Functions

log (x−2)
1. The domain of f (x) = e2 log(x+1)
e x −(2x+3)
, x ∈ R is (a) R − {1, 3} (b) (2, ∞) − {3} (c)
(−1, ∞) − {3} (d) R − {3} Jee Main 2023
√ √ √ √   √ √ 
2. The range of the function f (x) = 3 − x + 2 + x is (a) 5, 10 (b) 2 2, 11
√ √  √ √ 
(c) 5, 13 (d) 2, 7 Jee Main 2023

3. Let the sets A and B denote the domain and range respectively of the function
f (x) = √ 1 ,where [x] denotes the smallest integer greater than or equal to x.Then
[x]−x
among the statements (S1 ) : A ∩ B = (1, ∞) − N and (S2 ) : A ∪ B = (1, ∞).
(a) Only (S1 ) is true. (b) both (S1 ) and (S2 ) are true. (c) neither (S1 ) nor (S2 ) is
true. (d) Only (S2 ) is true. Jee Main 2023
  
4. Let D be the domain of the function f (x) = sin−1 log3x 6+2−5x log3 x
.If the range
of the function g : D → R defined by g(x) = x − [x] is (α, β),then α2 + β5 is equal
to (a) 46 (b) 135 (c) 136 (d) 45 Jee Main 2023

5. The domain of the function f (x) = √ 1

is (a) (−∞, −2) ∪ (5, ∞) (b)
[x]2 −3[x]−10
(−∞, −3] ∪ [6, ∞) (c) (−∞, −2) ∪ [6, ∞) (d) (−∞, −3] ∪ (5, ∞) Jee Main 2023
2
6. f : R − {2, 6} → R be real valued function defined as f (x) = xx2 −8x+12
+2x+1
.Then range
21 21 21

  21

of f is (a)  −∞, − 4 ∪ [0, ∞) (b) −∞, − 4 ∪ (0, ∞) (c) −∞, − 4 ∪ 4 , ∞ (d)
−∞, − 214
∪ [1, ∞) Jee Main 2023
√
7. Let f : R → R be a function defined by f (x) = log√m 2(sin x − cos x) + m − 2 ,for
some m,such that the range of f is [0, 2].Then the value of m is (a) 5 (b) 3 (c) 2 (d)
4 Jee Main 2023
 2 
8. The range of f (x) = 4 sin−1 x2x+1 is (a) [0, π] (b) [1, 2π) (c) [0, π) (d) [0, 2π).
. Jee Main 2023
[x]
9. If the domain of definition of the function f (x) = 1+x 2 is [2, 6),then its range is (a)
5 2 9 27 18 9 5 2 5 2 9 27 18 9 5 2
     
,
26 5
− , , ,
29 109 89 53
(b) ,
26 5
(c) ,
37 5
− , ,
29 109 89 53
, (d) ,
37 5
. Jee Main 2023

10. The absolute minimum value,of the function f (x) = |x2 − x + 1| + [x2 − x + 1] in
the interval [−1, 2] is (a) 34 (b) 32 (c) 14 (d) 45 Jee Main 2023

11. If the domain of function f (x) = loge (4x2 + 11x + 6) + sin−1 (4x + 3) + cos−1 10x+6


3
is (α, β],then 36|α + β| is equal to (a) 63 (b) 45 (c) 72 (d) 54 Jee Main 2023
 2   2 
−3x+4
12. If the domain of the function loge 6x 2x−1 +5x+1
+cos−1 2x 3x−5 is (α, β)∪(γ, δ],then
18(α2 + β 2 + γ 2 + δ 2 ) is equal to ———— Jee Main 2023
√ √ √ √
13. [ 1] + [ 2] + [ 3] + · · · + [ 20] is equal to ——– Jee Main 2023

14. Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}.Then the number of functions f :

