Relations and Functions

Let A = {1, 2, 3}. Which of the following relations is an equivalence relation?

1. {(1,1), (2,2), (3,3)}
2. {(1,2), (2,3), (3,1)}
3. {(1,2), (2,1)}
4. {(1,1), (1,2), (2,2)}

If f(x) = x² - 4, then the value of f(-3) is

1. 5
2. 9
3. 13
4. -13

The total number of functions from a set A containing 2 elements to a set B containing 3 elements is

1. 6
2. 8
3. 9
4. 27

Which of the following functions is one-one on R?

1. f(x) = x²
2. f(x) = |x|
3. f(x) = 2x + 1
4. f(x) = x² + 1

Let f: R → R be defined by f(x) = 5x - 4. The inverse function f⁻¹(x) is

1. (x + 4)/5
2. (x - 4)/5
3. 5x + 4
4. (4 - x)/5

Which of the following relations on the set A = {1, 2, 3} is symmetric?

1. {(1,2), (2,3)}
2. {(1,2), (2,1), (3,3)}
3. {(1,1), (2,2)}
4. {(1,2), (2,3), (3,1)}

The domain of the function f(x) = 1/(x - 5) is

1. All real numbers
2. x ≠ 0
3. x ≠ 5
4. x > 5

If f(x) = x² and g(x) = x + 1, then (f o g)(2) equals

1. 4
2. 8
3. 9
4. 16

Let f(x) = x + 2 and g(x) = 3x. Then (g o f)(2) is

1. 6
2. 8
3. 10
4. 12

Let f: R → R be defined by f(x) = 3x - 7. The value of f(-2) is

1. -1
2. -13
3. 13
4. -6

If f(x) = 2x + 5, then the value of f(3) is

1. 6
2. 8
3. 11
4. 13

Which of the following is a function from R to R?

1. f(x) = ±√x
2. f(x) = 1/x
3. f(x) = x²
4. f(x) = √x

If A = {1, 2} and B = {3, 4, 5}, then the number of elements in A x B is

1. 5
2. 6
3. 7
4. 8

Let R = {(1,1), (2,2), (3,3)} on the set A = {1,2,3}. Which property does R satisfy?

1. Symmetric only
2. Reflexive only
3. Reflexive, symmetric and transitive
4. Transitive only

Let A = {1, 2, 3} and B = {a, b}. How many relations are possible from A to B?

1. 8
2. 16
3. 64
4. 4096

If f(x) = (2x-1)/(x+5) (x ≠ -5), then f⁻¹(x) is equal to

1. (x+5)/(2x-1)
2. (5x+1)/(2-x)
3. (x+1)/(5x-2)
4. none of these

If f(x) = px/(p-1) where p is odd function, then the value of parameter P is

1. P ≥ 1/2
2. P ≤ 1/2
3. P > 5
4. none of these

Let f(x) = sin[x²/9] and g(x) = sin(x) where [.] denotes the greatest integer function. The value of R(f∘g) is

1. [-1, 1]
2. {sin(n²/9): n ∈ Z}
3. (-1, 1)
4. {0}

Let f(x) = |x|, then which of the following is true

1. f(x+y) = f(x)+f(y)
2. f(xy) = f(x)·f(y)
3. f(|x|) = f(x)
4. none of these

If f(x) = (x+1)/(x-1) and g(x) = 1/f(x), then fog and gof is

1. fog = gof = identity
2. fog ≠ gof
3. fog = identity, gof ≠ identity
4. none of these

Which of the following is not an invertible function?

1. f: R → R, f(x) = x
2. f: R → R⁺, f(x) = eˣ
3. f: R⁺ → R⁺, f(x) = x²
4. none of these

Let f: R → R be defined by f(x) = 2x+sinx for x ∈ R. Then f is

1. one-one and onto
2. one-one but not onto
3. onto but not one-one
4. neither one-one nor onto

The domain of f(x) = sin⁻¹(log₃(x/3)) is

1. [1, 9]
2. [-1, 9]
3. [-9, 1]
4. [-9, -1]

If f and g are two functions such that f(g(x)) = x for all x in domain of g, then

1. f and g are inverse of each other
2. fog = gof = x
3. f is one-one and g is onto
4. none of these

If f(x) = (a-xⁿ)^(1/n) and g(x) = (b-xⁿ)^(1/n), then f(g(x)) + g(f(x)) is equal to

1. a+b
2. 0
3. a-b
4. none of these

If f(x+y) = f(x)·f(y) and f(1) = 2, then f(n) is equal to

1. 2ⁿ
2. n·2
3. 2^(n-1)
4. an odd function

If f(x) = x - [x] where [.] denotes greatest integer function, then f(x) is periodic with period 1 — is this true, and what is the range?

