binary operation on the set Q of rational numbers as

follows :

2
a*b = ab
Find which of the binary operations are commutative and which are

associative.

Watch Video Solution

16. Find which of the operations given above has identity.

a*b = a − b

Watch Video Solution

17. Find which of the operations given above has identity.

2 2
a*b = a + b

Watch Video Solution

18. For which values of p does the pair of equations given below has

unique solution ?
4x + py + 8 = 0

2x + 2y + 2 = 0

Watch Video Solution

19. Find which of the operations given above has identity.

2
a*b = (a − b)

Watch Video Solution

20. Find which of the operations given above has identity.

ab
a*b =
4

Watch Video Solution

21. Find which of the operations given above has identity.

2
a*b = ab

Watch Video Solution

22. L A = N × N and ∗ be the binary operation on A defined by

(a, b)*(c, d) = (a + c, b + d) Show that ∗ is commutative and

associative . Find the identity element for ∗ on A , if any.

Watch Video Solution

23. State whether the following statements are true or false . Justify .

For an arbitrary binary operation ∗ on a set N, a*a = a ∀a ∈ N .

Watch Video Solution

24. State whether the following statements are true or false . Justify .

If ∗ is commutative binary operation on N, then a*(b*c) = (c*b)*a .

Watch Video Solution

 25. Consider a binary operation ∗ on N defined as a*b = a
3 3
+ b .

Choose the correct answer.

A. Is ∗ both associative and commutative ?

B. Is ∗ commutative but not associative ?

C. Is ∗ associative but not commutative ?

D. Is ∗ neither commutative nor associative ?

Answer: B

Watch Video Solution

Miscellaneous Exercise 1

1. Let f:R → R be defined as f(x) = 10x + 7. Find the function

g: R → R such that gof = fog = I

g

Watch Video Solution

2. Let f : W → W be defined as f(n) = n - 1, if n is odd and f(n) = n + 1 , if n

even. Show that f is invertible. Find the inverse of f. Here, W is the set all

whole numbers.

Watch Video Solution

3. If f : R → R is defined by f(x) = x
2
− 3x + 2 , find f(f(x)).

Watch Video Solution

4. Show that the function f : R → {x ∈ R : − 1 < x < 1} defined by

x
f(x) = , x ∈ R is one one and onto function.
1 + |x|

Watch Video Solution

5. Show that the function f : R → R given by f(x) = x

3
is injective.

Watch Video Solution

6. Give examples of two functions f : N → Z and g : Z → Z such that

gof is injective but g is not injective.

(Hint : Consider f(x) = x and g(x) = |x| ) .

Watch Video Solution

7. Give examples of two function f : N → N and g : N → N such that

gof is onto but f is not onto. (Hint: Consider f(x) = x+1 " and "g(x) = {x-1 if

x>1 1 if x=1.

Watch Video Solution

8. Given a non empty set X , consider P(X) which is the set of all subsets of

X. Define the relation R in P(X) as follows : For subsets A , B in P(X) ARB if

and only if A ⊂ B . Is R an equivalence relation on P(X) ? Justify your

answer.

Watch Video Solution

9. Given a non - empty set, X , consider the binary operation

∗ : P (X) × P (X) → P (X) given by A ∗ B = A ∩ B, ∀ A, B in P(X) ,

where P(X) is the power set X. Show that X is the identity element for this

operation and X is the only invertible element in P(X) with respect to the

operation ∗ .

Watch Video Solution

10. Find the number of all onto functions from the set {1,2,3,.......,n} to itself.

Watch Video Solution

11. Let S = {a,b,c} and T = {1,2,3} . Find F

−1
of the following functions F

from S to T , if it exists .

F = {(a, 3), (b, 2), (c, 1)}

Watch Video Solution

12. Let S = {a,b,c} and T = {1,2,3} . Find F
−1
of the following functions F

from S to T , if it exists .

F = {(a, 2), (b, 1), (c, 1)}

Watch Video Solution

13. Consider the binary operations ∗ R × R → R and o : R × R → R

defined as a ∗ b|a − b| and a o b = a, ∀ a, b ∈ R. Show that ∗ is

commutative but not associative , o is associative but not commutative.

Further, show that ∀ a, b, c ∈ R, a ∗ (b o c) = (a ∗ b)o(a ∗ c) . [If it is so

, we say that the operation ∗ distributes over the operation o] . Does o

distribute over ∗ ? Justify your answer.

View Text Solution

14. Given a non - empty set X , let ∗ : P (X) × P (X) → P (X) be defined

as A ∗ B = (A − B) ∪ (B − A), ∀ A, B ∈ P (X) . Show that the empty

set ϕ is the identity for the operation ∗ and all the elements A of P(X)

are invertible with A

−1
= A. (Hint :

(A − ϕ) ∪ (ϕ − A) = a and (A − A) ∪ (A − A) = A ∗ A = ϕ)

View Text Solution

15. Define a binary operation ∗ on the set {0, 1, 2, 3, 4, 5} as

a + b, if a + b < 6
a ∗ b = { Show that zero is the identity for
a + b − 6, If a + b ≥ 6

this operation and each element a ≠ 0 of the set is invertible with 6 - a

being the inverse of a.

Watch Video Solution

16. Let A = { − 1, 0, 1, 2}, B = { − 4, − 2, 0, 2} and f, g : A → B be

functions defined
1 ∣
. Are f and g
2
∣
f(x) = x − x, x ∈ R and g(x) = 2∣ x − ∣ − 1, x ∈ R
∣ 2 ∣

equal ? Justify your answer.

(Hint : One may note that two functions f : A → B and g : A → B such

that f (a) = g(a) Aa ∈ A, are called equal functions).

Watch Video Solution

17. Let A = {1,2,3}. Then number of relations containing (1,2) and (1,3) which

are reflexive and symmetric but not transitive is

A. 1

B. 2

C. 3

D. 4

Answer: A

Watch Video Solution

18. Let A = {1,2,3}. Then number of equivalence relations containing (1,2) is

 A. 1

B. 2

C. 3

D. 4

Answer: B

Watch Video Solution

19. Let f:R → R be the Signum Function defined as

⎧ 1 x > 0
⎪

f(x) = ⎨ 0 x = 0 and g: R → R be the Greatest Integer Function

⎩
⎪
−1 x < 0

given by g(x) = [x] , where [x] is greatest integer less than or equal to x.

Then , does fog and gof coincide in (0,1] ?

A. Yes

B. No

C. Nothing can be said

 D. Composite function does not exists

Answer: B

Watch Video Solution

20. Number of binary operations on the set {a,b} are

A. 10

B. 16

C. 20

D. 8

Answer: B

Watch Video Solution

Practice Work
1. The relation R defined in the set of real number R is as follow :

R{(x, y) : x − y + √2 is an irrational number}

Is R transitive relation ?

Watch Video Solution

2. Let R be relation defined on the set of natural number N as follows :

R = {(x, y) : x ∈ N , y ∈ N , 2x + y = 41} . Find the domian and range

of the relation R . Also verify whether R is reflexive, symmetric and

transitive.

Watch Video Solution

3. A = {(1, 2, 3, ......10} The relation R defined in the set A as R

= {(x, y) : y = 2x} . Show that R is not an equivalence relation.

Watch Video Solution

4. The relation R difined the set Z as R = {(x, y) : x − y ∈ Z} show that

R is an equivalence relation.

Watch Video Solution

5. Show that the relation R defined by (a, b)R(c, d) ⇒ a + d = b + c on

the set N × N is an equivalence relation.

Watch Video Solution

6. R is relation in N × N as (a,b) R (c,d) . Show that R is an

⇔ ad = bc

equivalence relation.