A → B satisfying f (1) + f (2) = f (4) − 1 is equal to Jee Main 2023

15. If the domain of the function f (x) = sec−1 5x+3

2x


is [α, β) ∪ (γ, δ],then |3α + 10(β +
γ) + 21δ| is equal to ——— Jee Main 2023
16. The total number of functions f : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that f (1) +
f (2) = f (3) is equal to (a) 60 (b) 90 (c) 108 (d) 126 Jee Main 2022
 
17. The domain of the function f (x) = sin−1 [2x2 − 3] + log2 log 1 (x2 − 5x + 5) , is (a)
2
 q √   √ √   √  h √ 
5 5− 5 5− 5 5+ 5 5− 5 5+ 5
− 2, 2 (b) 2
, 2 (c) 1, 2 (d) 1, 2 Jee Main 2022

18. Let f (x) = ax2 + bx + c be such that f (1) = 3, f (−2) = λ and f (3) = 4.If f (0) +
f (1)+f (−2)+f (3) = 14,then λ is equal to (a) −4 (b) 132
(c) 23
2
(d) 4Jee Main 2022
 −1  1  
2 sin
−1 4x2 −1
is (a) R− − 21 , 12 (b) (−∞, −1]∪

19. The domain of the function cos π
 i 
1 1
[1, ∞) ∪ {0} (c) −∞, − 2 ∪ 2 , ∞ ∪ {0} (d) −∞, − 2 ∪ 21 , ∞ ∪ {0}
1




√

. Jee Main 2022

x2 −5x+6
 
cos−1
x2 −9
20. The domain of the function f (x) = log is (a) (−∞, −1)∪(2, ∞) (b) (2, ∞)
(x2 −3x+2)
e n √ √ o
(c) − 2 , 1 ∪ (2, ∞) (d) − 2 , 1 ∪ (2, ∞) − 3+2 5 , 3−2 5
 1
 1

Jee Main 2022

21. Let f (x) = 2 cos−1 x + 4 cot−1 x − 3x2 − 2x + 10, x ∈ [−1, 1].If [a, b] is the range of the
function f ,then 4a−b is equal to (a) 11 (b) 11−π (c) 11+π (d) 15−πJee Main 2022
 2 
22. The domain of the function f (x) = sin−1 xx2 +2x+7−3x+2
is : (a) [1, ∞) (b) [−1, 2] (c)
[−1, ∞) (d) (−∞, 2] Jee Main 2022

23. The number of functions f ,from the set A = {x ∈ N : x2 − 10x + 9 ≤ 0} to the set
B = {n2 : nN} such that f (x) ≤ (x − 3)2 + 1,for every x ∈ A, is —–
. Jee Main 2022
 2 
24. The domain of the function f (x) = sin−1 3x(x−1)+x−1
+cos−1 x+1
x−1
is : (a) 41 , 12 ∪{0}

 
2

(b) 0, 12 (c) 0, 14 (d) [−2, 0] ∪ 41 , 12

     
Jee Main 2021

25. The range
of2 ax
∈ R for which the function f (x) = (4a − 3)(x + loge 5) + 2(a −
7) cot x2 sin
 4 2 , x ̸= 2nπ, nN has critical points is : (a) (−∞, −1] (b) (−3, 1) (c)
[1, ∞) (d) − 3 , 2 Jee Main 2021

26. Let M and m respectively be the maximum and minimum values of the function
−1 π

f (x) = tan (sin x + cos x) in 0, 2 .Then the value of tan(M − m) is equal to (a)
√ √ √ √
3 − 2 2 (b) 2 + 3 (c) 3 + 2 2 (d) 2 − 3 Jee Main 2021

27. The range of the function

f (x) = log√5 3h + cos i3π + x + cos π4 + x + cos π 3π





4 4
− x − cos 4
−x is
√ √
(a) [−2, 2] (b) √15 , 5 (c) (0, 5) (d) [0, 2] Jee Main 2021

