Give an example of a function

Which is one – one but not onto.

Give an example of a function

Which is not one – one but onto.

Give an example of a function

Which is neither one – one nor onto.

Which of the following functions from A to B are one – one and onto?

f₁ = {(1, 3), (2, 5), (3, 7)}; A = {1, 2, 3}, B = {3, 5, 7}

Which of the following functions from A to B are one – one and onto?

f₂ = {(2, a), (3, b), (4, c)}; A = {2, 3, 4}, B = {a, b, c}

Which of the following functions from A to B are one – one and onto?

f₃ = {(a, x), (b, x), (c, z), (d, z)}; A = {a, b, c, d}, B = {x, y, z}

Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one – one but not onto.

Let A = {–1, 0, 1} and f = {(x, x²) : x ∈ A}. Show that f : A → A is neither one – one nor onto.

Classify the following functions as injection, surjection or bijection:

f : N → N given by f(x) = x²

Classify the following functions as injection, surjection or bijection:

f : Z → Z given by f(x) = x²

Classify the following functions as injection, surjection or bijection:

f : N → N given by f(x) = x³

Classify the following functions as injection, surjection or bijection:

f : Z → Z given by f(x) = x³

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = |x|

Classify the following functions as injection, surjection or bijection:

f : Z → Z, defined by f(x) = x² + x

Classify the following functions as injection, surjection or bijection:

f : Z → Z, defined by f(x) = x – 5

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = sin x

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = x³ + 1

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = x³ – x

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = sin²x + cos²x

Classify the following functions as injection, surjection or bijection:

f: Q – {3} → Q, defined by

Classify the following functions as injection, surjection or bijection:

f : Q → Q, defined by f(x) = x³ + 1

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = 5x³ + 4

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = 3 – 4x

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = 1 + x²

Classify the following functions as injection, surjection or bijection:

f: R → R, defined by

If f: A → B is an injection such that range of f = {a}. Determine the number of elements in A.

Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:

Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:

g(x) = |x|

Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:

h(x) = x²

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective:

{(x, y): x is a person, y is the mother of x}

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective:

{(a, b) : a is a person, b is an ancestor of a}

Let A = {1, 2, 3}. Write all one – one from A to itself.

If f : R → R be the function defined by f(x) = 4x³ + 7, show that f is a bijection.

Show that the exponential function f: R → R, given by f(x) = e^x, is one – one but not onto. What happens if the co – domain is replaced by R₀ ⁺ (set of all positive real numbers).

Show that the logarithmic function f : R₊⁰→ R given by f(x) = log_a x, a > 0 is a bijection.

If A = {1, 2, 3}, show that a one – one function f : A → A must be onto.

If A = {1, 2, 3}, show that an onto function f : A → A must be one – one.

Find the number of all onto functions from the set A = {1, 2, 3, …., n} to itself.

Give examples of two one – one functions f₁ and f₂ from R to R such that f₁ + f₂ : R → R, defined by (f₁ + f₂)(x) = f₁(x) + f₂(x) is not one – one.

Give examples of two surjective function f₁ and f₂ from Z to Z such that f₁ + f₂ is not surjective.

Show that if f₁ and f₂ are one – one maps from R to R, then the product f₁ × f₂ : R → R defined by (f₁ × f₂)(x) = f₁(x)f₂(x) need not be one – one.

Suppose f₁ and f₂ are non – zero one – one functions from R to R. Is necessarily one – one? Justify your answer. Here, is given by for all x ∈ R.

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

an injective map from A to B

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

a mapping from A to B which is not injective

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

a mapping from A to B.

Show that f : R → R, given by f(x) = x – [x], is neither one – one nor onto.

Let f : N → N be defined by

Show that f is a bijection.