Watch Video Solution

7. The relation R defined in the set N of natural number as ∀ n, min N if

on division by 5 each of the integers n and m leaves the remainder less

than 5. Show that R is equivalence relation. Also obtain the pairwise

disjoint subset determined by R.

Watch Video Solution

8. Find the domain and range of the following function :

f : R → R, f(x) = − |x|

Watch Video Solution

9. Find the domain and range of the following function :

2
x − 1
f : R → R, f(x) = , x ≠ 1
x − 1

Watch Video Solution

10. Find the domain and range of the following function :

1
f : R → R, f(x) = , x ≠ ± 1
1 − x2

Watch Video Solution

11. Find the domain and range of the following function :

2
f : R → R, f(x) = √9 − x

Watch Video Solution

12. Find the domain and range of the following function :

2
f : R → R, f(x) =
x − 2

Watch Video Solution

12x + 5 x > 1
13. f : R → R, f(x) = { then find
x − 4 x ≤ 1

1
f(0), f( − ), f(3), f( − 5) .
2

Watch Video Solution

14. Check the injectivity and surjectivity of the following functions .

2
f : R → R, f(x) = x + 7

Watch Video Solution

15. Check the injectivity and surjectivity of the following functions .

3
f : R → R, f(x) = x

Watch Video Solution

16. Check the injectivity and surjectivity of the following functions .

2
f : R → R, f(x) = x − 2

Watch Video Solution

17. Show that the function f : R → {x ∈ R : − 1 < x < 1} defined by

x
f(x) = , x ∈ R is one one and onto function.
1 + |x|
 Watch Video Solution

(n + 2) if n is even
18. f : Z → Z, f(n) = {
(2n + 1) if n is odd

State whether the function f is one - one and onto .

Watch Video Solution

19. f : N × N → N , f((m,n)) = m + n . If f one one and onto ?

Watch Video Solution

x
20. Show that f : R → R, f(x) =
2
is not one one and onto
x + 1

function.

Watch Video Solution

21. f : R → R, f(x) = x
2
. Find the preimage of 17 and -3.
+ 1
 Watch Video Solution

⎧ 2x x > 3
⎪

22. f : R → R, f(x) = ⎨ x
2
1 < x ≤ 3 then find f (-1) + f(2) + f(4) .
⎩
⎪
3x x ≤ 1

Watch Video Solution

23. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function, justify. If this is

described by the relation, g(x) = αx + β , then what values should be

assigned to α and β ?

Watch Video Solution

24. The functions f and g are defined as follow :

f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)} . Find the

range of f and g . Also find the composition function fog and gof .

Watch Video Solution

25. For functions f : A → B and g : B → A, gof = IA . Prove that f is

one one and g onto functions .

Watch Video Solution

x
26. f : R → R, f(x) = x
2
+ 2 and g : R → R, g(x) = then find
x − 1

fog and gof.

Watch Video Solution

27. f : N → R, f(x) = 4x
2
+ 12x + 5 . Show that f:N → R is invertible

function . Find the inverse of f.

Watch Video Solution

28. f and g are real valued function

f(x) = x
2
+ x + 7, x ∈ R and g(x) = 5x − 3, x ∈ R . Find fog and
gof. Also find (fog)(2) and (gof)(1).

Watch Video Solution

29. If f is greatest integer function and g is a modulus functions the find .

1 1
(gof)( − ) − (fog)( − ) .
3 3

Watch Video Solution

x
30. f : R → R, f(x) = , ∀x ∈ R . Then find (fofof) (x).
√1 + x 2

Watch Video Solution

31. f : Z → Z and g : Z → Z . Defined as f(n) = 3n and

n
, If n is a multiple of 3
g(n) = {
3
∀n ∈ Z Then show that
0, If n is not a multiple of 3

gof = Iz but fog ≠ Iz

Watch Video Solution

 x
32. f : R → R be defined by f(x) = + 3, g : R → R be defined by g(x)
2

= 2x − K . If fog = gof then find the value of K.

Watch Video Solution

33. f : Z → Z and g : Z → Z are defined as follow :

n + 2 n even 2n n even
f(n) = { , g(n) = { n−1
Find fog and gof.
2n − 1 n odd n odd
2

Watch Video Solution

34. ∗ is a binary operation on the set Q.

2a + b
a*b = then find 2*3.
4

Watch Video Solution

35. ∗ is a binary operation on the set Q.
1
a*b = a + 12b + ab then find 2*
3

Watch Video Solution

36. ∗ is a binary operation on the set Q.

a b 1 4
a*b = + then find * .
2 3 2 5

Watch Video Solution

37. ∗ is a binary operation o Z. If x*y = x

2
+ y
2
+ xy then find

.
2
[(1*2) + (0*3)]

Watch Video Solution

38. ∗ be a binary operation on R defined by

a b
a*b = + , a, b ∈ R .
4 7
Show that ∗ is not commutative and associative.

Watch Video Solution

39. Show that addition and multiplication are associative binary

operation on R. But subtraction and division is not associative on R.

Watch Video Solution

40. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On R defined a*b = √a
2 2
− b , |a| > |b| .

Watch Video Solution

41. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On Z defined a*b = a + b − 2 .
 Watch Video Solution

42. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On R - {1} defined a*b = a + b − ab .

Watch Video Solution

43. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

ab
On Q -{0} defined a*b = .
2

Watch Video Solution

44. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On Q - {-1} defined a*b = a + b + ab.

Watch Video Solution

45. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On P(X) defined A*B = A ∩ B , where X ≠ ϕ .

Watch Video Solution

46. Find the identity element , if it exists for the following operation . Also

find the inverse if it exists.

On P(X) defined A*B = A ∪ B , where X ≠ ϕ .

Watch Video Solution

47. On R - {-1}, a binary operation ∗ defined by a*b = a + b + ab then

find a −1
.

Watch Video Solution

48. ∗ be a binary operation on a set {0, 1, 2, 3, 4} defined by
a + b if a + b < 6
a ∗ b = {
a + b − 6 if a + b ≥ 6

Then find identity element of ∗ .

Watch Video Solution

49. On Z ∗ defined by a ∗ b = a + b + 1 . Is ∗ associative ? Find identity

element and inverse if it exists.

Watch Video Solution

50. be binary operation defined on a set R by .

2
∗ a ∗ b = a + b − (ab)

Show that ∗ is commutative, but it is not associative. Find the identity

element for ∗ .

Watch Video Solution

51. A binary operation ∗ be defined on the set R by a ∗ b = a + b + ab .

Show that ∗ is commutative, and it is also Associative.

Watch Video Solution

52. Show that if f : A → B and g : B → C are onto, then gof : A → C is

also onto.

Watch Video Solution

53. Show that if f : A → B and g : B → C are one- one, then gof:

A → C is also one-one.

Watch Video Solution

54. f : R → R, f(x) = cos x and g : R → R, g(x) = 3x

2
then find the

composite functions gof and fog.

 Watch Video Solution

55. Check the injectivity and surjectivity of the following function .

−x + 1 x ≥ 0
f : R → R, f(x) = {
2
x x < 0

Watch Video Solution

56. Check the injectivity and surjectivity of the following function .

2x + 1 x ≥ 0
f : R → R, f(x) = {
2
x x < 0

Watch Video Solution

57. Check the injectivity and surjectivity of the following function .

x
f : R × R − {0} → R, f(x, y) =
y

Watch Video Solution

58. Check the injectivity and surjectivity of the following function .

f : [ − 1, 1] → [ − 1, 1], f(x) = x|x|

Watch Video Solution

59. Check the injectivity and surjectivity of the following function .

.
n
f : N → N ∪ {0}, f(n) = n + ( − 1)

Watch Video Solution

60. Check the injectivity and surjectivity of the following function .

f : N − {1} → N , f(n) = Greatest prime factor of n .

