Sets, Relations and Functions — practice questions

150 curated practice questions from Sets, Relations and Functions (Mathematics JEE Advanced, NTA Exams) — 102 hard, 48 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• The domain of the function f (x) = [4] _ 3 ( 1 | x| ) is :
• If a 2 + b 2 + c 2 = 1, then ab + bc + ca lies in the interval
• Let A = (x 1 , x 2 ,…..,x 8 ), B = (y 1 , y 2 , y 3 ), the total no. of functions f : A → B that are onto and there are…
• The equation | | x – 1| + a | = 4 can have real solutions for x if ‘a’ belongs to the interval
• The function f ( x ) = | px – q | + r | x |, x ∈ (–∞, ∞) where p > 0, q > 0, r > 0 assumes its minimum value only on…
• Find all possible values of x satisfying : [x] [x-2] - [x-2] [x] = 8 x +12 [x-2][x] (where [ ] denotes the greatest…
• Let f ( x ) = x 2 and g ( x ) = sin x for all x ∈ R . Then the set of all x satisfying ( fogogof )( x ) = ( gogof )( x…
• Let f : (–∞, 2] → (–∞, 4] be a function defined by f (x) = 4x – x 2 . Then f –1 (x) is :
• Let f : R – n → R be a function defined by f(x) = x-m x-n , where m ≠ n . This function is-
• If f ( x + y 8 , x - y 8 )= xy , then f (m, n) + f (n, m) = 0
• The function f : [0, 3] → [1, 29], defined by f ( x ) = 2 x 3 –15 x 2 + 36 x + 1, is
• If f (x) = x-1 x+1 , then f (2x) is :
• 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 (…
• Assertion : ^ -1 [ ^ -1 x+ ^ -1 (1-x) ]= 2 has no non-zero integral solution. Reason : The greatest and least values of…
• Assertion : Let f (x) be a function satisfying f (x – 1) + f (x + 1) = 2 f (x) for all x ∈ R. Then f (x) is periodic…
• Let f be a real valued function defined by f (x) = e ^ x - e ^ -| x | e ^ x + e ^ | x | then range of f (x) is :
• Solution of 2 x + 2 |x| > 2 2 is
• Let f (x) = max 1 + sin x, 1 – cos x, 1 x ∈ [0, 2π] and g (x) = max 1, | x –1| x ∈ R. Then gof is :
• If x ^ 4 f (x)- 1- 2 x =| f (x)|-2 f (x), then f(-2) equals:
• Let R = (x, y) : x, y ∈ R. x 2 + y 2 R^ = (x, y): x, y R, y 4 9 x^ 2 then is
• If f (x) = _ t (1+ x)^ t -1 (1+ x)^ t +1 , then range of f (x) is
• Let f ( x ) be defined for all x > 0 and be continuous. If f ( x ) satisfy f ( x y ) = f ( x ) – f ( y ) for all x , y…
• The range of the function f (x) = |x – 1| + |x – 2|, –1 < x < 3, is
• Let f (x) and g (x) be two real valued function given by, f (x) = – l nx and g (x) = e –x . Let h (x) = f (x) – x and m…
• Let f : R → R be a function such that f ( x + y ) = f ( x ) + f ( y ), ∀ x , y ∈ R . If f ( x ) is differentiable at x…
• Let f , g and h be real-valued functions defined on the interval [0, 1] by f ( x ) = e x 2 + e – x 2 , g ( x ) = xe x 2…
• Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞) Let h ( x ) = f ( g ( x )). If h…
• The number points (x, y), where curves |y| = l n |x| and (x –1) 2 + y 2 – 4 = 0 cut each other, is
• Let f (x) = min x, x 2 , for every x ∈ R. Then :
• Let f : R → R is a function satisfying f (2 – x) = f (2 + x) and f (20 – x) = f (x), x ∈ R. For this function f answer…
• f(x)= array ll x, & if x is rational 0, & if x is irrational array , . and g(x)= array l 0, if x is rational x, if x is…
• A certain polynomial P(x), x ∈ R when divided by x – a, x – b, x – c leaves remainder a, b, c respectively. The…
• If the function f : R → R be such that f (x) = x – [x], where [x] denotes the greatest integer less than or equal to x…
• Let f : R → R, g : R → R be two functions given by f (x) = 2x – 3, g (x) = x 3 + 5 . Then ( f og) –1 (x) is equal to :
• If 3 x = 4 x – 1 , then x =
• If g ( f ( x )) =|sin x | and , then
• Let f: R R be defined by f(x)= x^ 2 -3 x-6 x^ 2 +2 x+4 Then which of the following statements is (are) TRUE?
