## JEE Advanced Archive on Functions Multiple Choice Single Correct Type Questions

1. the number of points in (-∞, ∞), for which x² – xSinx – Cosx = 0 is/are. {IIT JEE 2013}.

(A) 0
(B) 2
(C) 4
(D) 6

Answer: Option (B) is the correct choice.

1. The function f: [0, 3] → [1, 29] is defined by f(x) = 2x³ – 15x² + 36x + 1 is. {IIT JEE 2012}.

(A) one – one & onto
(B) onto but not one – one
(C) one – one but not onto
(D) neither one – one nor onto

Answer: Option (B) is the correct choice.

1. Let f(x) = x² and g(x) = Sinx for all x in R. Then the set of all x satisfying (fogogof)(x) = (gogof)(x), where (fog)(x) = f(g(x)) is. {IIT JEE 2011}.

(A) ±√nx, n = {0, 1, 2,…}
(B) ±√nx, n = {1, 2,…}
(C) 2nπ + π/2, n = {…, -2, -1, 0, 1, 2,…}
(D) 2nπ, n = {…, -2, -1, 0, 1, 2,…}

Answer: Option (A) is the correct choice.

1. If f”(x) = -f(x) and g(x) = f'(x) anf F(x) = (f(x/2))² + (g(x/2))² and given that F(5) = 5, then the value of F(10) is. {IIT JEE 2006}.

(A) 0
(B) 5
(C) 10
(D) 15

Answer: Option (B) is the correct choice.

1. If the functions f(x) and g(x) are defined on R → R such that f(x) = {0 if x is rational; x if x is irrational} and g(x) = {0 if x is irrational; x if x is rational}, then (f – g)(x) is. {IIT JEE 2005}.

(A) one – one & onto
(B) neither one – one nor onto
(C) one – one but not onto
(D) onto but not one – one

Answer: Option (A) is the correct choice.

1. X and Y are two sets and f: X → Y. If f(c) = {y: c ⊂ X, y ⊂ Y} and f⁻¹(d) = {x: d ⊂ Y, x ⊂ X}, then the true statement is. {IIT JEE 2005}.

(A) f(f⁻¹(b)) = b
(B) f⁻¹(f(a)) = a
(C) f(f⁻¹(b)) = b, b ⊂ Y
(D) f⁻¹(f(a)) = a, a ⊂ X

Answer: Option (D) is the correct choice.

1. If f(x) = Sinx + Cosx, g(x) = x² – 1, the g(f(x)) is invertible in the domain. {IIT JEE 2004}.

(A) [0, π/2]
(B) [-π/4, π/4]
(C) [-π/2, π/2]
(D) [0, π]

Answer: Option (B) is the correct choice.

1. If f: [0, ∞) → [0, ∞) is defined as f(x) = x/(1 + x), then the function f(x) is. {IIT JEE 2003}.

(A) one – one & onto
(B) neither one – one nor onto
(C) one – one but not onto
(D) onto but not one – one

Answer: Option (C) is the correct choice.

1. Range of the functions f(x) = (x² + x + 2)/(x² + x + 1) for all x in R, is. {IIT JEE 2003}.

(A) (1, ∞)
(B) (1, 11/7)
(C) (1, 7/3)
(D) (1, 7/5)

Answer: Option (C) is the correct choice.

1. The domain of definition of the function f(x) = √(Sin⁻¹(2x) + π/6) is. {IIT JEE 2003}.

(A) [-1/4, 1/2]
(B) [-1/2, 1/2]
(C) (-1/2, 1/9)
(D) [-1/4, 1/4]

Answer: Option (A) is the correct choice.

1. Suppose f(x) = (x + 1)² for x ≥ -1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y = x, the g(x) is equal to. {IIT JEE 2002}.

(A) -√x – 1, x ≥ 0
(B) 1/(x + 1)², x ≥ -1
(C) √(x + 1), x ≥ -1
(D) √x – 1, x ≥ 0

Answer: Option (D) is the correct choice.