Watch Video Solution

x x
10 − 10
61. f : R → ( − 1, 1), f(x) =
x −x
. If inverse of f −1
exists then
10 + 10

find it .
 Watch Video Solution

62. f : R +
∪ {0} → R
+
∪ {0}, f(x) = √x .

g : R → R, g(x) = x
2
− 1 then find fog .

Watch Video Solution

2 4x + 3
63. f : R − { } → R, f(x) = . Prove that (fof) (x) = x , what is
3 6x − 4

about f −1
?

Watch Video Solution

64. A = {1,2,3,4} , B = {1,5,9,11,15,16}

f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

Is f a relation from A to B ?

Give reason for your answer.

Watch Video Solution

 65. A = {1,2,3,4} , B = {1,5,9,11,15,16}

f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

Is f a function from A to B ?

Give reason for your answer.

Watch Video Solution

66. Let f be the subset of Z × Z defined by .

f = {(ab, a + b) : a, b ∈ Z}

Is f a function from Z to Z? Justify your answer.

Watch Video Solution

Textbook Based Mcqs

1. If a set A has m elements and a set B has n elements then the number

of relation from a to B is ...........

A. 2m+n
 B. 2 mn

C. m + n

D. mn

Answer: B

Watch Video Solution

2. A relation R on a finite set having n elements is reflexive. If R has m

pairs then ............

A. m ≥ n

B. m ≤ n

C. m = n

D. None of these

Answer: A

Watch Video Solution

3. x and y are real numbers . If xRy ⇔ x − y + √5 is on irrational

number then R is ......... Relation .

A. Reflexive

B. Symmetric

C. Transitive

D. None of these

Answer: A

Watch Video Solution

4. A = {1,2,3,4} . A relation R is A is given by

F = {(2, 2), (3, 3), (4, 4), (1, 2)} . Then R is relation.

A. Reflexive

B. Symmetric
 C. Transitive

D. None of these

Answer: C

Watch Video Solution

5. A relation R is form set A to B , and a relation S is from set B to C . Then

relation SOR is from ........

A. Set C to A

B. Set A to C

C. Does not exist

D. None of these

Answer: B

Watch Video Solution

6. Relation R = {(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)} then R −1
OR = .......

A. {(1, 1), (4, 4), (7, 4), (4, 7), (7, 7)}

B. {(1, 1), (4, 4), (4, 7), (7, 4), (7, 7), (3, 3)}

C. {(1, 5), (1, 6), (3, 6)}

D. None of these

Answer: B

Watch Video Solution

7. Which of the graphs is not a graph of functions ?

A.
 B.

C.

D.

Answer: B

Watch Video Solution

8. If f(1) = 1, f(n + 1) = 2f(n) + 1, n ≥ 1 then f(n) = .........

A. 2 n
+ 1

B. 2 n
 C. 2
n
− 1

D. 2
n−1
− 1

Answer: C

Watch Video Solution

9. A function y = f(x) satisfies the condition

1 1
f(x + ) = x
2
+ (x ≠ 0) then f(x) = ........
2
x x

A. − x 2
+ 2

B. x 2
− 2

C. x 2
− 2, x ∈ R − {0}

D. x 2
− 2, |x| ∈ [2, ∞)

Answer: D

Watch Video Solution

10. If f(x + ay, x − ay) = axy then f(x,y) = ......

A. xy

B. x 2
− a y
2 2

2 2
x − y
C.
4

2 2
x − y
D.
2
a

Answer: C

Watch Video Solution

αx
11. For function f(x) = , x ≠ − 1 if fof(x) = x then α = ..........
x + 1

A. √2

B. − 1

1
C.
2

D. − √2
Answer: B

Watch Video Solution

π
12. For real valued functions f and g, f(x) = 2sin ( ) and g(x) = √x .
x

Then fog(4) - gof (6) = ..............

A. 0

1
B.
2

C. 1

√3
D.
2

Answer: C

Watch Video Solution

13. The domain of the real value function

f(x) = √5 − 4x − x
2
+ x
2
log(x + 4) is ...........
 A. − 5 ≤ x ≤ 1

B. − 5 ≤ 4 and n ≥ 1

C. − 4 < x ≤ 1

D. ϕ

Answer: C

Watch Video Solution

x
14. The domian of sin −1
[log3 ( )] is .......
3

A. [1,9]

B. [ − 1, 9]

C. [ − 9, 1]

D. [ − 9, − 1]

Answer: A

Watch Video Solution

 2
x + x + 2
15. Range of the function f(x) = is……
2
x + x + 1

A. (1, ∞)

11
B. (1, )
7

7
C. (1, )
3

7
D. (1, )
5

Answer: C

Watch Video Solution

1
16. If g(x) = x
2
+ x − 2 and (gof)(x) = 2x
2
− 5x + 2 then f(X) =........
2

A. 2x − 3

B. 2x + 3

C. 2x 2
+ 3x + 1
 D. 2x 2
− 3x − 1

Answer: B

Watch Video Solution

17. g(x) = 1 + √x and f(g(x)) = 3 + 2√x + x then f(x) = ......

A. 1 + 2x 2

B. 2 + x 2

C. 1 + x

D. 2 + x

Answer: B

Watch Video Solution

18. If real function f(x) = (x + 1)
2
and g(x) = x
2
+ 1 then (fog) (-3) =

..........

A. 121

B. 112

C. 211

D. 111

Answer: A

Watch Video Solution

19. f(x) = cot

−1
x: R
+
→ (0, π) and g(x) = 2x − x
2
:R → R then

the range of f(g(x)) is ...........

π
A. (0, )
2

π
B. (0, )
4

π π
C. [ , )
4 2
 π
D. { }
4

Answer: C

Watch Video Solution

20. The domian of f is [-5,7] and g(x) = |2x+5| then the domian of (fog) (x)

is ............

A. [ − 4, 1]

B. [ − 5, 1]

C. [ − 6, 1]

D. None of these

Answer: C

Watch Video Solution

21. A set A has 3 elements and a set B has 4 elements . The number of one

one function defined from set A to B is .........

A. 144

B. 12

C. 24

D. 64

Answer: C

Watch Video Solution

22. f : R → R, f(x) = (x − 1)(x − 2)(x − 3) then f is ........

A. One - one but not onto.

B. Onto but not one - one

C. One - one and onto.

D. Neither one one nor onto.

Answer: B

Watch Video Solution

23. f : N 2
→ N , f(n) = (n + 5) , n ∈ N , then the function f is ............

A. Neither one one nor onto

B. One one and onto

C. One one but not onto

D. Onto but not one one.

Answer: B

Watch Video Solution

x
24. f : [0, ∞) → [0, ∞), f(x) = then the function f is .........
1 + x

A. One one and onto

 B. One one but not onto

C. Onto but not one one

D. Neither one one nor onto.

Answer: B

Watch Video Solution

3 2
x x
25. f(x) = + + ax + b, ∀ x ∈ R. If (x) is one one function then
3 2

the minimum value of a is ........

1
A.
4

B. 1

1
C.
2

1
D.
8

Answer: A

Watch Video Solution

26. f(x) = x
2
− 2x − 1, ∀ x ∈ R, f : ( − ∞, ∞] → [b, ∞) is one one

and onto function then b = .............