• Solution set of the inequality : 1 2^ x -1 > 1 1-2^ ( x -1) is
• If f (n + 1) = 2 f( n )+1 2 , n = 1, 2, ... and f (1) = 2, then f (101) equals
• Assertion : ^ -1 x- x^ 2 2 + x^ 3 4 - = d 2 - ^ -1 x^ 2 - x^ 4 2 + x^ 6 4 for 0 Reason : ^ -1 x(x+1) + ^ -1 x^ 2 +x+1 =…
• Assertion : Let A and B be two sets each with a finite number of elements. Assume that there is an injective mapping…
• Let f ( x ) = sin x and g ( x ) = ln | x |. If the ranges of the composition functions fog and gof are R 1 and R 2…
• The number of roots of the question 1 + l og 2 (1 – x) = 2 –x is :
• Let f be a real valued function with domain R satisfying 0 f (x) f (x + a) = 1 2 - f( x )-(f( x ))^ 2 x ∈ R then the…
• Solution of the inequality x > (1-x) is given by
• If f is an even function defined on the interval [–5, 5], then the real values of x satisfying the equation f (x) = f (…
• If f(x)= x^ 2 -1 x^ 2 +1 , for every real number x, then the minimum value of f
• The domain of definition of the function f ( x ) given by the equation 2 x + 2 y = 2 is
• If for x R, 1 3 x^ 2 -2 x+4 x^ 2 +2 x+4 3, then 9.3^ 2 x +6.3^ x +4 9.3^ 2 x -6.3^ x +4 lies b/w
• Let f(x)= ( 6 ( 2 x ) ) for all x ∈ R and g(x)= 2 x for all x ∈ R . Let ( fog )( x ) denote f ( g ( x )) and ( gof )( x…
• The number of solutions of log 4 ( x – 1) = log 2 ( x –3) is
• The domain of the function f (x) = X ^ 1 x is :
• Let f ( x ) = [ x ] = the greatest integer less than or equal to x and g ( x ) = x – [ x ]. Then for any two real…
• If f ( x ) = cos [ π 2 ] x + cos [– π 2 ] x , where [ x ] stands for the greatest integer function, then
• Assertion : If a is twice the tangent of the arithmetic mean of sin –1 x and cos –1 x, be is the geometric mean of tan…
• If f : [1 , ∞ ) → [2 , ∞ ) is given by then f –1 ( x ) equals
• Let f (x) be invertible function and let f –1 (x) be its inverse. Let equation f f –1 (x) = f –1 (x) has two real roots…
• Let f: (- 2 , 2 ) R be given by f ( x ) = (log(sec x + tan x )) 3 . Then
• The domain of definition of the function y= 1 _ 10 (1-x) + x+2 is
• A function f from a set X to Y is called onto, if for every y ∈ Y there exist x ∈ X such that f (x) = y. Unless the…
• Assertion : The function a x+b c x+d , (ad – bc ≠ 0) cannot attain the value a c . Reason : The domain of the function…
• The domain of definition of f(x)= _ 2 (x+3) x^ 2 +3 x+2 is
• If f is an even function defined on the interval (–5, 5) then a value of x satisfying the equation f (x) = f ( x+1 x+2…
• Column–I Column–II (A) odd function (P) x – [x] (B) even function (Q) (x+ 1+x^ 2 ) (C) neither even nor odd (R) x 1+x…
• If f (x) + 2 f (1 – x) = x 2 + 2, x ∈ R, then f (x) is given as:
• Let E = 1, 2, 3, 4 and F = 1, 2 , then the number of onto functions from E to F is
• Let f : (0, ∞) → R be given by Then
• If x = l og a (bc), y = l og b (ca) and z = l og c (ab) then which of the following is equal to 1?
• Let f : R → R is a function satisfying f (2 – x) = f (2 + x) and f (20 – x) = f (x), x ∈ R. For this function f answer…
• Complete solution set of the equation |x 2 – 1 + cos x| = |x 2 – 1| + |cos x| belonging to (–2π, π) is
• If [x] denotes the greatest integer ≤ x, then [ 2 3 ]+ [ 2 3 + 1 99 ]+ [ 2 3 + 2 99 ]+ + [ 2 3 + 98 99 ] is equal to
• The domain of function f(x)= 1 x^ 2 - x ^ 2 , where x denotes fraction part of x .