−1 1+x 1

  1

28. Thedomain of the function cosec x
is : (a) −1, − 2
∪(0, ∞) (b) − 2
, ∞ −{0}
1 1



(c) − 2 , 0 ∪ [1, ∞) (d) − 2 , ∞ − {0} Jee Main 2021
√ √
29. If the functions are defined as f (x) = x and g(x) = 1 − x,then what is the
common domain of the following functions: f + g, f − g, f /g, g/f, g − f .
(a) 0 ≤ x ≤ 1 (b) 0 ≤ x < 1 (c) 0 < x < 1 (d) 0 < x ≤ 1 Jee Main 2021
−1
√
cos x2 −x+1
30. If the domain of the function f (x) = q is the interval (α, β],then α + β
sin ( 2x−1
−1
2 )
1 3
is equal to : (a) 1 (b) 2
(c) 2 (d) 2
Jee Main 2021
q
|[x]|−2
31. If the domain of the real valued function f (x) = |[x]|−3
is (−∞, a) ∪ [b, c) ∪
[4, ∞), a < b < c,then the value of a + b + c is : (a) −3 (b) 8 (c) −2 (d)
1 Jee Main 2021

32. Let S = {1, 2, 3, 4, 5, 6, 7}.The number of possible function f : S → S such that

f (m.n) = f (m).f (n) for every m, n ∈ S and m.n ∈ S is equal to ———–
. Jee Main 2021
x[x]
33. Let f : (1, 3) → R be a function defined by f (x) = 1+x 2 .Then the range of f is : (a)
3 4 2 1 2 3 3 4 2 4




,
4 5
(b) 5 , 2 (c) 5 , 5 ∪ 4 , 5 (d) 5 , 5 Jee Main 2020
 
−1 |x|+5
34. The domain of the function f (x) = sin x2 +1
is (−∞, −a] ∪ [a, ∞).Then a is
√ √ √ √
17−1 17 17 1+ 17
equal to : (a) 2
(b) 2
(c) 2
+ 1 (d) 2
Jee Main 2020

35. The equation [x]2 − 2[x + 2] − 7 = 0 has : (a) infinitely many solutions (b) Exactly
two solutions (c) No integral solution (d) Exactly four integral solution.
. Jee Main 2020

36. Let A = {a, b, c} and B = {1, 2, 3, 4}.Then the number of elements in the set
C = {f : A → B|2 ∈ f (A) and f is not one-one} is ——— Jee Main 2020

37. If the function f : R → R is defined by f (x) = |x|(x − sin x),then which of the
following statements is true ?
(a) f is one-one, but not onto. (b) f is onto but not one-one. (c) f is both one-one
and onto. (d) f is neither one-one nor onto. Jee Main 2020

38. Let f : [0, 2]
→ R be the function defined by f (x) = (3 − sin(2πx)) sin πx − π4 −

sin 3πx + π4 .If α, β ∈ [0, 2] are such that {x ∈ [0, 2] : f (x) ≥ 0} = [α, β].Then the
value of β − α is ——— Jee Advanced 2020
x
 1 1
39. Let f : R → R be defined by f (x) = 1+x 2 , x ∈ R.Then the range of f is : (a) − 2 , 2

(b) R − [−1, 1] (c) R − − 12 , 12 (d) (−1, 1) − {0}

 
Jee Main 2019

40. Let A function f : (0, ∞) → [0, ∞) be defined by f (x) = 1 − x1 .Then f is (a) not
injective but it is surjective. (b) bijective only. (c) neither injective nor surjective.
(d) both injective as well as surjective. Jee Main 2019
1 3
41. The domain of the definition of the function f (x) = 4−x 2 + log10 (x − x) is (a)

(−1, 0) ∪ (1, 2) ∪ (3, ∞) (b) (−2, −1) ∪ (−1, 0) ∪ (2, ∞) (c) (−1, 0) ∪ (1, 2) ∪ (2, ∞)
(d) (1, 2) ∪ (2, ∞) Jee Main 2019