A. − 2

B. − 1

C. 0

D. 1

Answer: B

Watch Video Solution

x −x
e − e
27. f(x) =
x −x
+ 2 . The inverse of f(x) is ........
e + e

x − 2 2

A. log e
( )
x − 1

x − 1
B. log
2

( )
e
3 − x
1

x 2

C. log e
( )
2 − x
 1
−
x − 1 2

D. log e
( )
x + 1

Answer: B

Watch Video Solution

x
28. f : (2, 4) → (1, 3), f(x) = x − [ ] , where [.] is a greatest integer
2

function then f −1
(x) = ......

A. 2x

x
B. x + [ ]
2

C. x + 1

D. does not exist

Answer: C

Watch Video Solution

29. f : [2, ∞) → y, f(x) = x
2
− 4x + 5 is a one and Onto function . If

y ∈ [a, ∞) then the value of a is .........

A. 2

B. 1

C. − ∞

D. − 1

Answer: B

Watch Video Solution

30. f : N → N , f(x) = x + ( − 1)
x−1
then f −1
(x) = .......

A. xy

B. x − 1

C. x
x−1
− ( − 1)

D. x
x−1
+ ( − 1)
Answer: D

Watch Video Solution

31. a > 1 is a real number f(x) = loga x

2
, where x > 0 If f
−1
(x) is a

inverse of f(x) and b and c are real numbers then f −1

(b + c) = ......

A. f −1
(b). f
−1
(c)

B. f −1
(b) + f
−1
(c)

1
C.
f(b + )

D. None of these

Answer: A

Watch Video Solution

32. f : R → R, f(x) = 2x + |cos x| then f is ......... function .

 A. One one and onto

B. One one but not onto

C. Neither one one nor onto

D. Not one one but onto

Answer: A

Watch Video Solution

33. The number of onto function from set {1, 2, 3, 4} to {3, 4, 7} is .......

A. 18

B. 36

C. 64

D. None of these

Answer: B

Watch Video Solution

34. Match the Section (A) with the Section (B) properly.

A. 1 → A, 2 → D, 3 → B, 4 → C

B. 1 → C, 2 → A, 3 → D, 4 → B

C. 1 → A, 2 → C, 3 → B, 4 → D

D. 1 → C, 2 → B, 3 → D, 4 → A

Answer: B

View Text Solution

35. f : [0, 3] → [1, 29], f(x) = 2x
3
− 15x
2
+ 36x + 1 then f is ........

function.

A. One one and onto

B. One one but not onto

C. Neither one one nor onto

D. Not one one but onto

Answer: B

Watch Video Solution

36. f(x,y) = and g(x,y)= max (x,y) - min (x,y) then

( min ( x , y ) )
( max (x, y))

3
f(g( − 1, − ), g( − 4, − 1.75)) = .........
2

A. 0.5

B. − 0.5

C. 1
 D. 1.5

Answer: D

Watch Video Solution

37. Let A = {1,2,3}. Then number of equivalence relations containing (1,2) is

A. 1

B. 2

C. 3

D. 8

Answer: B

Watch Video Solution

38. S is defined in Z by (x, y) ∈ S ⇔ |x − y| ≤ 1. S is ........

 A. Reflexive and transitive but not symmetric.

B. Reflexive and symmetric but not transitive.

C. symmetric and transitive but not reflexive.

D. an equivalence relation

Answer: B

Watch Video Solution

39. If S is defined on R by (x,y) ∈ R ⇔ xy ≥ 0 . Then S is ...........

A. an equivalence relation

B. reflexive only

C. symmetric only

D. transitive only

Answer: A

Watch Video Solution

40. Which of the following defined on Z is not an equivalence relation ?

A. (x, y) ∈ S ⇔ x ≥ y

B. (x, y) ∈ S ⇔ x = y

C. (x, y) ∈ S ↔ x − y is a multiple of 3

D. (x, y) ∈ S if |x-y| is even

Answer: A

Watch Video Solution

ab
41. If a ∗ b = on Q +
then the inverse of a(a ≠ 0) for ∗ is ......
3

3
A.
a

9
B.
a

1
C.
a
 2
D.
a

Answer: B

Watch Video Solution

42. The number of binary operation on {1, 2, 3, ......, n} is ..........

A. 2n

B. n
2
n

C. n 3

D. n 2n

Answer: B

Watch Video Solution

43. If a ∗ b = a + b on R - {1} , then a −1

is ........
 A. a 3

1
B.
a

−a
C.
a + 1

1
D.
2
a

Answer: C

Watch Video Solution

44. For a ∗ b = a + b + 10 on Z , the identity element is ........

A. 0

B. − 5

C. − 10

D. 1

Answer: C

Watch Video Solution

 x − p
45. f : R − {q} → R − {1}, f(x) = , then f is .........
, p ≠ q
x − q

A. one - one and onto .

B. many - one and not onto .

C. one - one and not onto .

D. many - one and onto .

Answer: A

Watch Video Solution

46. Check the injectivity and surjectivity of the following function .

f : [ − 1, 1] → [ − 1, 1], f(x) = x|x|

A. one - one and onto .

B. many - one and onto .

C. many - one and not onto .

 D. one - one and not onto.

Answer: A

Watch Video Solution

π π
47. f : [ − , ] → [ − 1, 1] is a bijection , if ......
2 2

A. f(x) = |x|

B. f(x) = sin x

C. f(x) = x
2

D. f(x) = cos x

Answer: B

Watch Video Solution

48. f : R → R, f(x) = x
2
+ 2x + 3 is .......
 A. one - one but not onto.

B. onto but not one - one

C. onto but not one one

D. many - one and not onto .

Answer: D

Watch Video Solution

49. If a ∗ b = a
2 2
+ b on Z , then ∗ is ..........

A. commutative and associative.

B. commutative and not associative.

C. not commutative and associative.

D. neither commutative nor associative.

Answer: B

Watch Video Solution

50. If a ∗ b = a + b − ab on Q
+
, then the identity and the inverse of a

for ∗ are respectively ..........

a
A. 0 and
a − 1

a − 1
B. 1 and
a

C. − 1 and a

1
D. 0,
a

Answer: A

Watch Video Solution

ab 1 1
51. If a ∗ b = on Q +
, then 3 ∗ ( ∗ ) is .......
3 5 2

5
A.
160

1
B.
30

3
C.
160
 3
D.
60

Answer: B

Watch Video Solution

52. If Δ is defined on P (X)(X ≠ ϕ) by , AδB = (A ∪ B) − (A ∩ B) ,

then ..........

A. identity for Δ is ϕ and inverse of A is A.

B. identity for Δ is A and inverse of A is ϕ .

C. identity for Δ is A' and inverse of A is A.

D. identity for Δ is X and inverse of A is ϕ .

Answer: A

Watch Video Solution

53. S is defined on N × N by ((a, b), (c, d) ∈ S ⇔ a + d = b + c....... .

A. S is reflexive , but not symmetric

B. S is reflexive , and transitive only

C. S is an equivalence relation

D. S is transitive only

Answer: C

Watch Video Solution

x
54. If f : R +
→ R, f(x) = is .......
x + 1

A. one - one and onto .

B. one - one and not onto .

C. not one - one and not onto.

D. Onto but not one - one.

Answer: B

Watch Video Solution

55. If

f : R → R, f(x) = [x], g : R → R, g(x) = sin x, h : R → R, h(x) = 2x

, then ho(gof) = ..........

A. sin[x]

B. [sin 2x]

C. 2(sin[x])

D. sin 2[x]

Answer: C

Watch Video Solution

 ⎧ −1 x < 0
⎪

56. f : R → R, f(x) = ⎨ 0 x = 0 g : R → R,g(x) = 1 + x - [x]

⎩
⎪
1 x > 0

then for all x, f (g(x)) = ........

