Important Relations And Functions Questions for CBSE Class 12th

Let S be the set of all subsets of first 10 natural numbers. If a relation R is defined on set S such that (A, B) ∈ R if A ∩ B ≠ ϕ ; w h e r e A , B ∈ S , then R is

Let R be a relation defined on the set of real numbers by a R b ⇔ 1 + a b > 0 . Then R is

Let P = ( x , y ) ∣ x 2 + y 2 = 1 , x , y ∈ R . Then, P is

Which one of the following relations on R is an equivalence relation?

Let R be an equivalence relation on a finite set A having ‘n’ elements. Then, the number of ordered pairs in R is

Let R be a relation on a set A such that R = R − 1 , then R is

The domain of definition of the function f ( x ) given by the equation 2 x + 2 y = 2 is

The domain of definition of the function f ( x ) = sin − 1 ⁡ ( 2 x ) + π 6 for real-valued x is

Let R be an equivalence relation on a finite set A having n elements. Then, the number of ordered pairs in R is

The function f ( x ) = sec − 1 ⁡ x x − [ x ] , where [ ⋅ ] denotes the greatest integer less than or equal to x is defined for all x belonging to

Let R be the relation on the set of all real numbers defined by a R b iff | a − b | ≤ 1 . Then, R is

The domain of the function f ( x ) = log 3 + x ⁡ x 2 − 1 is

If f ( x ) is continuous and increasing function such that domain of g ( x ) = f ( x ) − x be R and h ( x ) = 1 1 − x , then the domain of ϕ ( x ) = f ( f ( f ( x ) ) ) − h ( h ( h ( x ) ) ) is

Let f ( x ) = ( x + 1 ) 2 − 1 , x ≥ − 1 . Then the set x : f ( x ) = f − 1 ( x ) is

Find the range of the function f ( x ) = x + x 2

If f ( x ) = sin ⁡ x + cos ⁡ x , g ( x ) = x 2 − 1 , then g ( f ( x ) ) is invertible in the domain

Let A = { 1 , 2 , 3 } . Then find the number of relations containing (1,2) and (1, 3) which are reflexive and symmetric but not transitive.

the relation R in the set R of real numbers, defined as R = ( a , b ) : a ≤ b 2 is

Which of the following functions is periodic?

If the period of cos ⁡ ( sin ⁡ ( n x ) ) tan ⁡ x n , n ∈ N , is 6 π , then n =

Period of f ( x ) = sgn ⁡ ( [ x ] + [ − x ] ) is equal to (where [.] denotes greatest integer function)

Find the domain of the function f ( x ) = x − 3 ( x + 3 ) x 2 − 4

The range of the y = x 2 − 4 is

The range of f ( x ) = sgn ⁡ x 2 − 2 x + 3 i s

The domain of f(x) = log ⁡ { x } , (where { } represents the fractional part function).

if a,b,c are non-zero rational numbers then the sum of all the possible values of a a + b b + c c

The domain of f ( x ) = sin ⁡ x + 16 − x 2 is

The domain of the function 1 1 + 2 sin ⁡ x is

The range of f ( x ) = log 3 ⁡ 5 − 4 x − x 2 is

The domain of f(x) = sin − 1 ⁡ [ x ] (where [ ] represents the greatest integer function).

Let A = x 1 , x 2 , x 3 , x 4 , x 5 and f : A A . The number of bijective functions such that f x i = x i for exactly two of the x i ′ s is ( i = 1 to 5 )

The number of integral elements in the domain of the function f ( x ) = sin − 1 ⁡ x 2 − 2 x 3 + [ x ] + [ − x ] , where [.]denotes greatest integer function, is

Consider a real-valued function f ( x ) satisfying 2 f ( x y ) = ( f ( x ) ) y + f ( y ) x ∀ x , y ∈ R and f ( 1 ) = a where a ≠ 1 . then ( a − 1 ) ∑ l = 1 n f ( i ) =

Domain of definition of the function f ( x ) = log ( | x | − 1 ) ⁡ x 2 + 4 x + 4 is

Let f 1 ( x ) = x , 0 ≤ x ≤ 1 1 , x > 1 0 , otherwise f 2 ( x ) = f 1 ( − x ) for all x f 3 ( x ) = − f 2 ( x ) for all x f 4 ( x ) = f 3 ( − x ) for all x which of the following is necessarily true?

Let P = x , y | x 2 + y 2 = 1 , x , y ∈ R . Then P is

Which one of following best represent the graph of y = x log , π ?

Range of the function f x = cos K sin x is [-1,1], then the least positive integral value of K will be

The domain of the function f x = sin x + cos x + 7 x – x 2 – 6 is

If R ⊂ A × B and S ⊂ B × C be two relations, then ( SoR ) – 1 =

Let X be a family of set and R be a relation on X defined by ‘A is disjoint from B’. Then R is

Let L denote the set of all straight lines in a plane. Let a relation R be defined by αRβ ⇔ α ⊥ β , α , β ∈ L . Then R is

Let n be a fixed positive integer. Define a relation R on the set Z of integers by, aRb ⇔ n a – b . Then R is

x 2 = xy is a relation which is

Let R be a relation over the set N × N and it is defined by a , b R c , d ⇒ a + d = b + c . Then R is

n/m means that n is factor of m, then the relation f is

If f ( x ) = cos 2 ⁡ x + sin 4 ⁡ x sin 2 ⁡ x + cos 4 ⁡ x for x ∈ R then f ( 2018 ) =

The domain of the function f ( x ) = cos ⁡ log ⁡ 16 − x 2 3 − x is

The domain of the function f ( x ) = 24 − x C 3 x − 1 + 40 − 6 x C 8 x − 10 is ,

The domain of the function f ( x ) = log 1 / 2 ⁡ x − 1 2 + log 2 ⁡ 4 x 2 − 4 x + 5 is

The domain of definition of the function y (x) given by the equation 2 x + 2 y = 2 is

If g [ f ( x ) ] = | sin ⁡ x | and f [ g ( x ) ] = ( sin ⁡ x ) 2 , then

Which of the following is a function ([.] denotes the greatest integer function, {.} denotes the fractional part function)?