42. Let f (x) = ax (a > 0) be written as f (x) = f1 (x) + f2 (x),where f1 (x) is an even
function and f2 (x) is an odd function.Then f1 (x+y)+f1 (x−y) equals (a) 2f1 (x)f2 (y)
(b) 2f1 (x)f1 (y) (c) 2f1 (x + y)f2 (x − y) (d) 2f1 (x + y)f1 (x − y) Jee Main 2019

43. Let S = {1, 2, 3, 4, 5, 6}.Then the number of one-one functions f : S → P (s) such
that f (n) ⊂ f (m) where n < m is Jee Main 2023
2
44. Let f : R → R be a function such that f (x) = x x+2x+1
2 +1 .Then (a) f (x) is many-one
in (−∞, −1) (b) f (x) is many-one in (1, ∞) (c) f (x) is one-one in [1, ∞) but not in
(−∞, ∞) (d) f (x) is one-one in (−∞, ∞) Jee Main 2023
45. Let f (x) = 3x+2 , x ∈ R − − 32 .For n ≥ 2,define f n (x) = f ◦ f n−1 (x).If f 5 (x) =

2x+3
ax+b
bx+a
, gcd(a, b) = 1,then a + b is equal to ——- Jee Main 2023
46. Let A = {1, 2, 3, 5, 8, 9}.Then the number of possible functions f : A → A such that
f (m.n) = f (m).f (n) for every m, n ∈ A with m.n ∈ A is equal to ——
. Jee Main 2023
47. Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}.Total number of possible functions f :
R → S such that f (a) ̸= 1, is equal to —— Jee Main 2023
48. Let A = {x ∈ R : |x + 1| < 2} and B = {x ∈ R : |x − 1| ≥ 2}.Then which one of
the following statements is not true ? (a) A − B = (−1, 1) (b) B − A = R − (−3, 1)
(c) A ∩ B = (−3, −1] (d) A ∪ B = R − [1, 3) Jee Main 2022

2n, n = 2, 4, 6, 8, . . .
49. Let a function f : N → N be defined by f (n) = n − 1, n = 3, 7, 11, 15, . . . Then

n+1
2
, n = 1, 5, 9, 13, . . .
f is (a) one-one but not onto. (b) onto but not one-one. (c) neither non-one nor
onto (d) one-one and onto. Jee Main 2022
50. Let f, g : N − {1} → N be functions defined by f (a) = α,where α is the maximum
of the powers of those primes p such that pα divides a, and g(a) = a + 1,for all
a ∈ N − {1}.Then,the function f + g is (a) one-one but not onto (b) onto but not
one-one. (c) both one-one and onto (d) neither one-one nor onto. Jee Main 2022
51. The number of bijective functions f : {1, 3, 5, 7, . . . , 99} → {2, 4, 6, 8, . . . , 100},such
that f (3) ≥ f (9) ≥ f (15) ≥ f (21) ≥ · · · ≥ f (99) is (a) 50 p17 (b) 50 p33 (c) 33! × 17!
(d) 50!
2
Jee Main 2022
52. The number of distinct real roots of x4 − 4x + 1 = 0 is : (a) 4 (b) 2 (c) 1 (d)
0 Jee Main 2022
53. The number of one-one functions f : {a, b, c, d} → {0, 1, 2, . . . , 10} such that 2f (a)−
f (b) + 3f (c) + f (d) = 0 is ——- Jee Main 2022

