Relations and Functions Assignment - One Shot

Q1. Let R be a relation in N defined by R = {(1 + x, 1 + x2) : x ≤ 5, x ∈ N}.Which of the following is false?

R = {(2, 2), (3, 5), (4, 10), (5, 17), (6, 25)}

Range of R = {2, 5, 10, 17, 26}

Domain of R = {2, 3, 4, 5, 6}

A

B

C

D None of the above

Q2. The relation R defined on set A = {x : | x | < 3, x ∈ I} by R = {(x, y) : y = | x |} is

{(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}

{(0, 0), (1, 1), (2, 2)}

{(-2, -2), (-2, 2), (-1, 1), (0, 0), (1, -2), (1, 2), (2, -1), (2, -2)}

None of the above

A

B

C

D

Q3. Let a relation R be defined by R = {(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)}. The relation R-1 oR is given by

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

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

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

None of the above

A

B

C

D

Q4. Let a function f : R ➝ A is defined as If f is onto function, then find the set of values of A.

Q5. Let a function f : (2, ∞) ➝ [0, ∞) defined as then f is

Injective and surjective

Injective but not surjective

Not injective but surjective

Neither injective nor surjective

A

B

C

D

Q6. Let a function f : (0, ∞) ➝ [0, ∞) be defined by Then f is

Injective only

Not injective but it is surjective

Both injective as well as surjective

Neither injective nor surjective

JEE Main - 2019A

B

C

D

Q7. The function f : [0, 7] ➝ [0, 70] where f(x) = x3 - 12x2 + 45x, is

one-one & onto

one-onto & into

many-one & onto

many-one & into

A

B

C

D

Q8. A function f : R - {-1} ➝ R - {1} is defined as Prove that f is onto function.

Q9. Find fog and gof for the functions f(x) = sin x and

Q10. Let g(x) = 1 + x - [x] and then for all x,find f(g(x))

Q11. A function f(x) is defined as x > 0, n ∈ I+ . Then find

Q12. If f : R ➝ R, g : R ➝ R and h : R ➝ R are such that f(x) = x2, g(x) = tan x and

h(x) = log x, then the value of (ho(gof)(x), if will be

0

-1

1

π

A

B

C

D

Q13. If f(x) = sin x + cos x and g(x) = x2 - 1, then g(f(x)) is invertible in the domain

[0, π]

A

B

C

D

Q14. Let f : R ➝ R be a function given by f(x) = x2 + 1. Find: (i) f-1 {-5} (ii) f-1{26} (iii) f-1{10, 37}

Q15. If f : [1, ∞) ➝ [2, ∞) is given by then f-1(x) is :

A

B

C

D

Q16. If g is the inverse of a function f and then g’(x) is equal to

1 + x5

1 + {g(x)}5

5x4

JEE Main - 2014A

B

C

D

Q17. A function f : [1, ∞) ➝ [1, ∞) is defined as f(x) = 2x(x - 1). Find f -1 (x).

Q18. If a function f is bijective such that Find f -1(x)

Relations and Functions Assignment - One Shot Solutions

Q1. Let R be a relation in N defined by R = {(1 + x, 1 + x2) : x ≤ 5, x ∈ N}.Which of the following is false?

R = {(2, 2), (3, 5), (4, 10), (5, 17), (6, 25)}

Range of R = {2, 5, 10, 17, 26}

Domain of R = {2, 3, 4, 5, 6}

None of the above

A

B

C

D

Solution:

Q2. The relation R defined on set A = {x : | x | < 3, x ∈ I} by R = {(x, y) : y = | x |} is

{(-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2)}

{(0, 0), (1, 1), (2, 2)}

{(-2, -2), (-2, 2), (-1, 1), (0, 0), (1, -2), (1, 2), (2, -1), (2, -2)}

None of the above

A

B

C

D

Solution:

Q3. Let a relation R be defined by R = {(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)}. The relation R-1 oR is given by

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

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

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

None of the above

A

B

C

D

Solution:

Q4. Let a function f : R ➝ A is defined as If f is onto function, then find the set of values of A.

Solution:

Q5. Let a function f : (2, ∞) ➝ [0, ∞) defined as then f is

Injective and surjective

Injective but not surjective

Not injective but surjective

Neither injective nor surjective

A

B

C

D

Solution:

Solution:

Q6. Let a function f : (0, ∞) ➝ [0, ∞) be defined by Then f is

Injective only

Not injective but it is surjective

Both injective as well as surjective

Neither injective nor surjective

JEE Main - 2019A

B

C

D

Solution:

Graphically f(x) is not injective but surjective.

Q7. The function f : [0, 7] ➝ [0, 70] where f(x) = x3 - 12x2 + 45x, is

one-one & onto

one-onto & into

many-one & onto

many-one & into

A

B

C

D

Solution:

Q8. A function f : R - {-1} ➝ R - {1} is defined as Prove that f is onto function.

Solution:

Q9. Find fog and gof for the functions f(x) = sin x and

Solution:

Q10. Let g(x) = 1 + x - [x] and then for all x,find f(g(x))

Solution:

Q11. A function f(x) is defined as x > 0, n ∈ I+ . Then find

Solution:

Q12. If f : R ➝ R, g : R ➝ R and h : R ➝ R are such that f(x) = x2, g(x) = tan x and

h(x) = log x, then the value of (ho(gof)(x), if will be

0

-1

1

π

A

B

C

D

Solution:

Q13. If f(x) = sin x + cos x and g(x) = x2 - 1, then g(f(x)) is invertible in the domain

[0, π]

A

B

C

D

Solution:

Q14. Let f : R ➝ R be a function given by f(x) = x2 + 1. Find: (i) f-1 {-5} (ii) f-1{26} (iii) f-1{10, 37}

Recall that if f : A ➝ B is a function and y ∈ B.

Then, f-1 {y} = {x ∈ A : f(x) = y}.

In other words, f-1 {y} is the set of pre-images of y.

(i) Let f-1(-5) = x. Then,

f(x) = -5 ⇒ x2 + 1 = -5 ⇒ x2 = -6

Clearly, this equation is not solvable in R.

Therefore, there is no pre-image of -5. So, f-1 {-5} = 𝜙.

(ii) Let f-1(26) = x. The,

f(x) = 26 ⇒ x2 + 1 = 26 ⇒ x = ±5

So, pre-images of 26 are -5 and 5.

∴ f-1{26} = {-5, 5}.

Solution:

(iii) Let f-1{10} = x. Then,

f(x) = 10 ⇒ x2 + 1 = 10 ⇒ x2 = 9 ⇒ x = ±3

So, pre-images of 10 are -3 and 3

Let f-1(37) = x. Then,

f(x) = 37 ⇒ x2 + 1 = 37 ⇒ x2 = 36 ⇒ x = ±6

So, pre-images of 37 are -6 and 6.

Hence, f-1{10, 37} = {3, -3, 6, -6}.

Solution:

Q15. If f : [1, ∞) ➝ [2, ∞) is given by then f-1(x) is :

A

B

C

D

Solution:

Q16. If g is the inverse of a function f and then g’(x) is equal to

1 + x5

1 + {g(x)}5

5x4

JEE Main - 2014A

B

C

D

Solution:

Q17. A function f : [1, ∞) ➝ [1, ∞) is defined as f(x) = 2x(x - 1). Find f -1 (x).

Solution:

Solution:

Q18. If a function f is bijective such that Find f -1(x)

Solution:

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