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = 2x + 3 and g(x) = x² + 5

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = 2x + x² and g(x) = x³

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = x² + 8 and g(x) = 3x³ + 1

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = x and g(x) = |x|

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = x² + 2x – 3 and g(x) = 3x – 4

Find gof and fog when f: R → R and g: R → R is defined by

f(x) = 8x³ and g(x) = x^1/3

Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3), (4, 9), (5, 9)}. Show that gof and fog are both defined, Also, find fog and gof.

Let f = {(1, – 1), (4, – 2), (9, – 3), (16, 4)} and g = {(– 1, – 2), (– 2, – 4), (– 3, – 6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.

Let A = {a, b c}, B = {u, v, w} and let f and g be two functions from A to B and from B to A respectively defined as: f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.

Show that f and g both are bijections and find fog and gof.

Find fog (2) and gof (1) when: f: R → R; f(x) = x² + 8 and g: R → R; g(x) = 3x³ + 1.

Let R⁺ be the set of all non – negative real numbers. If f: R⁺ → R⁺ and g: R⁺ → R⁺ are defined as f(x) = x² and g(x) = + √x. Find fog and gof. Are they equal functions.

Let f: R → R and g: R → R be defined by f(x) = x² and g(x) = x + 1. Show that fog ≠ gof.

Let f: R → R an g: R → R be defined by f(x) = x + 1 and g(x) = x – 1. Show that fog = gof = I_R.

Verify associativity for the following three mappings: f: N → Z₀ (the set of non – zero integers), g: Z₀→ Q and h: Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = e^x.

Consider f: N → N, g: N → N and h: N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.

Give examples of two functions f: N → N and g: N → N such that gof is onto, but f is not onto.

Give examples of two functions f: N → Z and g: Z → Z such that gof is injective, but g is not injective.

If f: A → B and g: B → C are one – one functions show that gof is a one – one function.

If f: A → B and g: B → C are onto functions show that gof is an onto function.

Find fog and gof, if

f(x) = e^x, g(x) = log_ex

Find fog and gof, if

f(x) = x², g(x) = cos x

Find fog and gof, if

f(x) = |x|, g(x) = sin x

Find fog and gof, if

f(x) = x + 1, g(x) = e^x

Find fog and gof, if

f(x) = sin^–1 x, g(x) = x²

Find fog and gof, if

f(x) = x + 1, g(x) = sin x

Find fog and gof, if

f(x) = x + 1, g(x) = 2x + 3

Find fog and gof, if

f(x) = c, c ∈ R, g(x) = sin x²

Find fog and gof, if

f(x) = x² + 2,

Let f(x) = x² + x + 1 and g(x) = sin x. Show that fog ≠ gof.

If f(x) = |x|, prove that fof = f.

If f(x) = 2x + 5 and g(x) = x² + 1 be two real functions, then describe each of the following functions:

(i) fog

(ii) gof

(iii) fof

(iv) f²

Also, show that fof ≠ f².

If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?

Let f, g, h be real functions given by f(x) = sin x, g(x) = 2x and h(x) = cos x. Prove that fog = go(fh).

Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.

If and g(x) = log_ex are two real functions, then describe functions fog and gof.

If and g(x) = x² + 1 be two real functions, then find fog and gof.

Let f be a real function given by . Find each of the following:

fof

Let f be a real function given by . Find each of the following:

fofof

Let f be a real function given by . Find each of the following:

(fofof)(38)

Let f be a real function given by . Find each of the following:

f²

Also, show that fof ≠ f².

Let Find fof.

If f, g: R → R be two functions defined as f(x) = |x| + x and

g(x) = |x|–x for all x ∈ R. Then, find fog and gof. Hence, find fog (–3),

fog (5) and gof(–2).

State with reasons whether the following functions have inverse:

(i) f : [1, 2, 3, 4] → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Find f^-1 if it exists for f: A → B where

(i) A = {0, –1, –3, 2}; B = {–9, –3, 0, 6} & f(x) = 3x

(ii) A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} & f(x) = x²

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Show that f, g and gof are invertible. Find f^-1, g^-1, (gof)^-1 and show that (gof)^-1 = f^-1og^-1.

Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x² – 2. Express (gof)^-1 and f^-1og^-1 as the sets of ordered pairs and verify (gof)^-1 = f^-1og^-1.

Show that the function f : Q → Q defined by f(x) = 3x + 5 is invertible. Also, find f^-1.

Show that the function f : R → R defined by f(x) = 4x + 3 is invertible. Find the inverse of f.

Consider f : R⁺→ [4, ∞) given by f(x) = x² + 4. Show that f is invertible with f^-1 of f given by, where R⁺ is the set of all non-negative real numbers.

Consider f : R⁺→ [–5, ∞) given by f(x) = 9x² + 6x – 5. Show that f is invertible with.

If f : R → R be defined by f(x) = x³ – 3, then prove that f^-1 exists and find a formula for f^-1. Hence, find f^-1(24) and f^-1(5).

A function f : R → R is defined as f(x) = x³ + 4. Is it a bijection or not? In case it is a bijection, find f^-1(3).

If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)^-1 = f^-1og^-1.

Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by. Show that f is one-one and onto and hence find f^-1.

Consider the function f : R⁺→ [–9, ∞) given by f(x) = 5x² + 6x – 9. Prove that f is invertible with.

Let f : N → N be a function defined as f(x) = 9x² + 6x – 5. Show that f : N → S, where S is the range of f, is invertible. Find the inverse of f and hence find f^-1(43) and f^-1(163).

Let f : R – → R be a function defined as. Show that f : R – → range(f) is one-one and onto. Hence, find f^-1.

If f: R → (–1, 1) defined by is invertible, find f^-1.

If f: R → (0, 2) defined by is invertible, find f^-1.

Let f : [–1, ∞) → [–1, ∞) is given by f(x) = (x + 1)² – 1. Show that f is invertible. Also, find the set S = {x: f(x) = f^-1(x)}

Let A = {x ϵ R | –1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x² and g(x) = sin πx/2. Show that g^-1 exists but f^-1 does not exist. Also, find g^-1.

Let f be a function from R to R such that f(x) = cos (x + 2). Is f invertible? Justify your answer.

If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also, give their inverse function.

Let A and B be two sets each with finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.

If f : A → A and g : A → A are two bijections, then prove that

(i) fog is an injection

(ii) fog is a surjection

Which one of the following graphs represent a function?

Which one of the following graphs represent a one-one function?

If A = {1, 2, 3} and B = {a, b}, write total number of functions from A to B.

If A = {a, b, c} and B = {–2, –1, 0, 1, 2}, write total number of one-one functions from A to B.

Write total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b c}.

If f: R→ R is defined by f(x) = x², write f^–1(25).

If f : C → C is defined by f(x) = x², write f^–1(–4). Here, C denotes the set of all complex numbers.

If f: R → R is given by f(x) = x³, write f^–1(1).

Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x³.

Write f^–1(1).

Let f be a function from C (set of all complex numbers) to itself given by f(x) = x³. Write f^–1(–1).

Let f : R → R be defined by f(x) = x⁴, write f^–1(1).

If f : C → C is defined by f(x) = x⁴, f^–1(1).

If f :R→R is defined by f(x) = x², f^–1(–25).

If f : C → C is defined by f(x) = (x – 2)³, write f^–1(–1).

If f :R→R is defined by f(x) = 10x –7, then write f^–1(x).

If f : R → R defined by f(x) = 3x – 4 is invertible then write f^–1(x).

If f : R → R, g : R → R are given by f(x) = (x + 1)² and g(x) = x² + 1, then write the value of fog (–3).

Let A = {x ϵR : –4 ≤ x ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by Write the range of f.

Let be defined by f(x) = sinx. If f is a bijection, write set A.

Let f : R → R⁺ be defined by f(x) = a^x, a > 0 and a ≠ 1. Write f^–1(x).

Let f : R – {–1} → R – {1} be given by Write f^–1(x).