A. 1

B. 2

C. 0

D. − 1

Answer: A

Watch Video Solution

1
57. If f : {x ∣ x ≥ 1, x ∈ R} → {x ∣ x ≥ 2, x ∈ R} f(x) = x + then
x

f
−1
(x) = .........

2
x + √x − 4
A.
2

2
x − √x − 4
B.
2
 2
x + 1
C.
x

D. √x 2
− 4

Answer: A

Watch Video Solution

x
58. f : R → R, f(x) = , ∀x ∈ R . Then find (fofof) (x).
√1 + x 2

x
A. 2
1 + x

2
1 + x
B.
x

x
C.
√1 + 2x 2

x
D.
√1 + 3x 2

Answer: D

Watch Video Solution

59. 2
f : R → R, f(x) = x , g : R → R, g(x) = 2
x
, then

{x ∣ (fog)(x) = (gof)(x)} = ............

A. {0}

B. {0, 1}

C. R

D. {0, 2}

Answer: D

Watch Video Solution

60. The relation S on set {1, 2, 3, 4, 5} is

S = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} . The S is .........

A. Only symmetric

B. Only reflexive

C. Only transitive
 D. Equivalence relation

Answer: D

Watch Video Solution

61. The function f : R → R, f(x) = 5x + 7 then the function f is ..........

A. One one and onto

B. One one and not onto

C. Onto but not one one

D. Neither one one nor onto.

Answer: A

Watch Video Solution

62. The number of binary operation on set {1, 2} is .......

 A. 8

B. 16

C. 2

D. 4

Answer: B

Watch Video Solution

63. The function

1
+ + 3 + +
f:R → R , f(x) = x , g : R → R , g(x) = x 3

then (fog)(x) = .........

A. x 3

1
B.
x

C. √x3

D. x

Answer: D
 Watch Video Solution

64. a ∗ b = a
2 2
+ b + ab + 2 on Z then 3 ∗ 4 = ......

A. 39

B. 40

C. 25

D. 41

Answer: A

Watch Video Solution

Textbook Illustrations For Practice Work

1. Let A be the set of all students of a boys school. Show that the relation

R in A given by R = {(a, b) : a is sister of b} is the empty relation and

R' = {(a, b) : the difference between heights of a and b is less than 3

meters } is the universal relation.

Watch Video Solution

2. Let T be the set of all triangles in a plane with R a relation in T given by

R = {(T1 , T2 ) : T1 is congruent to T2 } Show that R is an equivalence

relation.

Watch Video Solution

3. Let L be the set of all lines in a plane and R be the relation in L defined

as R = {(L1 , L2 ) : L1 is perpendicular to L2 }. Show that R is symmetric

but neither reflexive nor transitive.

Watch Video Solution

4. Show that the relation R in the set {1, 2, 3} given by

R = {(1, 1), (2, 2), (3, 3), (1, 2). , (2, 3)} is reflexive but neither

symmetric nor transitive.

Watch Video Solution

5. Show that the relation R in the set Z of intergers given by

R = {(a, b) : 2 divides a-b }

is an equivalence relation.

Watch Video Solution

6. Let R be the realtion defined in the set A = {1, 2, 3, 4, 5, 6, 7} by

R = {(a, b) : both a and b are either odd or even}. Show that R is an

equivalance relation. further, show that all the elements of the subset

{1, 3, 5, 7} are related to each other and all elements of subset {2, 4, 6}

are related to each other, but no element of the subset {1, 3, 5, 7} is

related to any element of the subset {2, 4, 6}.

 Watch Video Solution

7. Let A be the set of all 50 students of Class X in a school. Let f : A → N

be function defined by f(x) = roll number of the student x. Show that f is

one-one but not onto.

Watch Video Solution

8. Show that the function f:N → N , given by f(x) = 2x, is one-one but

not onto.

Watch Video Solution

9. Prove that the function f:R → R , given by f(x) = 2x, is one-one and

onto.

Watch Video Solution

10. Show that the function f:N → N , given by f(1) = f(2) = 1 and f(x) = x -

1, for every x > 2 , is onto but not one-one.

Watch Video Solution

11. Show that the function f: R → R , defined as f(x) = x

2
, is neither

one-one nor onto.

Watch Video Solution

12. Show that f : N → N , given by

x + 1 if x is odd
f(x) = { is both one - one and onto .
x − 1 if x is even

Watch Video Solution

13. Show that an onto function f : {1, 2, 3} → {1, 2, 3} is always one-one.

Watch Video Solution

14. Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto.

Watch Video Solution

15. Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be

functions defined as f (2) = 3, f(3) = 4, f(4) = f(5) = 5 and g(3) = g(4) = 7 and

g(5) = g(9) = 11. Find gof.

Watch Video Solution

16. Find gof and fog, if f : R → R and g : R → R are given by

f(x) = cos x and g(x) = 3x

2
. Show that gof ≠ fog.

Watch Video Solution

7 3
17. Show that if f:R − { } → R − { } is defined by
5 5

3x + 4 3 7
f(x) = and g : R − { } → R − { } is defined by
5x − 7 5 5
 7x + 4
g(x) = , then fog = IA and gof = IB , where
5x − 3

3 7
A = R − { }, B = R − { }, IA (x) = x, ∀ x ∈ A, IB (x) = x, ∀ x ∈ B
5 5

are called identity functions on sets A and B , respectively .

Watch Video Solution

18. Show that if f : A → B and g : B → C are one- one, then gof:

A → C is also one-one.

Watch Video Solution

19. Show that if f : A → B and g : B → C are onto, then gof : A → C is

also onto.

Watch Video Solution

20. Consider functions f and g such that composite gof is defined and is

oneone. Are f and g both necessarily one-one.

 Watch Video Solution

21. Are fand g both necessarily onto, if gof is onto?

Watch Video Solution

22. Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1)

= a, f(2) = b and f(3) = C. Show that there exists a function

g : {a, b, c} → {1, 2, 3} such that gof = Ix and fog = IY , where X =

{1,2,3} and Y = {a,b,c} ,

Watch Video Solution

23. Let f:N → Y be a function defined as f(x) = 4x + 3, where, Y = {

y ∈ N : y = 4x + 3 for some x ∈ N }. Show that f is invertible. Find the

inverse.

Watch Video Solution

24. Let Y = {n 2
: n ∈ N} ⊂ N Consider f : N → Y as f(n) = n . Show that
2

f is invertible. Find the inverse of f.

Watch Video Solution

25. Let f' : N → R be a function defined as f' (x) = 4x

2
+ 12x + 15 .

Show that f:N → S , where, S is the range of f, is invertible. Find the

inverse of f.

Watch Video Solution

26. Consider f : N → N , g : N → N and h : N → R defined as

f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z, ∀ x, y and z in N. Show

that ho(gof) = (hog)of .

Watch Video Solution

27. Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat}

defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple,g(b) = ball and g

(c )= cat. Show that f, g and gof are invertible. Find out

and show that (gof)

−1 −1 −1 −1 −1 −1
f , g and (gof) = f og .

Watch Video Solution

28. Let S = {1, 2, 3}. Determine whether the functions f:S → S defined as

below have inverses. Find f −1

, if it exists.

Note : Here we accept that inverse at function is unique.

f = {(1, 1), (2, 2), (3, 3)}

Watch Video Solution

29. Let S = {1, 2, 3}. Determine whether the functions f:S → S defined as

below have inverses. Find f −1

, if it exists.