• Let f ( x ) = ( x + 1) 2 –1, x ≥ –1. Then the set x : f ( x ) = f –1 ( x ) is
• The domain of the function f(x)= x^ 2 -[x]^ 2 , where [ x ] = the greatest integer less than or equal to x is
• If [.] denotes the greatest integer function then the domain of the real-valued function _ [x+1 / 2] |x^ 2 -x-2 | is
• Consider the statements : P : There exists some x ∈ R such that f ( x ) + 2 x = 2(1 + x 2 ) Q : There exists some x ∈ R…
• If f ( x ) = 3 x – 5, then f –1 (x)
• The domain of the function f (x) = 1 x^ 12 -x^ 9 +x^ 4 -x+1 is
• Let f ( x ) = x – [ x ], for every real number x , where [ x ] is the integral part of x , then is
• The function f ( x ) = [ x ] 2 – [ x 2 ] (where [ y ] is the greatest integer less than or equal to y ) is…
• Assertion : A function y = f (x) is defined by x 2 – arc cos y = π, then domain of f (x) is R. Reason : cos –1 y ∈ [0…
• X and Y are two sets and f : X → Y . If f ( c ) = y ; c ⊂ X, y ⊂ Y , then the true statement is
• Let f 1 (n) = 1 + 1 2 + 1 3 + + 1 n , then f 1 (1) + f 1 (2) + f 1 (3) + ... + f 1 (n) is equal to :
• Column–I Column–II (A) f (x + y) = f (x) + f (y) (P) log 3 x (B) f (xy) = f (x) + f (y) (Q) tan –1 x (C) f (x + y) = f…
• The domain of the function y = l og 10 l og 10 l og 10 ... l og 10 x is ....... n times .......
• If graph of y = f ( x ) is Then f ( x ) can be
• Total number of solutions of the equation sin πx = | l n e | x|| are :
• f( x )= | x |+ x 3 x is defined for :
• If log 0.3 ( x – 1) 0.09 ( x – 1), then x lies in the interval
• A function f from a set X to Y is called onto, if for every y ∈ Y there exist x ∈ X such that f (x) = y. Unless the…
• If l og k x . l og 5 k = l og x 5, k ≠ 1, k > 0, then x is equal to
• Let denote the set of all natural numbers, and denote the set of all integers. Consider the functions defined by ext…
• The range of the function f (x) = cos [x], for −π/2 < x < π/2 contains.
• Assertion : The domain of a function y = f ( x ) will be all reals if for every real x there exist real y. Reason : The…
• The domain of the real-valued function f (x) = l og e | l og e x | is
• If f(x)= array ll |x|, & x 1 2-x, & x>1 array . , then f ( f (x)) is equal to
• If 2 2 < 3 then the number of positive roots of 1 x = x^ 2 , . denotes the fractional part of x, is :
• Let y 1 : [0, ∞) R , y 2 : [0, ∞ ] R , f : [0, ∞) R , and g : [0, ∞] R , be functions such that f (0) = g (0) = 0, y 1…
• The number of positive integers satisfying the equation x + log 10 (2 x + 1) = x log 10 5 +log 10 6 is
• The domain of definition of the function y(x) is given by the equation 2 x + 2 y = 2 is
• The minimum value of 2 (x 2 –3) 3 + 27 is :
• The function f : R → R defined by f ( x ) = 6 x + 6 |x| is
• If f(x)= 4^ x 4^ x +2 then f ( 1 1997 )+f ( 2 1997 )+ .+f ( 1996 1997 ) is equal to
• If the function f : [1, ∞) → [1, ∞) is defined by f ( x ) = 2 x ( x – 1) , then f –1 ( x ) is
• Let = _ k=1 ^ ^ 2 k ( 6 ) Let g : [0, 1] → R be the function defined by g ( x ) = 2 α x + 2 α (1 – x ) Then, which of…
• Let f : x, y, z → a, b, c be a one-one function and only one of the conditions (i) f (x) ≠ b, (ii) f (y), = b (iii) f…
• Let S = (0, 1) ∪ (1, 2) ∪ (3, 4) and T = 0, 1, 2, 3 . Then which of the following statements is(are) true?