The period of the function f (x) = cos x 2 is

The domain of the function f ( x ) = log 10 ⁡ 1 − log 10 ⁡ x 2 − 5 x + 16 is

The domain of the function f ( x ) = log x + 1 2 ⁡ x 2 − 5 x + 6 is

If f : R R, g : R R be two given functions then f (x) = 2 min { f (x) – g (x), 0} equals

The domain of the function f ( x ) = 1 | sin ⁡ x | + sin ⁡ x is

Let f be a function defined on [0, 1] such that f ( x ) = x x ∈ Q 1 − x , x ∉ Q Then for all x ∈ [0, 1], fof (x) is

Let f : R R be a function defined by, f ( x ) = x + x 2 , then f is

The minimum number of elements that must be added to the relation R = { ( 1 , 2 ) , ( 2 , 3 ) } on the sub set { 1 , 2 , 3 } of natural numbers so that it is an equivalence relation is

The solution of 8 x ≡ 6 ( mod 14 ) is where [ a ] = { a + 14 k : k ∈ I }

The relation ‘less than’ on the set of natural numbers is

Let f : R 0 , π 2 defined by f ( x ) = tan − 1 ⁡ x 2 + x + a , then the set of values of a for which/is onto is

If f : R R , f ( f ( x ) ) = ( f ( x ) ) 2 , then f f f ( x ) is not equal to

The range of the function f ( x ) = tan ⁡ π 2 9 − x 2 is

If f : R R and g : R R are given by f ( x ) = | x | and g ( x ) = [ x ] for each x ∈ R , then { x ∈ R : g ( f ( x ) ) ≤ f ( g ( x ) ) } =

Range of the function f ( x ) = cos − 1 ⁡ log 4 ⁡ x − π 2 + sin − 1 ⁡ 1 + x 2 4 x is

If ∑ k = 0 n f ( x + k a ) = 0 , where a > 0 , then the period of f ( x ) i s

Range of function f ( x ) = log 2 ⁡ π + 2 sin − 1 ⁡ 3 − x 7 π is

The function f : ( − ∞ , − 1 ) 0 , e 5 defined by f ( x ) = e x 3 − 3 x + 2 is

f : R R defined by f ( x ) = 1 2 x | x | + cos ⁡ x + 1 is

Period of f ( x ) = cos ⁡ ( | sin ⁡ x | − | cos ⁡ x | ) is

If g ( x ) = x 2 + x − 2 and 1 2 ( gof ) ( x ) = 2 x 2 − 5 x + 2 , then f ( x ) is equal to

Let f : { x , y , z } { 1 , 2 , 3 } be a one-one mapping such that only one of the following three statements is true and remaining two are false: f ( x ) ≠ 2 , f ( y ) = 2 , f ( z ) ≠ 1 , then

Which of the following functions is an injective (one-one) function in its respective domain?

The range of the function y = 1 x 2 + 2

N u m b e r o f i n t e g e r a l v a l u e s o f x f o r w h i c h x 2 – x + 4 – 2 – 3 = x 2 + x – 12 i s

The domain of f ( x ) = sin − 1 ⁡ x 2 2 is

Number of integral values of x satisfyng the inequality 3 4 6 x + 10 – x 2 < 27 64 i s

Find the range of f(x) f ( x ) = sin − 1 [ x ]

Let f be a real-valued invertible function such that f 2 x – 3 x – 2 = 5 x – 2 , x ≠ 2 . T h e n t h e v a l u e o f f – 1 ( 13 ) i s

The function f is continuous and has the property f(f(x))=1-x, then the value of f 1 4 + f 3 4 i s

The relation R defined on the set N of natural numbers by x R y ⇔ 2 x 2 − 3 x y + y 2 = 0 is

If f ( x ) is a polynomical function such that f ( x ) ⋅ f ( 1 / x ) = f ( x ) + f ( 1 / x ) and f ( 2 ) = 9 then the value of f ( 4 ) =

Let f ( x ) = x 3 − 1 , x < 2 x 2 + 3 , x ≥ 2 . Then

The number of linear functions f satisfying f ( x + f ( x ) ) = x + f ( x ) ∀ x ∈ R is

If f ( x ) = sin ⁡ x + cos ⁡ x , g ( x ) = x 2 − 1 , then g ( f ( x ) ) is invertible in the domain

Let f : N N where N is set of natural numbers be a function such that f ( x + y ) = f ( x y ) ∀ x ≥ 4 and y ≥ 4 , then

The number of elements in the domain of the function f ( x ) = sin − 1 ⁡ x 2 − 2 x 3 + [ x ] + [ − x ] , (where [.] denotes the greatest integer function) is equal to

L e t X = a 1 , a 2 , . . . . , a 6 a n d Y = b 1 , b 2 , b 3 . The number of functions f from X to Y such that it is onto and there are exactly three elements x in X such that f x = b 1 is

The function f x = sec – 1 x x – x , where [ x ] denotes the greatest integer less than or equal to x , is defined for all x ∈

The domain of definition of the function f x given by the equation 2 x + 2 y = 2 is

The domain of the following function is f x = log 2 – log 1 / 2 1 + 1 x 1 / 4 – 1

The exhaustive domain of the following function is f x = x 12 – x 9 + x 4 – x + 1

The range of the following function is f x = 1 – cos x 1 – cos x 1 – cos x . . . . ∞

Let R be the relation on the set R of all real numbers defined by a R b if a – b ≤ 1 . Then R is

Let a relation R on the set N of natural number be defined as ( x , y ) ∈ R If and only If x 2 − 4 xy + 3 y 2 = 0 for all, x , y ∈ N and the relation is

Let R be a relation defined by R={(a,b):a≥b}, where a and b are real numbers, then R is

Let A = x , y , z , B = u , v , w and f : A B be defined by f(x)=u, f(y)=v, f(z) = w. Then f is

In the set X = a , b , c , d , which of the following functions in X ?

