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Relations and Functions Assignment - One Shot
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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
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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
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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
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Q4. Let a function f : R ➝ A is defined as If f is onto function, then find the set of values of A.
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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
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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
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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
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Q8. A function f : R - {-1} ➝ R - {1} is defined as Prove that f is onto function.
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Q9. Find fog and gof for the functions f(x) = sin x and
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Q10. Let g(x) = 1 + x - [x] and then for all x,find f(g(x))
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Q11. A function f(x) is defined as x > 0, n ∈ I+ . Then find
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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
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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
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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}
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Q15. If f : [1, ∞) ➝ [2, ∞) is given by then f-1(x) is :
A
B
C
D
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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
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Q17. A function f : [1, ∞) ➝ [1, ∞) is defined as f(x) = 2x(x - 1). Find f -1 (x).
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Q18. If a function f is bijective such that Find f -1(x)
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Relations and Functions Assignment - One Shot Solutions
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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
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Solution:
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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
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Solution:
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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
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Solution:
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Q4. Let a function f : R ➝ A is defined as If f is onto function, then find the set of values of A.
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Solution:
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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
![Page 30: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/30.jpg)
Solution:
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Solution:
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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
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Solution:
Graphically f(x) is not injective but surjective.
![Page 34: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/34.jpg)
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
![Page 35: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/35.jpg)
Solution:
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Q8. A function f : R - {-1} ➝ R - {1} is defined as Prove that f is onto function.
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Solution:
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Q9. Find fog and gof for the functions f(x) = sin x and
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Solution:
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Q10. Let g(x) = 1 + x - [x] and then for all x,find f(g(x))
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Solution:
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Q11. A function f(x) is defined as x > 0, n ∈ I+ . Then find
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Solution:
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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
![Page 45: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/45.jpg)
Solution:
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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
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Solution:
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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}
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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:
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(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:
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Q15. If f : [1, ∞) ➝ [2, ∞) is given by then f-1(x) is :
A
B
C
D
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Solution:
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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
![Page 54: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/54.jpg)
Solution:
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Q17. A function f : [1, ∞) ➝ [1, ∞) is defined as f(x) = 2x(x - 1). Find f -1 (x).
![Page 56: Relations and Functions Assignment - One Shot](https://reader034.vdocuments.mx/reader034/viewer/2022051813/62830b0089c9ca54d3450b16/html5/thumbnails/56.jpg)
Solution:
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Solution:
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Q18. If a function f is bijective such that Find f -1(x)
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Solution:
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