• If 1 2 l og 0.1 x < 2 then
• The domain of the function f( x )= x - 1- x ^ 2 is
• Let f be a real valued function with domain R satisfying f (x + k) = 1 + [2 – 5 f (x) + 10 ( f (x)) 2 – 10 ( f (x)) 3 +…
• If f : [0, ∞) → [0, ∞), and then f is
• Let the function g:(- , ) (- 2 , 2 ) be given by g(u)=2 ^ -1 (e^ u )- 2 Then, g is
• A rational function is defined as quotient of two polynomials, p( x ) and q( x ). The domain of the rational function…
• If | x – 1| + | x | + | x + 1| > 6 ; then x lies in
• If 1 , then number of solutions of the equation tan –1 (x – 1) + tan –1 x + tan –1 (x + 1) = tan –1 3x, is/are
• If f (x) = cos –1 ( 2-|x| 4 )+[ (3-x)]^ -1 , then its domain is :
• If f ( x ) = cos (log x ), then f ( x ) f ( y ) - 1 2 [f ( x y )+f(x y) ] has the value
• The graph of a real-valued function f ( x ) is the following. The function is
• Range of the function f defined by f (x) = [ 1 x ] (where [.] and . respectively denote the greatest integer and the…
• Let f : [- 3 , 2 3 ] be a function defined as f (x) = 3 sin x – cos x + 2 . The f –1 (x) is given by
• f (x) = |x – 1|, f : R + → R and g (x) = e x , g : [–1, ∞) → R If the function fog (x) is defined, then its domain and…
• A function f from a set X to Y is called onto, if for every y ∈ Y there exist x ∈ X such that f (x) = y. Unless the…
• If f ( x ) = cos [π] x + cos [π x ], where [y] is the greatest integer function of y then f (π/2) is equal to
• Let f (x) be defined on [–π, π] and is given by, f(x)= array ll x & - x 0 x & 0 Let g (x) = f |x| + | f (x)|, x ∈ [–π…
• Let f be a function satisfying 2 f (xy) = ( f (x)) y + ( f (y)) x and f (1) = k ≠ 1, then _ r =1 ^ n f( r ) is equal to…
• A rational function is defined as quotient of two polynomials, p( x ) and q( x ). The domain of the rational function…
• Let g ( x ) = 1 + x –[ x ] and f(x)= array cc -1, & x 0 array . then for all x, f ( g ( x )) is equal to
• The domain of definition of f (x) = log 2 (- _ 1 / 2 (1+ 1 x ^ 1 / 4 )-1 ) is
• Let f : R → R be a function defined by f (x) = x^ 2 -8 x^ 2 +2 , then f is :
• A rational function is defined as quotient of two polynomials, p( x ) and q( x ). The domain of the rational function…
• If 2 f (x – 1) – f ( 1-x x ) = x, then f (x) is :
• The domain of the function f (x) = ( 1 x -1 ) is :
• Domain of definition of the function for real valued x , is
• The function f (x) = ( x) + sin –1 ( 1+x^ 2 2 x ) is defined for :
• Let f(x)= x x+1 , x ≠ –1, then for what value of α is f ( f ( x )) = x ?
• Let f : (0, 1) → R be defined by where b is a constant such that 0 b < 1. Then
• The graph of the function y = f (x) is as shown in the figure. Then which one of the following graphs are correct?
• The domain of the function f(x)= _ 3 [- _ 1 / 2 (1+ 1 x^ 1 / 5 )-1 ] is
• If f (x + y) = f (x) . f (y) for all real x, y and f (0) ≠ 0 then the function g (x) = f( x ) 1+(f( x ))^ 2 is
• The complete solution set of sin –1 (sin 5) > x 2 – 4x is
• The function f ( x ) = 2| x | + | x + 2| – || x + 2| – 2| x || has a local minimum or a local maximum at x =
• Assertion : cosec ^ -1 3 2 + ^ -1 2 3 -2 ^ -1 1 7 - ^ -1 7= ^ -1 7 Reason : sin –1 x + cos –1 x = 2 , tan –1 x + cot –1…
• Let function f : R → R be defined by f ( x ) = 2 x + sin x for x ∈ R . Then f is
• If the function f : ℝ → ℝ is defined by f ( x ) = | x |( x − sin x ), then which of the following statements is TRUE?
• The number of pairs, (x, y), x, y ∈ R, satisfying 4x 2 – 4x + 2 = sin 2 y and x 2 + y 2 < 3 are
• Let f : R → R is a function satisfying f (2 – x) = f (2 + x) and f (20 – x) = f (x), x ∈ R. For this function f answer…
• Range of the function f(x)= x^ 2 +x+2 x^ 2 +x+1 ; x R is
• If the function f:[1, ) [1, ) is defined by f (x) = 2 x(x–1) , then f –1 (x) is
• If 0 x x ] is the greatest integer less than or equal to x, the number of possible values of x is
• If f ( x ) = sin x + cos x , g ( x ) = x 2 – 1, then g ( f ( x )) is invertible in the domain

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