If f : R R is defined by f ( x ) = 2 x + x , then f 3 x – f – x – 4 x equals

For real numbers x and y, we write x R y ⇔ x – y + 2 is an irrational number. Then, the relation R is

The domain of the function f ( x ) = sin – 1 ( sin x ) – cos – 1 ( cos x ) in 0 , 2 π is

The minimum number of elements that must be added to the relation R = {(1, 2), (2. 3)} on the set {1, 2, 3} so that it is an equivalence relation

The domain of the function f ( x ) = x − 1 − x 2 is

The domain of the function f ( x ) = cos − 1 ⁡ 1 − | x | 2 is

The domain of the function f (x) = log 2 log 3 log 4 x is

The domain of definition of f ( x ) = 1 − | x | 2 − | x | is

The domain of the function f ( x ) = cos − 1 ⁡ 2 − | x | 4 + [ log ⁡ ( 3 − x ) ] − 1 is

The domain of the function f ( x ) = e sin − 1 ⁡ log 16 ⁡ x 2 is

The domain of the function f ( x ) = 1 [ x ] 2 − [ x ] − 6 is

The range of the function f ( x ) = 3 x 2 − 4 x + 5 is

The range of the function y = 3 sin ⁡ π 2 16 − x 2 is

The value of the function f ( x ) = x 2 − 3 x + 2 x 2 + x − 6 lies in the interval

The domain of the function f ( x ) = cot − 1 ⁡ x x 2 − x 2 , x ∈ R is

Let f : (4, 6) (6, 8) be a function defined by f ( x ) = x + x 2 (where [ ⋅ ] denotes the greatest integer function), then f – 1 (x) is equal to

The function f : R R defined by, f (x) = 4 x + 4 |x| is

Let f (x) = (–1) [x] (where [ . ] denotes the greatest integer function), then

Let f : [4, ∞) [4, ∞) be a function defined by, f (x) = 5 x(x – 4) , then f –1 (x) is

Let f be a function with domain [–3, 5] and let g (x) = |3x + 4|. Then the domain of ( fog) (x) is

The period of the function 3 sin 2 ⁡ πx + x − [ x ] + sin 4 ⁡ πx , where [⋅] denotes the greatest integer function is 503 k then k=

The period of the function f (x) = a sin kx + b cos kx is

The period of the function f ( x ) = cos ⁡ πx n ! − sin ⁡ πx ( n + 1 ) ! is

If the period of the function f ( x ) = sin ⁡ ( [ n ] x ) , where [n] denotes the greatest integer less than or equal to n, is 2 π , then

The period of the function f ( x ) = tan ⁡ x is

Which of the following functions has period π ?

The function f (x) = k |cos x| + k 2 |sin x| + ϕ (k) has period π 2 if k is equal to

The period of the function f ( x ) = 3 x + 3 − [ 3 x + 3 ] + sin ⁡ πx 2 , where [x] denotes the greatest integer ≤ x, is

The period of the function f ( x ) = 1 , when x is a rational 0 , when x is irrational is

The period of the function f (x) = x [x] is

If f ( x ) = x 2 + 1 [ x ] , ([⋅] denotes the greatest integer function), 1 ≤ x < 4, then

If 3 f ( x ) + 5 f 1 x = 1 x − 3 , ∀ x ( ≠ 0 ) ∈ R , then f (x) =

If f : R R , defined by f (x) = x 3 + 7, then the value of f –1 (71) and f –1 (–1) respectively are

The domain of the function f ( x ) = sin x x – x is

Let f ( x ) = a x + b , x ∈ R , a n d g ( x ) = x + d , x ∈ R , then f o g = g o f if and only if

The range of y = 1 – sin x is

The domain of the function f ( x ) = log 2 sin x is

The function f ( x ) = sin πx n ! – cos πx ( n + 1 ) ! is

Let f(x) be a function such that f ( x − 1 ) + f ( x + 1 ) = 2 f ( x ) for all x ∈ R . If f ( 3 ) = 5 then ∑ r = 0 10 f ( 3 + 8 r ) is equal to

A function out of the following whose period is not π is

Which of the following functions is an odd function

Let R be the real line. Consider the following subsets of the plane R × R . S = { ( x , y ) : y = x + 1 and 0 < x < 2 } T = { ( x , y ) : x − y is an integer } Statement-1 : T is an equivalence relation on R but S is not an equivalence relation on R. Statement-2 : S is neither reflexive nor symmetric but T is reflexive, symmetric and transitive

Statement-1: f : R R is a function defined by f(x) = 5x + 3. If g = f – 1 , then g ( x ) = x – 3 5 . Statement-2: If f : A B is a bijection and g : B A is the inverse of f , then f o g is the identity function on A.

Let f be a function defined by f ( x ) = ( x − 1 ) 2 + 1 , ( x ≥ 1 ) Statement-1: The set {x : f(x) = f –1 (x)} = {1, 2} Statement-2: f is a bijection and f – 1 ( x ) = 1 + x – 1 , x ≥ 1

Let L be the set of all lines in a plane and R be a relation on L defined by l 1 R l 2 if and only if l 1 ⊥ l 2 then R is

Statement-1: The number of bijective functions from the set A containing 100 elements to itself is 2 100. Statement-2: The total number of bijections from a set containing n elements to itself is n !

Let f ( x ) = ( x + 1 ) 2 – 1 ( x ≥ – 1 ) . Then the set S = { x : f ( x ) = f – 1 ( x ) } contains

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}, the relation R is

Let R = {(3, 3), (6, 6), (9, 9) (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is

Let f : { x , y , z } { 1 , 2 , 3 } be a one-one function. If it is given that exactly one of the following statements is true, Statement-1: f ( x ) = 1 , Statement-2: f ( y ) ≠ 1 , Statement-3: f ( z ) ≠ 2 . then f − 1 ( 1 ) is

Let R be a relation on a set A such that R = R − 1 then R is

The value of n ∈ z for which the function f ( x ) = sin ⁡ n x sin ⁡ ( x / n ) has 4 π as its period is

Let R be a relation on N defined by R = { ( m , n ) : m , n ∈ N and m = n 2 } Which of the following is true.

Let f ( x ) = x − [ x ] 1 + x − [ x ] , where x denotes the greatest integer less than or equal to x , then the range of f is

If f ( x + 3 y , x − 3 y ) = 12 x y , then f ( x , y ) is

The domain of the function f ( x ) = sin − 1 ⁡ log 3 ⁡ x 3 is

If f . ( 0 , π ) R is given by f ( x ) = ∑ k = 1 n [ 1 + sin ⁡ k x ] , [ x ] denotes the greatest integer function, then the range of f ( x ) is

If f ( x ) is a polynomial satisfying f ( x ) f ( 1 / x ) = f ( x ) + f ( 1 / x ) a n d f ( 3 ) = 28 , then f ( 4 ) is given by

Let f : R R be a function defined by f ( x ) = e | x | − e − x e x + e − x . Then

The domain of the function f ( x ) = − log 0.3 ⁡ ( x − 1 ) − x 2 + 2 x + 8 is

Which of the following functions is not onto

Part of the domain of the function f ( x ) = cos ⁡ x − 1 / 2 6 + 35 x − 6 x 2 lying in the interval [–1, 6] is