2n, if n = 1, 2, 3, 4, 5
54. Let S = {1, 2, 3, . . . , 10}.Define f : S → S as f (n) =
2n
 − 11, if n = 6, 7, 8, 9, 10
n + 1, if n is odd
Let g : S → S be a function such that f ◦ g(n) = ,then
n − 1, if n is even
g(10)(g(1) + g(2) + g(3) + g(4) + g(5)) is equal to ——— Jee Main 2022
55. Let s = {1, 2, 3, 4}.Then the number of elements in the set {f : S × S → S :
f is onto and f (a, b) = f (b, a) ≥ a for all (a, b) ∈ S × S} is — Jee Main 2022
100
P h (−1)n n i
56. 2
is equal to : (a) 4 (b) −2 (c) 2 (d) 0 Jee Main 2021
n=8

57. Let f, g : N → N such that f (n + 1) = f (n) + f (1) for all n ∈ Nand g be any
arbitrary function.Which of the following statements is not true ?
(a) If f ◦ g is one-one,the g is one-one. (b) f is one-one (c) If g is onto,then f ◦ g is
one-one. (d) If f is onto,then f (n) = n for all n ∈ N Jee Main 2021
58. Let f : N → N be a function such that f (m+n) = f (m)+f (n) for every m, n ∈ N.If
f (6) = 18,then f (2).f (3) is equal to : (a) 18 (b) 36 (c) 54 (d) 6 Jee Main 2021

59. The values of x ∈ R satisfying the equation [ex]2 +
[ex + 1] − 3 = 0 lie in the interval
: (a) [1, e) (b) [loge 2, loge 3] (c) [0, loge 2) (d) 0, 1e Jee Main 2021

60. Let A = {0, 1, 2, 3, 4, 5, 6, 7}.Then the number of bijective functions f : A → A such

that f (1) + f (2) = 3 − f (3) is equal to —— Jee Main 2021

61. The number of real roots of the equation e4x − e3x − 4e2x − ex + 1 = 0 equal to
————– Jee Main 2021

62. If a + α = 1, β + b = 2 and af (x) + αf x1 = bx + βx , x ̸= 0,then the value of the

f (x)+f ( 1 )
function x+ 1 x is ———- Jee Main 2021
x

2x −2x
63. The inverse function of f (x) = 882x −8 1
log8 e loge 1−x


+8−2x
, x ∈ (−1, 1), is (a) 4 1+x
(b)
1 1−x 1 1+x 1 1+x




4
loge 1+x (c) 4 loge 1−x (d) 4 log8 e loge 1−x Jee Main 2020

64. Let f : R → R be a function which satisfies f (x + y) = f (x) + f (y)∀x, y ∈ R.If

n−1
P
f (1) = 2 and g(n) = f (k), n ∈ N,then the value of n for which g(n) = 20 is (a)
k=1
9 (b) 20 (c) 5 (d) 4 Jee Main 2020
4x
65. Let the function f : [0, 1] → R be defined by f (x) = 4x +2
.Then
the value of
1 2 3
+ · · · + f 39 − f 12 is ——






f 40 + f 40 + f 40 40
Jee Advanced 2019
x 2
66. If the function f : R − {−1, 1} → A defined by f (x) = 1−x 2 is surjective,then A is

equal to : (a) R − [−1, 0) (b) R − (−1, 0) (c) R − {−1} (d) [0, ∞)Jee Main 2019

 n+1 f and g be defined as f, g : N → N such that

67. Two functions
if n is odd
f (n) = 2
n and g(n) = n − (−1)n .Then f ◦ g is : (a) onto but not
2
if n is even
one-one. (b) one-one but not onto. (c) both one-one and onto. (d) neither one-one
nor onto. Jee Main 2019
√ √ √
68. The sum of the solutions of the equation | x − 2| + x ( x − 4) + 2 = 0, (x > 0)
is equal to : (a) 4 (b) 9 (c) 10 (d) 12 Jee Main 2019
2403 k
69. If the fractional part of the number 15
is 15
,then k is equal to : (a) 6 (b) 8 (c) 4
(d) 14 Jee Main 2019
10
f (a + k) = 16 (210 − 1),where the function f satisfies f (x + y) = f (x)f (y)
P
70. Let
k=1
for all natural numbers x, y and f (1) = 2.Then the natural number a is (a) 4 (b) 3
(c) 16 (d) 2 Jee Main 2019