Let be a function defined as

Let f : R → R, g : R → R be two functions defined by f(x) = x² + x + 1 and g(x) = 1 – x². Write fog(–2).

Let f : R →R be defined as Write fof^–1(1).

Let f be an invertible real function. Write (f^–1 of) (1) + (f^–1 of) (2) + … +(f^–1 of) (100).

Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write total number of onto functions from A to B.

Write the domain of the real function

Write the domain of the real function

Write the domain of the real function

Write whether f : R → R given by is one-one, many-one, onto or into.

If f(x) = x + 7 and g(x) = x – 7, x ϵ R, write fog(7).

What is the range of the function

If f : R → R be defined by f(x) = (3 – x³)^1/3, then find fof(x).

If f : R → R is defined by f(x) = 3x + 2, find f(f (x)).

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, find fog.

Let f : R → R be the function defined by f(x) = 4x – 3 for all x ϵ R. Then write f^–1.

Which one the following relations on A = {1, 2, 3} is a function?

f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)}.

Write the domain of the real function f defined by

Let A = {a, b, c, d} and f : A → A be given by f = {(a, b), (b, d), (c, a), (d, c)}, write f^–1.

Let f, g : R → R be defined by f(x) = 2x + 1 and g(x) = x² – 2 for all x ϵ R, respectively. Then, find gof.

If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by

f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3), write fog.

If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = αx + β, find the values of α and β.

If f(x) = 4 – (x – 7)³, write f^–1(x).

Mark the correct alternative in each of the following:

Let A = {x ϵ R : –1 ≤ x ≤ 1} = B and C = {x ϵ R : X ≥ 0} and let S = {(x, y) ϵ A × B : x² + y² = 1} and S₀ = {(x, y) ϵ A × C : x² + y² = 1} Then

Mark the correct alternative in each of the following:

f: R → R given by is

Mark the correct alternative in each of the following:

If f : A → B given by 3^f(x) + 2^–x = 4 is a bijection, then

Mark the correct alternative in each of the following:

The function f : R → R defined by f(x) = 2^x + 2^|x| is

Mark the correct alternative in each of the following:

Let the function f : R – {–b} → R – {1} be defined by then

Mark the correct alternative in each of the following:

The function f : A → B defined by f(x) = –x² + 6x – 8 is a bijection, if

Mark the correct alternative in each of the following:

Let A = {x ϵR : –1 ≤ x ≤ 1} = B. Then, the mapping f : A → B given by f(x) = x |x| is

Mark the correct alternative in each of the following:

Let f: R → R be given by f(x) = [x]² + [x + 1]–3, where [x] denotes the greatest integer less than or equal to x. Then, f(x) is

Mark the correct alternative in each of the following:

Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then the function f : M → R defined by f(A) = |A| for every A ϵ M, is

Mark the correct alternative in each of the following:

The function f : [0, ∞) → R given by f(x) is

Mark the correct alternative in each of the following:

The range of the function f(x) = ^7–xP_x–3 is

Mark the correct alternative in each of the following:

A function f from the set on natural numbers to integers defined by

..is

Mark the correct alternative in each of the following:

Let f be an injective map with domain {x, y, z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false.

f(x) = 1, f(y) ≠ 1, f(z) ≠ 2.

The value of f^–1(1) is

Mark the correct alternative in each of the following:

Which of the following functions from Z to itself are bijections?

Mark the correct alternative in each of the following:

Which of the following functions from A = {x : –1 ≤ x ≤ 1} to itself are bijections?