Note : Here we accept that inverse at function is unique.

f = {(1, 2), (2, 1), (3, 1)}

 Watch Video Solution

30. Let S = {1, 2, 3}. Determine whether the functions f:S → S defined as

below have inverses. Find f −1

, if it exists.

Note : Here we accept that inverse at function is unique.

f = {(1, 3), (3, 2), (2, 1)}

Watch Video Solution

31. Show that addition, subtraction and multiplication are binary

operations on R, but division is not a binary operation on R. Further, show

that division is a binary operation on the set R

∗
of nonzero real

numbers.

Watch Video Solution

32. Show that subtraction and division are not binary operations on N.
 Watch Video Solution

33. Show that ∗ :R × R → R given by (a, b) → a + 4b

2
is a binary

operation.

Watch Video Solution

34. Let P be the set of all subsets of a given set X. Show that

∪ :P × P → P given by (A, B) → A ∪ B and ∩ :P × P → P given

by (A, B) → rA ∩ B are binary operations on the set P.

Watch Video Solution

35. Show that the V V :R × R → R given by (a, b) → max {a, b} and the

∧ :R × R → R given by (a, b) → min {a, b} are binary operations.

Watch Video Solution

36. Show that + : R × R → R and × :R × R → R are commutative

binary operations, but -: R × R → R and ÷ id : R

∗
× R
∗
→ R
∗
are

not commutative.

Watch Video Solution

37. Show that ∗ :R × R → R defined by a

∗
b = a + 2b is not

commutative.

Watch Video Solution

38. Show that addition and multiplication are associative binary

operation on R. But subtraction is not associative on R . Division is not

associative on R ∗

Watch Video Solution

39. Show that ∗
:R × R → R given by a ∗
b → a + 2b is not associative.

Watch Video Solution

40. Show that zero is the identity for addition on R and 1 is the identity

for multiplication on R. But there is no identity element for the

operations − : R × R → R and ÷ idR

⋅
× R
⋅
→ R.

Watch Video Solution

41. Show that −a is the inverse of a for the addition operation '+' on R
1
and is the inverse of a ≠ 0 for the multiplication operation x on R.
a

Watch Video Solution

42. Show that -a is not the inverse of a ∈ N for the addition operation
1
+ on N and not the inverse of a ∈ N for multiplication operation on
a
N , for a ≠ 1 .

Watch Video Solution

43. If R 1
and R 2 are equivalence relations in a set A, show that R1 ∩ R2

is also an equivalence relation.

Watch Video Solution

44. Let R be a relation on the set A of ordered pairs of positive integers

defined by (x, y) R (u, v) if and only if xv= yu. Show that R is an equivalence

relation.

Watch Video Solution

45. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9). Let R1 be a relation in X given by

R 1 = {(x, y) : x − y is divisible by 3} and R, be another relation on X

given by R 2 = {(x, y) : {x, y} ⊂ {1, 4, 7}} or {x, y} ⊂ {2, 5, 8} or

{x, y} ⊂ {3, 6, 9} . Show that R 1

= R2 .

Watch Video Solution

46. Let f:X → Y be a function. Define a relation R in X given by

R = {(a, b) : f(a) = f(b)} . Examine whether R is an equivalence

relation or not.

Watch Video Solution

47. Determine which of the following binary operations on the set R are

associative and which are commutative :

∗
a b = 1, ∀ a, b ∈ R

Watch Video Solution

48. Determine which of the following binary operations on the set R are

associative and which are commutative :

(a + b)
∗
a b = , ∀ a, b ∈ R
2

Watch Video Solution

49. Find the number of all one-one functions from set A = {1, 2, 3} to itself.

Watch Video Solution

50. Let A = {1, 2, 3} Then show that the number of relations containing

(1, 2) and (2, 3) which are reflexive and transitive but not symmetric is

three.

Watch Video Solution

51. Show that the number of equivalence relation in the set {1, 2, 3}

containing (1, 2) and (2, 1) is two.

Watch Video Solution

52. Show that the number of binary operations on {1, 2} having 1 as

identity and having 2 as the inverse. of 2 is exactly one.

Watch Video Solution

53. Consider the identity function IN : N → N defined as

IN (x) = x, ∀ x ∈ N . Show that although IN is onto but

IN + IN : N → N defined as

(IN + IN )(x) = IN (x) + IN (x) = x + x = 2x is not onto.

Watch Video Solution

 π
54. Consider a function f : [0, ] → R given by f(x) = sin x and g :
2

π
[0, ] → R given by g(x) = cos x. Show that f and g are one-one, but f+g
2

is not one-one.

Watch Video Solution

Solutions Of Ncert Exemplar Problems Short Answer Type Questions

1. Let A = {a, b, c} and the relation R be defined on A as follows :

R = {(a, a), (b, c), (a, b)} Then , write minimum number of ordered

pairs to be added in R to make R reflexive and transitive.

Watch Video Solution

2. Let D be the domain of the real valued function f defined by

f(x) = √25 − x
2
. Then , write D .

Watch Video Solution

3. If f, g : R → R be defined by f(x) = 2x + 1 and g(x) = x
2
− 2, ∀ x ∈ R ,

respectively . Find gof .

Watch Video Solution

4. Let f:R → R be the function defined by f(x) = 2x − 2, ∀ x ∈ R .

Write f −1
.

Watch Video Solution

5. If A = {a,b,c,d} and the function f = {(a, b), (b, d), (c, a), (d, c)} . Write

f
−1
.

Watch Video Solution

6. If f : R → R is defined by f(x) = x
2
− 3x + 2 , find f(f(x)).
 Watch Video Solution

7. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function, justify. If this is

described by the relation, g(x) = αx + β , then what values should be

assigned to α and β ?

Watch Video Solution

8. Are the following set of ordered pairs functions ? If so examine whether

the mapping is injective or surjective .

{(x,y) : x is a person, y is the mother of x }

Watch Video Solution

9. Are the following set of ordered pairs functions ? If so examine whether

the mapping is injective or surjective .

{(x,y) : x is a person, y is the mother of x }

Watch Video Solution

10. If the mappings f and g are given be

f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)} , write fog .

Watch Video Solution

11. Let C be the set of complex numbers . Prove that the mapping

f:C → R given by f(z) = |z|, ∀ z ∈ C , is neither one - one nor onto .

Watch Video Solution

12. Let the function f:R → R be defined by f(x) = cos x , AA x in R ` .

Show that f is nether one - one nor onto .

Watch Video Solution

13. Let X = {1,2,3} and Y = {4,5} . Find whether the following subsets of

X × Y are functions form X to Y or not .

f = {(1, 4), (1, 5), (2, 4), (3, 5)}

Watch Video Solution

14. Let X = {1,2,3} and Y = {4,5} . Find whether the following subsets of

X × Y are functions form X to Y or not .

g = {(1, 4), (2, 4), (3, 4)}

Watch Video Solution

15. Let X = {1,2,3} and Y = {4,5} . Find whether the following subsets of

X × Y are functions form X to Y or not .

h = {(1, 4), (2, 5), (3, 5)}

Watch Video Solution

16. Let X = {1,2,3} and Y = {4,5} . Find whether the following subsets of

X × Y are functions form X to Y or not .

k = {(1, 4), (2, 5)}

Watch Video Solution

17. If functions f : A → B and g : B → A satisfy gof = IA , then show

that f is one one and g is onto.

Watch Video Solution

1
18. Let f : R → R be the function defined by f(x) = , ∀x ∈ R
2 − cos x

. Then , find the range of f .

Watch Video Solution

19. Let n be a fixed positive integer. Defiene a relation R in Z as follows :

∀ a, b ∈ Z, aRb if and only if a - b divisible by n. Show that R is

equivalance relation.