71. Let A = {x ∈ R : x is not a positive integer}.Define a function f : A → R as

2x
f (x) = x−1 .Then f is (a) injective but not surjective. (b) not surjective. (c)
surjective but not injective. (d) neither injective nor surjective. Jee Main 2019

72. The sum of the series − 31 + − 13 − 1001

  1 2
+ · · · + − 13 − 100
99
     
+ − 3 − 100 is (a) −153
(b) −133 (c) −131 (d) −135 Jee Main 2019
1◦ )x+loge (123)
73. If f (x) = x(tan 4



◦
loge (1234)−(tan 1 )
, x > 0,then the least value of f (f (x)) + f f x
is (a) 8
(b) 4 (c) 2 (d) 0 Jee Main 2023

74. The set of all a ∈ R for which the equation x|x − 1| + |x − 2| + a = 0 has exactly one
real root is : (a) (−6, −3) (b) (−∞, ∞) (c) (−6, ∞) (d) (−∞, −3)Jee Main 2023
√
75. For x ∈ R,two real√valued functions f (x) and g(x) are such that,g(x) = x + 1 and
f ◦ g(x) = x + 3 − x.Then f (0) is equal to (a) 1 (b) −3 (c) 5 (d) 0Jee Main 2023

76. Let f (x) = x−1

x+1
, x ∈ R−{0, −1, 1}.If f n+1 (x) = f (f n (x)) for all nN,then f 6 (6)+f 7 (7)
is equal to (a) 67 (b) − 32 (c) 12
7 11
(d) − 12 Jee Main 2022

77. Let f : R → R be defined by f (x) = x − 1 and g : R − {−1, 1} → R be defined as

2
g(x) = x2x−1 .Then the function f ◦ g is (a) one-one but not onto function. (b) onto
but not one-one function (c) both one-one and onto function.(d) neither one-one
nor onto function. Jee Main 2022

78. Let α, β and γ be three positive real numbers.Let f (x) = αx5 + βx3 + γx, x ∈ R
and g : R → R be such that g(f (x)) = x for all x ∈ R.If
 a1 , na2 , a3 , . 
. . , an be in
arithmetic progression with mean zero,then the value f g n1
P
f (ai ) is equal
i=1
to : (a) 0 (b) 3 (c) 9 (d) 27 Jee Main 2022
2x
79. Let f : R → R be a function defined by f (x) = e2e 1 2



2x +e .Then f 100
+ f 100
+
3 99



f 100 + · · · + f 100 is equal to ——— Jee Main 2022
    501
x25 25
80. Let f : R → R be a function defined by f (x) = 2 1 − 2 (2 + x ) .If the
function g(x) = f (f (f (x))) + f (f (x)),the greatest integer less than or equal to g(1)
is ——– Jee Main 2022

81. Let f (x) and g(x) be two real polynomials of degree 2 and 1 respectively.If f (g(x)) =
8x2 − 2x, and g(f (x)) = 4x2 + 6x + 1,then the value of f (2) + g(2) is ——-
. Jee Main 2022
2
82. Let f (x) = sin−1 x and g(x) = 2x x −x−2
2 .If g(2) = lim g(x),then the domain of the
 4 −x−6
x→2
function f ◦g is (a)
 3 (−∞,
−2]∪ − 3
, ∞ (b) (−∞, −1]∪[2, ∞) (c) (−∞, −2]∪[−1, ∞)
(d) (−∞, −2] ∪ − 2 , ∞ Jee Main 2021

k + 1 if k is odd
83. Let A = {1, 2, 3, . . . , 10} and f : A → A be defined as f (k) =
k, if k is even
Then the number of possible functions g ◦ f = f is (a) 105 (b) 10 c5 (c) 55 (d)
5! Jee Main 2021