Let f ( x ) = x − 3 x + 1 , x ≠ − 1 . Then f 2010 ( 2014 ) (where f n ( x ) = f o f . . . of ( x ) ( n times)) is

The function f : [ − 1 / 2 , 1 / 2 ] [ − π / 2 , π / 2 ] defined by f ( x ) = sin − 1 ⁡ 3 x − 4 x 3 is

The inverse function of f ( x ) = 8 2 x – 8 – 2 x 8 2 x + 8 – 2 x , x ∈ ( – 1 , 1 ) , is

The domain of definition of the function y = 1 log 10 1 – x + x + 2 is

On the set ‘ N ‘ of all natural numbers, define the relation ‘ R ‘ by aRb if the G.C.D of a and b is 2. Then R is

The number of real solutions of the equation ( 9 / 10 ) x = – 3 + x – x 2 is

Let R be a relation from ℝ (set of real numbers) to ℝ defined by R = { ( a , b ) ∣ a , b ∈ ℝ and a − b + 3 is an irrational number } . The relation R is

Complete set of range of the function f ( x ) = 1 x 6 + | x | 3 − 1 is equal to

If a f ( x ) + b f 1 x = x − 1 , x ≠ 0 , a ≠ b , then f ( 2 ) =

Let the function f : R R be defined by f ( x ) = 2 x + sin ⁡ x for x ∈ R . Then f is

The relation R = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 1 , 2 ) , ( 2 , 3 ) , ( 1 , 3 ) } on set A = { 1 , 2 , 3 } is

A relation R on the set of complex numbers defined by Z 1 R Z 2 ⇔ Z 1 − Z 2 Z 1 + Z 2 is real then which of the following is not true

Let R be a relation on the set N be defined by { ( x , y ) ∣ x , y ∈ N , 2 x + y = 41 } . Then, R is

The void relation on a set A is

The domain of f ( x ) = 1 x – 1 x – 2 1 – 2 x is

The domain of | x − 2 | − 1 is

The domain of definition of the function f ( x ) given by the equation 2 x + 2 y = 2 is

The range of the function f ( x ) = tan π 2 9 − x 2 is

The domain of definition of f ( x ) = log 2 ⁡ ( x + 3 ) x 2 + 3 x + 2 is

Let R be a retlexive relation on a finite set A having n-elements, and let there be m ordered pairs in R. Then,

Let L denote the set of all straight lines in a plane. Let a relation R be defined by α R β ⇔ α ⊥ β , α , β ∈ L . Then, R is

The range of the function f ( x ) = | x − 1 | + | x − 2 | , − 1 ≤ x ≤ 3 , is

Let f ( x ) = α x ( x + 1 ) , x ≠ − 1 . Then for what value of α is f ( f ( x ) ) = x ?

If f : [ 1 , ∞ ) [ 2 , ∞ ) is given by f ( x ) = x + 1 x , then f − 1 ( x ) equals

Let the function f : R R be defined by f ( x ) = 2 x + sin ⁡ x for x ∈ R . Then f is

If f x 2 − 6 x + 6 + f x 2 − 4 x + 4 = 2 x ∀ x ∈ R then f ( − 3 ) + f ( 9 ) − 5 f ( 1 ) =

A function f : R R satisfies the equation f ( x ) f ( y ) − f ( x y ) = x + y ∀ x , y ∈ R and f ( 1 ) > 0 , then

The domain of the function f ( x ) = 1 10 C x − 1 − 3 × 10 C x c o n t a i n s t h e p o i n t s

If f ( x ) = x , x is rational 1 − x , x is irrational , then f ( f ( x ) ) is

If f ( x ) = x 2 sin ⁡ π x 2 , | x | < 1 x | x | , | x | ≥ 1 , then f ( x ) is

If g : [ − 2 , 2 ] R , where f ( x ) = x 3 + tan ⁡ x + x 2 + 1 P is an odd function, then the value of parametric P, where [.] d e n o t e s t h e g r e a t e s t i n t e g e r f u n c t i o n , i s

If f ( x ) is an invertible function and g ( x ) = 2 f ( x ) + 5 , then the value of g − 1 ( x ) is

If f ( x ) = maximum ⁡ x 3 , x 2 , 1 64 ∀ x ∈ [ 0 , ∞ ) , then

If f ( x ) is odd function and f ( 1 ) = a , and f ( x + 2 ) = f ( x ) + f ( 2 ) then the value of f ( 3 ) is

Let f ( x ) be defined on [ − 2 , 2 ] and is given by f ( x ) = − 1 , − 2 ≤ x ≤ 0 x − 1 , 0 ≤ x ≤ 2 , then f ( | x | ) is defined as

Let g ( x ) = f ( x ) − 1 . If f ( x ) + f ( 1 − x ) = 2 ∀ x ∈ R , then g ( x ) is symmetrical about

Consider the real-valued function satisfying 2 f ( sin ⁡ x ) + f ( cos ⁡ x ) = x . Then, which of the following is not true?

The relation ‘less than’ on the set of natural numbers is

If R be a relation < from A = { 1 , 2 , 3 , 4 } to B = { 1 , 3 , 5 } , that is, ( a , b ) ∈ R ⇔ a < b , then R o R − 1 is

Let R be a relation on a set A such that R = R − 1 , then R is

The relation R = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 1 , 2 ) , ( 2 , 3 ) , ( 1 , 3 ) } on set A = { 1 , 2 , 3 } is

Let P = ( x , y ) ∣ x 2 + y 2 = 1 , x , y ∈ R . Then, P is

Let R be a relation on the set N be defined by { ( x , y ) ∣ x , y ∈ N , 2 x + y = 41 } . Then, R is

The void relation on a set A is

Which one of the following relations on R is an equivalence relation?