Watch Video Solution

Solutions Of Ncert Exemplar Problems Long Answer Type Questions

1. If A = {1, 2, 3, 4} , define relations on A which have properties of being

Reflexive , transitive but not symmetric

Watch Video Solution

2. If A = {1, 2, 3, 4} , define relations on A which have properties of

being :

Symmetric but neither reflexive nor transitive

Watch Video Solution

3. If A = {1, 2, 3, 4} , define relations on A which have properties of

being :

Reflexive , symmetric and transitive .

 Watch Video Solution

4. Let R be relation defined on the set of natural number N as follows :

R = {(x, y) : x ∈ N , y ∈ N , 2x + y = 41} . Find the domian and range

of the relation R . Also verify whether R is reflexive, symmetric and

transitive.

Watch Video Solution

5. Given A = {2,3,4} , B = {2,5,6,7} . Construct an example of each of the

following :

An injective mapping from A to B .

Watch Video Solution

6. Given A = {2,3,4} , B = {2,5,6,7} . Construct an example of each of the

following :

A mapping from A to B which is not injective.

 Watch Video Solution

7. Given A = {2,3,4} , B = {2,5,6,7} . Construct an example of each of the

following :

A mapping from B to A .

Watch Video Solution

8. Give an example of a map

Which is one - one but not onto

Watch Video Solution

9. Give an example of a map

Which is not one - one but onto

Watch Video Solution

10. Give an example of a map

Which is neither one - one nor onto.

Watch Video Solution

11. Let A = R - {3} , B = R - {1} . If f:A → B be defined

x − 2
f(x) = ∀x ∈ A . Then show that f is bijective.
x − 3

Watch Video Solution

12. Let A = [-1,1] . Then , discuss whether the following functions defined on

A are one - one , onto or bijective.

x
f(x) =
2

Watch Video Solution

13. Let A = [-1,1] . Then , discuss whether the following functions defined on

A are one - one , onto or bijective.

g(x) = |x|

Watch Video Solution

14. Check the injectivity and surjectivity of the following function .

f : [ − 1, 1] → [ − 1, 1], f(x) = x|x|

Watch Video Solution

15. Let A = [-1,1] . Then , discuss whether the following functions defined on

A are one - one , onto or bijective.

2
k(x) = x

Watch Video Solution

16. Each of the following defines a relation of N :

x is greater than y,x, y ∈ N .

Determine which of the above relations are reflexive , symmetric and

transitive .

Watch Video Solution

17. Each of the following defines a relation of N :

x + y = 10, x, y ∈ N

Determine which of the above relations are reflexive , symmetric and

transitive .

Watch Video Solution

18. Each of the following defines a relation of N :

x. y is square of an integer x, y ∈ N .

Determine which of the above relations are reflexive , symmetric and

transitive .

Watch Video Solution

19. Each of the following defines a relation of N :

x + 4y = 10, x, y ∈ N

Determine which of the above relations are reflexive , symmetric and

transitive .

Watch Video Solution

20. Let A = {1, 2, 3.......9} and R be the relation in A × A defined by

(a,b) R , (c,d) if a + d = b + c for (a,b) , (c,d) in A × A . Prove that R is an

equivalence relation and also obtain the equivalent class [(2,5)].

Watch Video Solution

21. Using the definition ,prove that the function F : A → B is invertible if

and only if f is both one -one and onto.

Watch Video Solution

22. Functions f, g : R → R are defined ,respectively, by

f(x) = x
2
+ 3x + 1, g(x) = 2x − 3, find

fog

Watch Video Solution

23. Functions f, g : R → R are defined ,respectively, by

f(x) = x
2
+ 3x + 1, g(x) = 2x − 3, find gof .

Watch Video Solution

24. Functions f, g : R → R are defined ,respectively, by

f(x) = x
2
+ 3x + 1, g(x) = 2x − 3, find fof .

Watch Video Solution

25. Functions f, g : R → R are defined ,respectively, by

f(x) = x
2
+ 3x + 1, g(x) = 2x − 3, find gog.

Watch Video Solution

26. Let ∗ be the binary operation defined on Q. Find which of the

following binary operations are commutative.

a ∗ b = a − b, ∀ a, b ∈ Q

Watch Video Solution

27. Let ∗ be the binary operation defined on Q. Find which of the

following binary operations are commutative.

2 2
a ∗ b = a + b , ∀ a, b ∈ Q

Watch Video Solution

28. Let ∗ be the binary operation defined on Q. Find which of the

following binary operations are commutative.

a ∗ b = a + ab, ∀ a, b ∈ Q

Watch Video Solution

29. Let ∗ be the binary operation defined on Q. Find which of the

following binary operations are commutative.

2
a ∗ b = (a − ab) , ∀ a, b ∈ Q

Watch Video Solution

30. If ∗ be binary operation defined on R by a ∗ b = 1 + ab, ∀ a, b ∈ R .

Then the operation ∗ is

(i) Commutative but not associative.

(ii) Associative but not commutative .

(iii) Neither commutative nor associative .

(iv) Both commutative and associative.

 Watch Video Solution

Solutions Of Ncert Exemplar Problems Objective Type Questions

1. Let T be set of all triangle in the Euclidean plane , and let a relation R on

T be defined as aRb if a is congruent tob, ∀ a, b ∈ T . Then, R is ....

A. Reflexive but not transitive

B. Transitive but not symmetric

C. Equivalence

D. None of these

Answer: C

Watch Video Solution

2. Consider the non-empty set consisting of children in a family and a

relation R defined as aRb, if a is brother of b. Then , R is ........

 A. Symmetric but not transitive

B. Transitive but not symmetric

C. Neither symmetric not transitive

D. Both symmetric and transitive

Answer: B

Watch Video Solution

3. The maximum number of equivalence relations on the set A = {1,2,3} are

..........

A. 1

B. 2

C. 3

D. 7

Answer: D
 Watch Video Solution

4. If the relation R on the set {1,2,3} be defined by R = {(1,2)}. Then , R is .........

A. Reflexive

B. Transitive

C. Symmetric

D. None of these

Answer: B

Watch Video Solution

5. Let us define a relation R in R as aRb if a ≥ b . Then, R is .......

A. an equivalence relation

B. reflexive , Transitive but not symmetric

C. symmetric , transitive but not reflexive

 D. neither transitive nor reflexive but symmetric .

Answer: B

Watch Video Solution

6. If A = {1,2,3} and consider the relation R = {(1,1), (2,2) , (3,3) , (1,2) ,(2,3),

(1,3)} . Then R is .......

A. reflexive but not symmetric

B. reflexive but not transitive

C. symmetric and transitive

D. neither symmetric, nor transitive

Answer: A

Watch Video Solution

7. The identity element for the binary operation ∗ defined on Q~{0} as
ab
a ∗ b = ' ∀ a, b ∈ Q − {0} is .........
2

A. 1

B. 0

C. 2

D. None of these

Answer: C

Watch Video Solution

8. If the set A contains 5 elements and the set B contains 6 elements ,

then the number of one -one and onto mapping from A to B is ....

A. 720

B. 120

C. 0
 D. None of these

Answer: C

Watch Video Solution

9. If A = {1, 2, 3..., n} and B = {a, b} Then , the number of subjection

from A into B is .........