84. Let f : R − α6 → R be defined by f (x) = 6x−α 5x+3


.Then the value of α for which
α
(f ◦ f )(x) = x,for all x ∈ R − 6 , is (a) 6 (b) 5 (c) No such α exists (d)
5 Jee Main 2021

85. Let f : R → R be defined s f (x) = 2x − 1 and g : R − {1} → R − {1} be defined

x− 1
by g(x) = x−12 .Then the composite mapping f ◦ g is (a) both one-one and onto. (b)
onto but not one-one. (c) neither one-one nor onto. (d) one-one but not onto.
. Jee Main 2021
86. Let g : N → N de defined as g(3n+1) = 3n+2, g(3n+2) = 3n+3, g(3n+3) = 3n+1
for all n ≥ 0.Then which of the following statements is true ?
(a) There exists a function f : N → N such that g ◦ f = f (b) g ◦ g ◦ g = g (c) There
exists a one-one function f : N → N such that f ◦ g = f (d) There exist an onto
function such that f ◦ g = f Jee Main 2021

87. If g(x) = x2 + x − 1 and (g ◦ f )(x) = 4x2 − 10x + 5,then f 54 is equal to (a) 12 (b)

− 32 (c) − 21 (d) 32 Jee Main 2020

88. For a suitably chosen real constant a,let a function f : R − {−a} → R be defined
a−x
by f (x) = a+x .Further suppose that for any real number x ̸= −a and f (x) ̸=
−a,(f ◦f )(x) = x.Then f − 12 is equal to (a) −3 (b) 13 (c) − 31 (d) 3Jee Main 2020

89. If g(x) = x2 + x − 1 and (g ◦ f )(x) = 4x2 − 10x + 5,then f 54 is equal to (a) 12 (b)

− 32 (c) − 21 (D) 32 Jee Main 2020

90. For x ∈ R − [0, 1],let f1 (x) = x1 , f2 (x) = 1 − x and f3 (x) = 1−x 1

be three given
functions.If a function,J(x) satisfies (f2 ◦ J ◦ f1 )(x) = f3 (x),then J(x) is equal to
(a) f3 (x) (b) x1 f3 (x) (c) f2 (x) (d) f1 (x) Jee Main 2019

91. Let f (x) = x2 , x ∈ R.For any A ⊆ R,define g(A) = {x ∈ R : f (x) ∈ A}.If S =

[0, 4],then which one of the following statements is not true ? (a) f (g(S)) ̸= f (S)
(b) f (g(S)) = S (c) g(f (S)) = g(S) (d) g(f (S)) ̸= S Jee Main 2019

92. Let f (x) = ex −x and g(x) = x2 −x,for all x ∈ R.Then the

 set
x ∈ R,where
 1 of all the
1 1
 
function  = (f ◦ g)(x) is increasing, is : (a) −1, − 2 ∪ 2 , ∞ (b) 0, 2 ∪ [1, ∞)
h(x)
(c) − 21 , 0 ∪ [1, ∞) (d) [0, ∞)

Jee Main 2019
√ 2
93. For x ∈ 0, 23 ,let f (x) = x, g(x) = tan x and h(x) = 1−x


1+x 2 .If ϕ(x) = (h ◦ f ◦
π
g)(x),then ϕ 3 is equal to : (a) tan 12 (b) tan 12 (c) tan 12 (d) tan 7π
11π π 5π


12
. Jee Main 2019

1

1
R2
94. Let 5f (x) + 4f x
= x
+ 3, x > 0.Then 18 f (x)dx is equal to (a) 10 loge 2 − 6 (b)
1
10 loge 2 + 6 (c) 5 loge 2 + 3 (d) 5 loge 2 − 3 Jee Main 2023

95. Let f (x) be a function such that f (x + y) = f (x) × f (y) for all x, y ∈ N.If f (1) = 3
Pn
and f (k) = 3279,then the value of n is (a) 6 (b) 8 (c) 7 (d) 9 Jee Main 2023
k=1