. Let R be a relation on the set N of natural numbers denoted by n R m ⇔ n is a factor of m (i.e., n ∣ m ) . Then, R is

The domain of f ( x ) = x 2 + | x + 3 | + x x + 2 − 1 is

The domain of | x − 2 | − 1 + 3 − | x − 2 | is

The domain of definition of the function y = 1 log 10 ⁡ ( 1 − x ) + x + 2 is

The domain of definition of f ( x ) = log 2 ⁡ ( x + 3 ) x 2 + 3 x + 2 is

The domain of the definition of the function f ( x ) = log 4 ⁡ log 5 ⁡ log 3 ⁡ 18 x − x 2 − 77 is

Let f ( x ) = cos − 1 ⁡ x 2 1 + x 2 . The range of f is

The domain of f ( x ) = log ⁡ | log ⁡ x | is

If f ( x ) = 1 ( x + 1 ) e x − 1 ( x − 4 ) ( x + 5 ) ( x − 6 ) , then the domain of f ( x ) is

If f ( x ) = cos ⁡ log e ⁡ x , then f ( x ) f ( y ) − 1 2 f x y + f ( x y ) h a s v a l u e

Range of the expression f ( x ) = x 3 + x 2 + x − 3 x − 1 is

Let f ( x ) = 9 x 9 x + 3 . Then f ( x ) + f ( 1 − x ) =

Let f ( x ) = min . { 4 x + 1 , x + 2 , − 2 x + 4 } . Then the maximum value of f ( x ) is

The function f ( x ) = tan ⁡ x 11 e x 5 sgn ⁡ x 11 ⋅ 1 3 x 2 + 2 , where [.] denotes greatest integer function, is

If f ( x + 2 a ) = f ( x − 2 a ) , then f ( x ) is

The period of the function sin 3 ⁡ x 2 + cos 5 ⁡ x 5 is

Let ⁡ f ( x ) = [ a ] 2 − 5 [ a ] + 4 x 3 − 6 { a } 2 − 5 { a } + 1 x − ( tan ⁡ x ) × sgn ⁡ x be an even function for all x ∈ R . Then the sum of all possible values of a is (where [ ] and { ⋅ } denote greatest integer function and fractional part function, respectively)

If f ( x ) and g ( x ) are periodic functions with periods 7 and 11 , respectively, then the period of F ( x ) = f ( x ) g x 5 − g ( x ) f x 3 is

If the graph of the function f ( x ) = a x − 1 x n a x + 1 is symmetrical about the y -axis, then n equals

If f ( x ) = sin ⁡ x + tan ⁡ x 2 + sin ⁡ x 2 2 + tan ⁡ x 2 3 + … + sin ⁡ x 2 n − 1 + tan ⁡ x 2 n is a periodic function with period k π , then k =

What is the fundamental period of f ( x ) = sin ⁡ x + sin ⁡ 3 x cos ⁡ x + cos ⁡ 3 x ?

If f and g arc one-one functions, then

If f ( x ) = sin ⁡ ( [ x ] π ) x 2 + x + 1 , where [.] denotes the greatest integer

Let f : R R and g : R R be two given functions such that f is injective and g is surjective. Then which of the following is injective?

Let f : R R be a continuous and differentiable function such that f x 2 + 1 x = 5 for ∀ x ∈ ( 0 , ∞ ) . Then the value of f 16 + y 2 y 2 4 y for y ∈ ( 0 , ∞ ) is equal to

If f ( x ) = 2 − x ; x < 0 x 2 − 4 x + 2 ; x ≥ 0 , then the value of f ( f ( f ( 1 ) ) ) i s

If f ( x ) = 5 log 5 ⁡ x then f − 1 ( α − β ) where α , β ∈ R is equal to

If f ( x ) = x 1 + x 2 , then ( fofof ) ( x ) =

Find the domain of the function f ( x ) = x − 3 ( x + 3 ) x 2 − 4

Find the domain of the function f ( x ) = 1 | x − 2 | − ( x − 2 )

Given a non empty set & consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only it A ⊂ B. Then R is not

f:R R, f(x 2 + x + 3 ) + 2 f ( x 2 – 3 x + 5 ) = 6 x 2 – 10 x + 17 ∀ x ∈ R t h e n t h e v a l u e o f f ( 100 ) i s

The range of the y = 9 − x 2 is

The range of the function y = x 2 − 2 x + 10 is

The function f(x)= x + 1 x 3 + 1 c a n b e w r i t t e n a s t h e s u m o f a n e v e n f u n c t i o n g ( x ) a n d a n o d d f u n c t i o n h ( x ) . T h e n t h e v a l u e o f g ( 0 ) i s

The range of the function f ( x ) = 1 − x 2 x 2 + 3 is

The range of f ( x ) = sin 2 ⁡ x − sin ⁡ x + 1 i s

The domain of f ( x ) = cos ( sin x ) is

The range of tan − 1 ⁡ 2 x 1 + x 2 is

Find the range of f ( x ) = cot − 1 ⁡ 2 x − x 2 .

The domain of f ( x ) = ( 0.625 ) 4 − 3 x − ( 1.6 ) x ( x + 8 )

The domain of f ( x ) = log 0.4 ⁡ x − 1 x + 5 is

T h e d o m a i n o f f ( x ) = ( [ x ] − 1 ) + ( 4 − [ x ] ) ( w h e r e [ ] r e p r e s e n t s t h e g r e a t e s t i n t e g e r f u n c t i o n ) .

The range of f(x) = log ⁡ { x } , where { } represents the fractional part function).

If f(x)= 4 – x 2 + x 2 – 1 , t h e n t h e m a x i m u m v a l u e o f ( f ( x ) ) 2 i s

Number of integral values of x satisfying the inequality 3 4 6 x + 10 − x 2 < 27 64 is

Let R be a relation on the set N of natural numbers denoted by n R m ⇔ n is a factor of m (i.e., n ∣ m ) . Then, R is

The domain of definition of the function y = 1 log 10 ⁡ ( 1 − x ) – x + 2 is

Let R be a relation over the set N × N and it is defined by ( a , b ) R ( c , d ) ⇒ a + d = b + c . Then, R is

The range of the function f ( x ) = e x − e | x | e x + e | x | is

Let f : − π 3 , 2 π 3 [ 0 , 4 ] be a function defined as f ( x ) = 3 sin ⁡ x − cos ⁡ x + 2 . Then f − 1 ( x ) is given by

If the functions f ( x ) and g ( x ) are defined on R R such that f ( x ) = 0 , x ∈ rational x , x ∈ irrational and g ( x ) = 0 , x ∈ irrational x , x ∈ rational then ( f − g ) ( x ) is

If f ( x ) = 2 x 3 + 7 x − 5 , then f − 1 ( 4 ) is

A relation R on the set of complex numbers defined by Z 1 R Z 2 ⇔ Z 1 − Z 2 Z 1 + Z 2 is real then which of the following is not true

Let R be the relation on the set of all real numbers defined by a R b iff | a − b | ≤ 1 . Then, R is