1. Let the function f: R → R be defined by f(x) = 2x + Sinx, then f(x) is. {IIT JEE 2002}.

(A) one – one & onto
(B) neither one – one nor onto
(C) one – one but not onto
(D) onto but not one – one

Answer: Option (A) is the correct choice.

1. Let f(x) = ax/(x + 1), x ≠ -1, then for what value of ‘a’ is f(f(x)) = x. {IIT JEE 2001}.

(A) -√2
(B) -1
(C) 1
(D) √2

Answer: Option (B) is the correct choice.

1. The function f: [1, ∞) → [2, ∞) is defined by f(x) = x + 1/x, then f⁻¹(x) is equal to. {IIT JEE 2001}.

(A) (x + √(x² – 4))/2
(B) x/(1 + x²)
(C) (x – √(x² – 4))/2
(D) 1 + √(x² – 4)

Answer: Option (A) is the correct choice.

1. The domain of definition of the function f(x) = log₂(x + 3)/(x² + 3x + 2) is. {IIT JEE 2001}.

(A) R – {-2, -1}
(B) (-2, ∞)
(C) R – {-3, -2, -1}
(D) (-3, ∞) – {-2, -1}

Answer: Option (D) is the correct choice.

1. Let g(x) = 1 + x + [x] {where [x] is the Greatest Integer Function} and f(x) = {-1 if x < 0; 0 if x = 0; 1 if x > 0}. Then for all x, f(g(x)) is equal to. {IIT JEE 2001}.

(A) x
(B) 1
(C) f(x)
(D) g(x)

Answer: Option (B) is the correct choice.

1. Let E = {1, 2, 3, 4} anf F = {1, 2}. Then the number of onto functions from E to F are. {IIT JEE 2001}.

(A) 8
(B) 10
(C) 12
(D) 14

Answer: Option (D) is the correct choice.

1. The domain of definition of the function y(x) given by the equation 2ˣ + 2ʸ = 2 is. {IIT JEE 2000}.

(A) 0 < x ≤ 1
(B) 0 ≤ x ≤ 1
(C) -∞ < x ≤ 0
(D) -∞ < x < 1

Answer: Option (D) is the correct choice.

1. If the function f: [1, ∞) → (1, ∞) is defined as f(x) = 2ˣ⁽ˣ⁻¹⁾, then f⁻¹(x) is equal to. {IIT JEE 1999}.

(A) (1/2)ˣ⁽ˣ⁻¹⁾
(B) (1 + √(1 + 4log₂x))/2
(C) (1 – √(1 + 4log₂x))/2
(D) Not Defined

Answer: Option (B) is the correct choice.

1. If f(x) = 3x – 5, then f⁻¹(x) is equal to. {IIT JEE 1998}.

(A) 1/(3x – 5)
(B) (x + 5)/3
(C) not defined since f is not one – one
(D) not defined since f is not onto

Answer: Option (B) is the correct choice.

1. If g(f(x)) = |Sinx| and f(g(x)) = (Sin√x)², then. {IIT JEE 1998}.

(A) f(x) = Sin²x, g(x) = √x
(B) f(x) = Sinx, g(x) = |x|
(C) f(x) = x², g(x) = Sin√x
(D) f and g can not be determined

Answer: Option (A) is the correct choice.

1. The graph of the function Cosx Cos(x + 2) – Cos²(x + 1) is. {IIT JEE 1997}.

(A) A straight line passing through (0, -Sin²1) with slope 2
(B) A straight line passing through (0, 0)
(C) A parabola with vertex at (0, -Sin²1)
(D) A straight line passing through (π/2, -Sin²1) and parallel to x-axis

Answer: Option (D) is the correct choice.

1. Let f(x) = [x] Sin(π/([x + 1])), where [.] denotes the greatest integer function, then domain of f is. {IIT JEE 1996}.

(A) (-∞, -1)
(B) [0, ∞)
(C) (-∞, -1) U [0, ∞)
(D) none of the above

Answer: Option (C) is the correct choice.