22x 1

2

2022


96. If f (x) = 22x +2
, x ∈ R,then f 2023
+f 2023
+ ··· + f 2023
is equal to (a) 2011
(b) 1010 (c) 2010 (d) 1011 Jee Main 2023

97. Let f (x) = 2xn + λ, λ ∈ R, n ∈ N,and f (4) = 133, f (5) = 225.Then the sum of all
positive integer divisors of f (3) − f (2) is (a) 61 (b) 60 (c) 58 (d) 59Jee Main 2023

98. Suppose f : R → (0, ∞) be a differentiable function such that 5f (x + y) =

5
P
f (x).f (y),for all x, y ∈ R.If f (3) = 320,then f (n) is equal to : (a) 6875 (b)
n=0
6575 (c) 6825 (d) 6528 Jee Main 2023
1


99. Let f : R − {0, 1} → R be a function such that f (x) + f 1−x
= 1 + x.Then f (2)
is equal to : (a) 92 (b) 94 (c) 74 (d) 73 Jee Main 2023
100. The number of functions f : {1, 2, 3, 4} → {a ∈ Z : |a| ≤ 8} satisfying f (n)+ n1 f (n+
1) = 1 for all n ∈ {1, 2, 3} is (a) 3 (b) 4 (c) 1 (d) 2 Jee Main 2023

101. Consider the function f : N → R,satisfying f (1) + 2f (2) + 3f (3) + · · · + xf (x) =

1 1
x(x + 1)f (x); x ≥ 2 with f (1) = 1.Then f (2022) + f (2028) is equal to (a) 8200 (b) 8000
(c) 8400 (d) 8100 Jee Main 2023

102. For some a, b, c ∈ N,let f (x) = ax − 3 and g(x) = xb + c, x ∈ R.If (f ◦ g)−1 (x) =

1
x−7 3
2
,then (f ◦ g)(ac) + (g ◦ f )(b) is equal to —— Jee Main 2023

103. Suppose f is a function satisfying f (x + y) = f (x) + f (y) for all x, y ∈ N and

m
f (n)
f (1) = 51 .If 1
P
n(n+1)(n+2)
= 12 ,then m is equal to —— Jee Main 2023
n=1

104. Let f : R → R be a continuous function such that f (3x) − f (x) = x.If f (8) = 7,then
f (14) is equal to : (a) 4 (b) 10 (c) 11 (d) 16 Jee Main 2022

105. Let f : N → R be a function such that f (x + y) = 2f (x)f (y) for natural numbers
10
f (α + k) = 512 (220 − 1) holds,
P
x and y.If f (1) = 2,then the value of α for which 3
k=1
is (a) 2 (b) 3 (c) 4 (d) 6 Jee Main 2022

106. Let f : R − {3} → R − {1} be defined by f (x) = x−2

x−3
.Let g : R → R be given as
g(x) = 2x − 3.Then,the sum of all values of x for which f −1 (x) + g −1 (x) = 13
2
is
equal to (a) 7 (b) 2 (c) 5 (d) 3 Jee Main 2021
∞
P f (4) 1
107. If f (x + y) = f (x)f (y) and f (x) = 2, x, y ∈ N,then the value of f (2)
is : (a) 9
n=1
4 1 2
(b) 9
(c) 3
(d) 3
Jee Main 2020

108. Suppose that a function f : R → R satisfies f (x + y) = f (x)f (y) for all x, y ∈ R

n
P
and f (a) = 3.If f (i) = 363,then n is equal to ——– Jee Main 2020
i=1

1−x 2x



109. If f (x) = loge 1+x , |x| < 1,then f 1+x2
is equal to (a) 2f (x) (b) (f (x))2 (c)
2
2f (x ) (d) −2f (x) Jee Main 2019