The range of the function, f ( x ) = cot − 1 ⁡ log 0.5 ⁡ x 4 − 2 x 2 + 3 i s

The domain of f ( x ) = 1 | cos ⁡ x | + cos ⁡ x is

If f : [ 0 , ∞ ) [ 0 , ∞ ) and f ( x ) = x 1 + x , then f is

f : N N , where f ( x ) = x − ( − 1 ) x . Then f is

If a function f : [ 2 , ∞ ) B defined by f ( x ) = x 2 − 4 x + 5 is a bijection, then B is equal to

Let f : R R , be defined as f ( x ) = e x 2 + cos ⁡ x , then f is

If f ( x ) = 2 x + | x | , g ( x ) = 1 3 ( 2 x − | x | ) and h ( x ) = f ( g ( x ) ) then domain of sin − 1 ⁡ ( h ( h ( h ( h … h ( x ) … ) ) ) ) ⏟ n times is

Let f ( x ) = sin ⁡ x and g ( x ) = log e ⁡ | x | . If the ranges of the composition functions fog and gof are R 1 and R 2 , respectively, then

Find the domain of the function f ( x ) = 2 x 2 − x + 1 − 1 x + 1 − 2 x − 1 x 3 + 1

The domain of f ( x ) = sin − 1 ⁡ log 9 ⁡ x 2 / 4 is

Number of integral values of a for which f(x)=log(log 1 3 ( l o g 7 ( s i n x + a ) ) ) b e d e f i n e d f o r e v e r y r e a l v a l u e o f x i s

Let A = A 1 , A 2 , A 3 , A 4 , A 5 , A 6 be the set of six unit circles with centres C 1 , C 2 , C 3 , … … . . C 6 arranged as shown in the diagram. The relation R on A is defined by A i , A j ∈ R ⇔ C i C j ≤ 2 2 then

Let f be a function satisfying f ( x y ) = f ( x ) y for all positive real numbers x and y . If f ( 30 ) = 20 , then the value of f ( 40 ) is

If A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3)} then the relation R on the set A is

f : R R f x 2 + x + 3 + 2 f x 2 − 3 x + 5 = 6 x 2 − 10 x + 17 ∀ x ∈ R then the value of f ( 100 ) is

Let f : ( 2 , 4 ) ( 1 , 3 ) where f ( x ) = x − x 2 (where [ ⋅ ] denotes the greatest integer function), then f − 1 ( x ) is

Let f ( x ) = a x + b and g ( x ) = c x + d , a ≠ 0 , c ≠ 0 . Assume a = 1 , b = 2 . If ( f ∘ g ) ( x ) = ( g o f ) ( x ) for all x , what can you say about c and d ?

f is a real-valued function not identically zero, satisfying f ( x + y ) + f ( x − y ) = 2 f ( x ) ⋅ f ( y ) ∀ x , y ∈ R ⋅ f ( x ) is definitely

Let f x + 1 y + f x − 1 y = 2 f ( x ) f 1 y ∀ x , y ∈ R y ≠ 0 and f ( 0 ) = 0 then the value of f ( 1 ) + f ( 2 ) =

If f x 2 − 6 x + 6 + f x 2 − 4 x + 4 = 2 x ∀ x ∈ R then f ( − 3 ) + f ( 9 ) − 5 f ( 1 ) =

g ( x ) is symmetrical about Let g ( x ) = f ( x ) − 1 . If f ( x ) + f ( 1 − x ) = 2 ∀ x ∈ R , then

which pair of functions is identical?

which of the following statements is incorrect

If f ( x ) = x 2 , for x ≥ 0 x , for x < 0 , then f o f ( x ) is given by

Let f ( x ) be defined for all x > 0 and be continuous. Let f ( x ) satisfies f x y = f ( x ) − f ( y ) for all x , y and f ( e ) = 1 . T h e n

If f ( x ) = a x 7 + b x 3 + c x − 5 , a , b , c are real constants, and f ( − 7 ) = 7 , then the range of f ( 7 ) + 17 cos ⁡ x is

If e f ( x ) = 10 + x 10 − x , x ∈ ( − 10 , 10 ) and f ( x ) = k f 200 x 100 + x 2 then k =

If f ( x ) is a real-valued function defined as f ( x ) = ln ⁡ ( 1 − sin ⁡ x ) , then the graph of f ( x ) is

Let f : ( − ∞ , 0 ] [ 1 , ∞ ) be defined as f ( x ) = ( 1 + − x ) − ( − x − x ) , then f ( x ) is

Let G ( x ) = 1 a x − 1 + 1 2 F ( x ) , where a is a positive real number not equal to 1 and F ( x ) is an odd function. Which of the following statements is true?

Which of the following statements are incorrect? I. If f ( x ) and g ( x ) are one-one then f ( x ) + g ( x ) is also one-one II. If f ( x ) and g ( x ) are one-one then f ( x ) ⋅ g ( x ) is also one-one III. If f ( x ) is odd then it is necessarily one-one.

Which of the following functions is one-one?

If f ( x ) = x 2 + x + 3 4 and g ( x ) = x 2 + a x + 1 be two real functions, then the range of a for which g ( f ( x ) ) = 0 has no real solution is

Let f : [ − 2 , 2 ] R be defined by f ( x ) = − 1 , − 2 ≤ x < 0 x − 1 , 0 ≤ x ≤ 2 . Then sum of the roots of the equation f ( | x | ) = x is

If the function f ( x ) = x + 1 if x ≤ 1 2 x + 1 if 1 < x ≤ 2 and g ( x ) = x 2 − 1 ≤ x ≤ 2 x + 2 2 ≤ x ≤ 3 then the number of roots of the equation f ( g ( x ) ) = 2 is

Range of the function f ( x ) = log 2 ⁡ 2 − log 2 ⁡ 16 sin 2 ⁡ x + 1 is :

The domain of the function f x = 1 C x – 1 10 – 3 × 10 C x is

The domain of f x = log 2 x + 3 x 2 + 3 x + 2 is

The domain of the function f x = x 2 – x 2 , where [ x ] is the greatest integer less than or equal to x , is

The domain of the function f x = log 10 5 x – x 2 4 1 / 2 is

The domain of f x = cos – 1 2 – x 4 + log 3 – x – 1 is

The domain of the function f x = log 1 sin x

Let f : R 0 , π 2 o. defined by f x = tan – 1 x 2 + x + a . Then the set of values of a for which f is onto is

The domain of the function f x = l n x – 1 x 2 + 4 x + 4 is

The domain of f x = l n a x 3 + a + b x 2 + b + c x + c , where a > 0 , b 2 – 4 a c = 0 , is (where [.] represents greatest integer function)