Mark the correct alternative in each of the following:

Let A = {x : –1 ≤ x ≤ 1} and f : A → A such that f(x) = x |x|, then f is

Mark the correct alternative in each of the following:

If the function f: R → A given by is a surjection, then A =

Mark the correct alternative in each of the following:

If a function f : [2, ∞) → B defined by f(x) = x² – 4x + 5 is a bijection, then B =

Mark the correct alternative in each of the following:

The function f : R → R defined by f(x) = (x – 1) (x – 2) (x – 3) is

Mark the correct alternative in each of the following:

The function f : [–1/2, 1/2] → [π/2, π/2] defined by f(x) = sin^–1 (3x – 4x³) is

Mark the correct alternative in each of the following:

Let f : R → R be a function defined by Then,

Mark the correct alternative in each of the following:

Let f : R –{n} → R be a function defined by where m ≠ n. Then,

Mark the correct alternative in each of the following:

Let f : R → R be a function defined by Then, f is

Mark the correct alternative in each of the following:

f : R → R is defined by is

Mark the correct alternative in each of the following:

The function f : R → R, f(x) = x² is

Mark the correct alternative in each of the following:

A function f from the set of natural, numbers to the set of integers defined by

Mark the correct alternative in each of the following:

Which of the following functions from A = {x ϵ R : –1 ≤ x ≤ 1} to itself are bijections?

Mark the correct alternative in each of the following:

Let f: Z → Z be given by Then, f is

Mark the correct alternative in each of the following:

The function f : R → R defined by f(x) = 6^x + 6^|x| is

Mark the correct alternative in each of the following:

Let f(x) = x² and g(x) = 2^x. Then the solution set of the equation fog(x) = gof (x) is

Mark the correct alternative in each of the following:

If f: R → R is given by f(x) = 3x – 5, then f^–1(x)

Mark the correct alternative in each of the following:

If g(f(x)) = |sin x| and f (g(x)) = , then

Mark the correct alternative in each of the following:

The inverse of the function f : R → [x ϵ R : x < 1] given by is

Mark the correct alternative in each of the following:

Let A = {x ϵR : x ≥ 1}. The inverse of the function f : A → A given by f(x) = 2^x(x–1), is

Mark the correct alternative in each of the following:

Let A = {x ϵ R: x ≤ 1} and f : A → A given by f(x) = x(2 – x).Then, f^–1(x) is

Mark the correct alternative in each of the following:

Let Then, {fo(fof)} (x)

Mark the correct alternative in each of the following:

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, then f^–1 (x) is

Mark the correct alternative in each of the following:

If F: [1, ∞) → [2, ∞) is given by then f^–1(x) equals.

Mark the correct alternative in each of the following:

Let g(x) = 1 + x – [x] and where [x] denotes the greatest integer less than or equal to x. Then for all x, f (g(x)) is equal to

Mark the correct alternative in each of the following:

Let Then, for what value of α is f(f(x)) = x?

Mark the correct alternative in each of the following:

The distinct linear functions which map [–1, 1] onto [0, 2] are

Mark the correct alternative in each of the following:

Let f : [2, ∞) → X be defined by f(x) = 4x – x². Then, f is invertible, if X =

Mark the correct alternative in each of the following:

If f : R → (–1, 1) is defined by then f^–1(x) equals

Mark the correct alternative in each of the following:

Let [x] denote the greatest integer less than or equal to x. If f(x) = sin^–1 x, g(x) = [x²] and then

Mark the correct alternative in each of the following:

If g(x) = x² + x – 2 and gof (x) = 2x² – 5x + 2, then f(x) is equal to

Mark the correct alternative in each of the following:

If f(x) = sin² x and the composite function g(f(x)) = |sin x|, then g(x) is equal to

Mark the correct alternative in each of the following:

If f : R → R is given by f(x) = x³ + 3, then f^–1(x) is equal to

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Let f(x) = x³ be a function with domain {0, 1, 2, 3}. Then domain of f^–1 is

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Let f : R → R be given by f(x) = x² – 3. Then, f^–1 is given by

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Let f : R → R be given by f(x) = tan x. Then, f^–1(1) is

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Let f: R → R be defined as

Then, find f(–1) + f(2) + f(4)

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Let A = {1, 2, …, n} and B = {a, b}. Then the number of subjections from A into B is

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If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

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If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is

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Let be defined by Then,