1. f(x) is periodic with period 1; range is [0,1)
2. f(x) is an even function
3. range of f(x) is given by (0,1)
4. f(x) is neither even nor odd

The image of the interval [-1, 3] under mapping f given by f(x) = 4x³ - 12x is

1. [8, 72]
2. [-8, 72]
3. [0, 8]
4. none of these

Let f(x) = sin(ln x) and g(x) = [x²-1] where [.] denotes the greatest integer function. The value of R(fog) is

1. [-1, 1]
2. [-sin1, sin1]
3. {sin(ln n): n ∈ Z}
4. none of these

Let f: R → R be defined by f(x) = 2x³-7 and g: R → R be defined by g(x) = ((x+7)/2)^(1/3). Then fog and gof are

1. fog = gof = identity
2. fog = identity, gof ≠ identity
3. fog ≠ identity, gof = identity
4. none of these

Let f: R → R be defined by f(x) = x³+5 and g: R → R be defined by g(x) = (x-5)^(1/3). Then fog and gof are

1. fog = gof = identity
2. fog = gof ≠ identity
3. fog ≠ gof
4. none of these

Let f: R → R and g: R → R be two functions given by f(x) = 2x-3, g(x) = x³+5. Then (fog)⁻¹(x) is equal to

1. ((x+7)/2)^(1/3)
2. ((x-7)/2)^(1/3)
3. ((x+2)/7)^(1/3)
4. ((x-2)/7)^(1/3)

The domain of f(x) = ¹⁰Cₓ + ¹⁰C₅₋ₓ / (x² - 5x + 4) is

1. (0, 1] ∪ [4, 5)
2. (0, 5)
3. {1, 4}
4. none of these

f(x) = sin⁻¹(x-1) + cos⁻¹(x-1) is equal to

1. π
2. π/2
3. 0
4. none of these

If f(x) = log((1+x)/(1-x)), then f(2x/(1+x²)) equals

1. 0
2. 1
3. x
4. 2f(x)

Function f(x) = sin⁻¹(2x/(1+x²)) is

1. one-one onto
2. one-one into
3. many one onto
4. many one into

If the real valued function f(x) = (xⁿ - sinx)/(xⁿ + sinx) is even, then n equals

1. 2
2. 2/3
3. 1/4
4. -1/3

The domain of the real-valued function f(x) = √(log₁/₂(x²-3x+3)) is

1. (1, 2)
2. [1, 2]
3. R
4. none of these

Range of f(x) = sin[x] (where [.] denotes the greatest integer function) is

1. {sin n : n ∈ Z}
2. [-1, 1]
3. [0, 1]
4. none of these

Period of the function f(x) = sin(x - [x])·π where [x] denotes the integral part of x is

1. 1
2. 2
3. 1/2
4. 2π

If f: R → R is defined by f(x) = x - [x] for all x ∈ R; then f is one-one and onto — is this true?

1. f is onto but not one-one
2. f is one-one but not onto
3. f is one-one and onto
4. f is neither one-one nor onto

If f(x) = [x] where [x] stands for the greatest integer function, then f(1/2) + f(-1/2) equals

1. 0
2. -1
3. 1
4. none of these

If f(x) = sin²x and g(x) = √x, where [.] denotes the greatest integer function, then Rf(gof) equals

1. [0, 1]
2. (0, 1)
3. [0, 1)
4. none of these

If f(x) = 1/(2-x) + 1/(x+2) for x ∈ R ~ {2, -2}, then f is

1. an increasing function
2. a decreasing function
3. both increasing and decreasing function
4. none of these

Which of the following functions is not periodic?

1. |sin x|
2. sin(x²)
3. sin²x
4. cos(sin x)

R is the set of real numbers and f and g are defined by f(x) = x² + 1 and g(x) = 2x for all x ∈ R. Then (fog)(x) is