A. n
P2

B. 2 n
− 2

C. 2 n
− 1

D. None of these

Answer: D

Watch Video Solution

 1
10. If f : R → R be defined by f(x) = , ∀x ∈ R . Then , f is .........
x

A. one-one

B. onto

C. bijective

D. f is not defined

Answer: D

Watch Video Solution

11. Let f:R → R be defined by

x
f(x) = 3x
2
− 5 and g : R → R, g(x) =
2
Then gof is ...........
x + 1

2
3x − 5
A. 4 2
9x − 30x + 26

2
3x − 5
B. 4 2
9x − 6x + 26

2
3x
C.
4 2
x + 2x − 4

2
3x
D. 4 2
9x + 30x − 2
Answer: A

Watch Video Solution

12. Which of the following functions from Z into Z are bijections ?

A. f(x) = x
3

B. f(x) = x + 2

C. f(x) = 2x + 1

D. f(x) = x
2
+ 1

Answer: B

Watch Video Solution

13. If f : R → R be the functions defined by f(x) = x

3
+ 5 , then f −1
(x)

is ........
 A. (x
1

+ 5) 3

B. (x
1

− 5) 3

C. (5 − x)
1

D. 5 − x

Answer: B

Watch Video Solution

14. If f : A → B and g : B → C be the bijective functions , then (gof)

−1

is ..........

A. f −1
og
−1

B. fog

C. g −1 −1
of

D. gof

Answer: A
 Watch Video Solution

3 3x + 2
15. If f : R − { } → R be defined by f(x) = , then .........
5 5x − 3

A. f −1
(x) = f(x)

B. f −1
(x) = − f(x)

C. fof(x) = − x

1
D. f −1
(x) = f(x)
19

Answer: A

Watch Video Solution

x if x is rational
16. If f : [0, 1] → [0, 1] be difined by f(x) = {
1 − x if x is irrational

then fof(x) is ............

A. constant

B. 1 + x
 C. x

D. None of these

Answer: C

Watch Video Solution

17. If f : [(2, ∞) → R be the function defined by f(x) = x

2
,
− 4x + 5

then the range of f is ...........

A. R

B. [1, ∞)

C. [4, ∞)

D. [5, ∞)

Answer: B

Watch Video Solution

18. If f:N → R be the function defined by
2x − 1
f(x) = and g : Q → R be another function defined by
2

3
g(x) = x + 2 . Then , gof ( ) is .......
2

A. 1

B. − 1

C. 3

D. None of these

Answer: D

Watch Video Solution

⎧ 2x x > 3
⎪

19. f : R → R, f(x) = ⎨ x
2
1 < x ≤ 3 then find f (-1) + f(2) + f(4) .
⎩
⎪
3x x ≤ 1

A. 9

B. 14
 C. 5

D. None of these

Answer: A

Watch Video Solution

20. If f : R → R be given by f(x) = tan x , then f −1

(1) is .......

π
A.
4

π
B. {nπ + : n ∈ Z}
4

C. Does not exist

D. None of these

Answer: A

Watch Video Solution

Solutions Of Ncert Exemplar Problems Fillers

1. Let the relation R be defined in N by aRb , if 2a + 3b = 30 . Then , R =

.............

Watch Video Solution

2. If the relation R be defined on the set A = {1, 2, 3, 4, 5} by

R = {(a, b) : ∣
2 2
∣
∣ a − b ∣ < 8} . Then , R is given by .............

Watch Video Solution

3. The functions f and g are defined as follow : f = {(1, 2), (3, 5), (4, 1)}

and g = {(2, 3), (5, 1), (1, 3)} . Find the range of f and g . Also find the

composition function fog and gof .

Watch Video Solution

 x
4. f : R → R, f(x) = , ∀x ∈ R . Then find (fofof) (x).
√1 + x 2

Watch Video Solution

5. If f(x) , then f
3 −1
= [4 − (x − 7) ] (x) = .........

Watch Video Solution

Solutions Of Ncert Exemplar Problems True False

1. Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set

A = {1, 2, 3} . Then , R is symmetric , transitive but not reflexive.

Watch Video Solution

2. If f:R → R be the function defined by f(x) = sin(3x + 2) ∀ x ∈ R .

Then , f is invertible .
 Watch Video Solution

3. Every relation which is symmetric and transitive is also reflexive.

Watch Video Solution

4. An integer m is said to be related to another integer n if m is a integral

multiple of n. This relation in Z is reflexive, symmetric and transitive.

Watch Video Solution

5. If A = {0, 1} and N be the set of natural numbers. Then , the mapping

f:N → A defined by f(2n − 1) = 0, f(2n) = 1, ∀ n ∈ N , is onto.

Watch Video Solution

6. The relation R on the set A = {1, 2, 3} defined as R =

{(1, 1), (1, 2), (2, 1), (3, 3)} is reflexive , symmetric and transitive.

Watch Video Solution

7. The composition of function is commutative .

Watch Video Solution

8. The composition of function is associative.

Watch Video Solution

9. Every function is invertible.

Watch Video Solution

 10. A binary operation on a set has always the identity element.

Watch Video Solution

Practice Paper 1 Section A

1. Which of the following defined on Z is not an equivalence relation ?

A. (x, y) ∈ S ⇔ x ≥ y

B. (x, y) ∈ S ⇔ x = y

C. (x, y) ∈ S ↔ x − y is a multiple of 3

D. If |x − y| is even ⇔ (x, y) ∈ S

Answer:

Watch Video Solution

2. The number of binary operation on {1, 2, 3, ......, n} is ..........

A. 2n

B. n
2
n

C. n 3

D. n 2n

Answer:

Watch Video Solution

3. If a ∗ b = a
2 2
+ b is on Z then , (2 ∗ 3) ∗ 4 = ......

A. 13

B. 16

C. 185

D. 31
Answer:

Watch Video Solution

4. A = {1,2} , the number of one - one functions on A → A is ........

A. 1

B. 2

C. 3

D. 4

Answer:

Watch Video Solution

5. ∗ is defined by a ∗ b = a + b − 1 on Z , then identity element for ∗ is

.........
 A. 1

B. 0

C. − 1

D. 2

Answer:

Watch Video Solution

Practice Paper 1 Section B

1. If f(x) = 8x then find gof and fog .

1
3
and g(x) = x 3

Watch Video Solution

2. Let ∗ be the binary operation on Q define . Is

a ∗ b = a + ab ∗

commutative ? Is ∗ associative ?
 Watch Video Solution

3. Let f:R → R be defined as f(x) = 10x + 7 . Find the function

g: R → R such that gof = fog = I

g

Watch Video Solution

4. Let A = {1,2,3}. Then number of relations containing (1,2) and (1,3) which

are reflexive and symmetric but not transitive is

Watch Video Solution

Practice Paper 1 Section C

1. Let T be the set of all triangles in a plane with R a relation in T given by

R = {(T1 , T2 ) : T1 is congruent to T2 } Show that R is an equivalence

relation.

Watch Video Solution

2. Prove that binary operation on set R defined as a ∗ b = a + 2b does

not obey associative rule.

Watch Video Solution

3. Let f' : N → R be a function defined as f' (x) = 4x

2
+ 12x + 15 .

Show that f:N → S , where, S is the range of f, is invertible. Find the

inverse of f.

Watch Video Solution

4. If f : A → B and g : B → C be the bijective functions , then (gof)

−1

is ..........

Watch Video Solution

 1
5. Show that f : R +
→ R + , f(x) = is one to one and onto function.
x

Watch Video Solution

Practice Paper 1 Section D

n+1
if n is odd
1. Let : f:N → N be defined by f(n) = {
n
2
for all
if n is even
2

n ∈ N .

State whether the function f is bijective . Justify your answer.

Watch Video Solution

2. f : Z → Z and g : Z → Z are defined as follow :

n + 2 n even 2n n even
f(n) = { , g(n) = { n−1
Find fog and gof.
2n − 1 n odd n odd
2

Watch Video Solution