The number of real solutions of the log 0 . 5 x = 2 x is

The domain of f x = 1 cos x + cos x is

If x is real, then the values of the expression x 2 + 14 x + 9 x 2 + 2 x + 3 are

The function f : R R is defined by f x = cos 2 x + sin 4 x or x ∈ R Then the range of f x is

The range of the function f x = x – 1 + x – 2 , – 1 ≤ x ≤ 3 is

The range of f x = sin – 1 x 2 + 1 x 2 + 2 is

The range of f x = sin x + cos x , where [.] denotes the greatest integer function, is

The range of the function, f x = e x – e x e x + e x

The range of the function f defined by f x = 1 sin x (where [.] and {.}, respectively, denote the greatest integer and the fractional part functions) is

The domain of definition of the function f x = x x + x x is (where {.} represents fractional part and [.] represents greatest integral function)

Let f x = x – x (where {.} denotes the fractional part of x ) and X, Y are its domain and range, respectively). Then

If [ x ] and { x } represent the integral and fractional parts of x , respectively, then the value of ∑ r = 1 2000 x + r 2000 is

Let, P = x , y | x 2 + y 2 = 1 , x , y ∈ R . Then P is

Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is

For real numbers x and y, we write xRy ⇔ x – y + 2 is an irrational number. Then the relation R is

If R be a relation < from A = 1 , 2 , 3 , 4 to B = 1 , 3 , 5 i.e., a , b ∈ R ⇔ a < b , then RoR – 1 is

Let R be a reflexive relation on a finite set A having elements, and let there be m ordered pairs in R. Then

Let R be reflexive relation on a set A and I be the identity relation on A. Then

Let A = 1 , 2 , 3 , 4 and R be a relation in A given by R = 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 1 , 2 , 2 , 1 , 3 , 1 , 1 , 3 . Then R is

An integer m is said to be related to another integer n if m is a multiple of n. Then the relation is

The relation R defined in N as aRb ⇔ b is divisible by a is

Let R be a relation on a set A such that R = R – 1 , then R is

Let R = { a , a } be a relation on a set A. Then R is

The relation “is subset of” on the power set P(A) of a set A is

The relation R defined on a set A is antisymmetric if ( a , b ) ∈ R ⇒ ( b , a ) ∈ R for

Let A be the non void set of the children in a family. The relation ‘x is a brother of y’ on A is

Let R 1 be a relation defined by R 1 = { a , b | a ≥ b , a , b ∈ R } . Then R 1 is

Which one of the following relations on R is an equivalence relation

If R is an equivalence relation on a set A, then R – 1 is

R is a relation over the set of real numbers and it is given by nm ≥ 0 . Then R is

In order that a relation R defined on a non-empty set A is an equivalence relation, it is sufficient, if R

The relation “congruence modulo m” is

Let R = 3 , 3 , 6 , 6 , 9 , 9 , 12 , 12 , 6 , 12 , 3 , 9 , 3 , 12 , 3 , 6 be a relation on the set A = 3 , 6 , 9 , 12 . The relation is

Let R = 1 , 3 , 4 , 2 , 2 , 4 , 2 , 3 , 3 , 1 be a relation on the set A = 1 , 2 , 3 , 4 . The relation R is

Let S be the set of all real numbers. Then the relation R = a , b : 1 + ab > 0 on S is

Let R and S be two non-void relations on a set A. Which of the following statements is false

Let R be a relation on the set N of natural numbers defined by nRm ⇔ n is a factor of m (i.e., n\m). Then R is

Let R be a relation on the set N be defined by x , y | x , y ∈ N , 2 x + y = 41 . Then R is

The number of symmetric relations defined on the set S = {a, b, c} is

In order that a relation R defined on a non-empty set A is an equivalence relation. It is sufficient, if R

If f : A B is a bijective function, then f – 1 o f is equal to

Let f : R R is a function defined by f ( x ) = 3 x – 4 , then f – 1 ( x ) is

The period of sin π x 12 + cos π x 4 + tan π x 3 , Where x represents the greatest integer less than or equal x is

If f (2x+3y, 2x-7y) =20x then f(x, y) equals

If f : { 1 , 2 , 3 , … . . } { 0 , ± 1 , ± 2 , … . } is defined by f ( n ) = n / 2 if n is even − n − 1 2 if n is odd then f − 1 ( − 100 ) is

The domain of f(x) = log sin – 1 x 2 + x + 1 log ( x 2 – x + 1 ) is

For, x ∈ R , x ≠ 0 , x ≠ 1 , let f 0 ( x ) = 1 1 − x and f n + 1 ( x ) = f 0 f n ( x ) , n = 0 , 1 , 2 , ….Then the value of f 100 ( 3 ) + f 1 2 3 + f 2 3 2 is equal to :

The number of one one onto functions that can be defined from {1, 2, 3, 4) onto set B is 24 then n(B) =

Let A be the set of first 10 natural numbers and let R = { ( x , y ) / x ∈ A , y ∈ N and x + 2 y = 10 } then n dom ⁡ R − 1 =

If f : R R and F(x) = sin ( π { x } ) X 4 + 3 x 2 + 7 , Where { } is a fractional part of x , then

If n ( A ) = 4 , n ( B ) = 3 , n ( A × B × C ) = 24 , then n(C) is equal to

If A={1, 2, 3}, the number of reflexive relation in A is

The domain of the function f(x) = Cos -1 (sec (cos -1 X)) + sin -1 (Cosec(sin -1 X)) is

The domain of definition of function f ( x ) = log x 2 – 5 x – 24 – x – 2 , is

Let f : X Y , f ( x ) = sin x + cos x + 2 2 be invertible then X Y is /are.