1. 4x² + 1
2. 2x² + 2
3. (2x+1)²
4. none of these

The domain of the real-valued function f(x) = √(2-|x|)/(x-1) is

1. [-√2, √2] ~ {1}
2. (-∞, 1) ∪ (1, ∞)
3. [-2, 2]
4. none of these

The solution set of |x-1| + |x+1| = 2 is

1. {0}
2. [-1, 1]
3. R
4. none of these

If x > 0, then the minimum value of the function f(x) = x + 1/x is

1. 2
2. 1/2
3. -2
4. none of these

The domain of the function f(x) = log(1-x) + √(x²-1) is

1. (-3, -2) ~ {-2.5}
2. [0, 1] ~ {0.5}
3. [-2, 1) ~ {0}
4. none of these

Range of the function f: R → R defined by f(x) = (x²+x+2)/(x²+x+1) is

1. (1, ∞)
2. (1, 3/2)
3. (1, 7/3]
4. (1, 7/5]

If f(x) = [x - 2] + [x + 2] + {x}, then f(x) (where [.] denotes greatest integer function)

1. is periodic with period 1
2. domain of f is given by R
3. range of f is given by Z
4. none of these

Let function f: R → R be defined by f(x) = 2x + sinx for x ∈ R. Then f is

1. one-one and onto
2. one-one but not onto
3. onto but not one-one
4. neither one-one nor onto

Let f: R → R. Then the value of x for which f(f(x)) = x, is

1. 1/2
2. 1/3
3. 1
4. -1

If f(x) = x/(1+|x|), then

1. f(x) is injective
2. f(x) is surjective
3. f(x) is bijective
4. none of these

Range of f(x) = [x] - x is (where [.] denotes the greatest integer function)

1. {0}
2. (-1, 0]
3. [0, 1)
4. none of these

Period of the function f(x) = sin(x/3) + cos(x/2) where [x] denotes the integral part of x is

1. 1
2. 2
3. 3
4. 6π

The period of the function f(x) = sin(2x)/3 + cos(3x)/2, is

1. 2π
2. 10π
3. 6π
4. 6π

If f(ax+b) = cx+d and f(cx+d) = ax+b for all x ∈ R, then f(x) + f(-x) is equal to

1. 2(ax+b)
2. 0
3. 2(cx+d)
4. constant

The function f(x) = log(x + √(x²+1)) satisfies the equation

1. f(x+2) - 2f(x+1) + f(x) = 0
2. f(x+y) = f(x)·f(y)
3. f(x+y) = f(x) + f(y)
4. none of these

Domain and range of f(x) = sin⁻¹(log₂x) are respectively

1. [1/2, 2], [-π/2, π/2]
2. [-1/2, 2], [-π, π]
3. [1/2, 2], [0, π]
4. none of these

If f(x) = ax/(ax+√a) (a > 0), then f(x) + f(1-x) is equal to

1. 1/2
2. 1
3. √a
4. a

The domain of definition of the function f(x) = √((x-1)(4-x)) is

1. [1, 4]
2. (1, 4)
3. [0, 5]
4. (0, 5)

Let f: [0,1] → [0,2] be defined by f(x) = 1 + x, then f is

1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto

All the values of m for which both roots of the equation x² - 2mx + m² - 1 = 0 are greater than -2 but less than 4, lie in the interval

1. m > 3
2. -1 < m < 3
3. 1 < m < 4
4. -2 < m < 0

Consider the following relations: (1) A ⊂ B and B ⊂ C ⟹ A ⊂ C, (2) A ∩ B = A ∩ C does not imply B = C, (3) A ∪ B = A ∪ C ⟹ B = C. Which of these is/are correct?

1. (1) and (3)
2. (2) only
3. (2) and (3)
4. (1) and (2)

Let W denote the words in English dictionary. Define the relation R by R = {(x,y) ∈ W × W | the words x and y have at least one letter in common}. Then R is

1. reflexive, symmetric and not transitive
2. reflexive, symmetric and transitive
3. reflexive, not symmetric and transitive
4. not reflexive, symmetric and transitive

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

1. reflexive and symmetric
2. reflexive and transitive
3. symmetric and transitive
4. equivalence relation

Which one of the following relations in Z is an equivalence relation?

1. xR₁y ⟺ |x| = |y|
2. xR₂y ⟺ x > y
3. xR₃y ⟺ x divides y
4. none of these

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

1. reflexive
2. symmetric
3. transitive
4. none of these

The relation R is defined on a plane between two points P and Q if OP = OQ where O is a fixed point on the plane. Then the above relation is

1. only symmetric
2. only reflexive and symmetric
3. symmetric but not transitive
4. reflexive, symmetric and transitive

If A ⊆ B and N is the universal set, then A ∪ (B ∩ N) is

1. A
2. N
3. B
4. none of these

If nRm means that n is a factor of m, then the relation is

1. reflexive and symmetric
2. transitive and symmetric
3. reflexive, transitive and symmetric
4. reflexive, transitive and not symmetric