Let A = {2,3,4,……..30} and ≃ be an equivalence relation on A × A , defined by a , b ≃ c , d , if and only if ad=bc . Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4,3) is equal to

If f ( x ) = sin 2 ⁡ x and the composite functions g ( f ( x ) ) = | sin ⁡ x | then g(x) is equal to

The domain of the function f ( x ) = sin − 1 ⁡ log 2 ⁡ 1 2 x 2 is

The domain of the function f ( x ) = 1 x 12 − x 9 + x 4 − x + 1 is

The domain of the function f ( x ) = cos − 1 ⁡ 3 4 + 2 sin ⁡ x is

The domain of the function f ( x ) = 1 − 1 − 1 − x 2 is

The domain of the function f ( x ) = log 2 ⁡ log 2 ⁡ log 2 ⁡ … log 2 ⁡ x ⏟ n times is

The domain of the function f ( x ) = log 3 ⁡ − log 3 ⁡ x 2 + 5 log 3 ⁡ x − 6 is

The range of the function y = sin − 1 ⁡ x 2 1 + x 2 is

If [2 sin x] + [cos x] = –3, then the range of the function f ( x ) = sin ⁡ x + 3 cos ⁡ x in [ 0 , 2 π ] (where [⋅] denotes the greatest integer function) is

The range of the function y = e − x 1 + [ x ] is

The inverse of the function f ( x ) = a x − a − x a x + a − x is

The function f ( x ) = sec ⁡ log ⁡ x + 1 + x 2 is

A function whose graph is symmetrical about the origin is given by

Let the function f (x) = 3 sin x – 4 cos x + log (|x| + 1 + x 2 be defined on the interval [0, 1]. The odd extension of f (x) to the interval [– 1, 1] is

The period of the function f ( x ) = sin ⁡ x is

The value of n ∈ I for which the function f ( x ) = sin ⁡ nx sin ⁡ x n has 4 π as its period is

π is the period of the function

The number of solutions of the equation a f (x) + g (x) = 0, a > 0, g (x) ≠ 0 and has minimum value 1/2 is

If f (x + y, x – y) = xy, then the arithmetic mean of f (x, y) and f ( y, x) is

If f : R R is a function satisfying the property f (2x + 3) + f (2x + 7) = 2, ∀ x ∈ R, then the period of f (x) 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 + 5 { f (x)} 4 – { f (x)}5] 1/5 for all real x and some positive constant k, then the period of the function f (x) is

If X = {1, 2, 3, 4,}, then a one-one onto mapping f : X X such that f ( 1 ) = 1 , f ( 2 ) ≠ 2 a n d f ( 4 ) ≠ 4 is given by

Let f(x) = (x + 1) 2 – 1, x ≥ –1 then the set {x : f(x) = f –1 (x)} is equal to

Let R be a relation on R defined as a R b if a ≤ b . Then, relation R is

The relation R defined by ‘ > ‘ on the set N is

The domain of the function f ( x ) = sin – 1 x – 3 9 – x 2

The relation a R b defined by a is factor of b on N is not

The domain of y = cos – 1 ( 1 – 2 x ) is

If f ( x ) = sin x – cos x is written as f 1 ( x ) + f 2 ( x ) where f 1 ( x ) is even and f 2 ( x ) is odd then

Let X = {1, 2, 3, 4}. The number of equivalence relations that can be defined on X is

The function f ( x ) = x 2 log 1 – x 1 + x is

The domain of y = cos – 1 1 – 2 x 4 is

Which of the following functions is bounded

If f ( x ) 2 f 1 – x 1 + x = x 3 , x ≠ – 1 , 1 and f ( x ) ≠ 0 , then { f ( – 2 ) } (the fractional part of f(-2) is equal to

let f ( θ ) = sin 2 ⁡ θ cos 2 ⁡ θ 1 + 4 sin ⁡ 4 θ sin 2 ⁡ θ 1 + cos 2 ⁡ θ 4 sin ⁡ 4 θ 1 + sin 2 ⁡ θ cos 2 ⁡ θ 4 sin ⁡ 4 θ then f is

Consider the following relations. R = {(x, y) | x, y are real numbers and x = wy for some rational number w} S = m n , p q m , n , p , q are integer such that n . q ≠ 0 and q m = p n } Statement-1: S is an equivalence relation but R is not an equivalence relation. Statement-2: R and S both are symmetric.

Let R be a relation on the set N of natural numbers defined by n R m ⇔ n is a factor of m (i.e. n | m). Statement-1: R is not an equivalence relation Statement-2: R is not symmetric

Let I be the set of integers. For a , b ∈ I , a R b if and only if | a – b | < 1 , then

Let W denote the words in the English Dictionary. Define the relation R b y R = { ( x , y ) ∈ W x W : the words x and y have at least one letter common}, then R is

The range of the function f ( x ) = 7 − x P x − 3 is

If f : R R defined by f ( x ) = x 4 + 2 then the value of f – 1 ( 83 ) and f – 1 ( – 2 ) respectively are

Let X be a non-empty set and P ( X ) be the set of all subsets of X . For A , B ∈ P ( X ) , A R B if and only if A ∩ B = ∅ then the relation

Let f be a function satisfying 2 f ( x ) − 3 f ( 1 / x ) = x 2 for any x ≠ 0 . Then the value of f ( 2 ) is

Let R = { ( x , y ) : x , y ∈ A , x + y = 5 } where A = { 1 , 2 , 3 , 4 , 5 } then

The domain of definition of the functions y ( x ) given by the equation a x + a y = a ( a > 1 ) is

If 2 f x 2 + 3 f 1 x 2 = x 2 − 1 , then f ( x 2 ) is

For x , y ∈ R , define a relation R by x R y if and only if x − y + 2 is an irrational number. Then R is

If n ( A ) = 3 and n ( B ) = 5 then number of one-one functions that can be defined from A to B is

If the function f : [ 1 , ∞ ) [ 1 , ∞ ) is defined by f ( x ) = 2 x ( x − 1 ) then f − 1 ( x ) is

Let x , y ∈ I and suppose that a relation R on I is defined by x R y if and only if x ≤ y then

If f : R R is defined by f ( x ) = x 2 + 1 then value of f − 1 ( 17 ) and f − 1 ( − 3 ) are, respectively.

If f : R R given by f ( x ) = a x + sin ⁡ x + a , then f is one-one and onto for all

The functions f and g are given by f ( x ) = { x } the fractional part of x and g ( x ) = 1 2 sin ⁡ [ x ] π , where [ x ] denotes the integral part of x . Then range of g o f is

The set of all x for which f ( x ) = log x − 2 x + 3 ⁡ 2 and g ( x ) = 1 x 2 − 9 are both not defined is

The period of the function f ( x ) = cos 2 ⁡ 3 x + tan ⁡ 4 x is

Let f : ( 1 , ∞ ) ( 1 , ∞ ) be defined by f ( x ) = x + 2 x − 1 . Then

Given f ( x ) = 1 | x | − x and g ( x ) = 1 x − | x | then

If f ( x + y ) = f ( x y ) ∀ x , y ε R , and f ( 2013 ) = 2013, then f(– 2013) equals

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