Relations and Functions: Class 12 Maths Practice Questions

A delivery company assigns each order a unique tracking code. The tracking system uses a function f : ℝ → ℝ defined by f(x) = (3x + 7)/2, where x ∈ ℝ represents the order number and f(x) represents the tracking code.

A delivery company assigns each order a unique tracking code. The tracking system uses a function f : ℝ → ℝ defined by f(x) = (3x + 7)/2.

(i) A logistics analyst claims that no two different orders can ever receive the same tracking code under this system. Verify whether the analyst's claim is correct by checking if f is one-one. [1]

(ii) The system administrator needs every real number to be a valid tracking code output. Check whether f is onto. [1]

(iii) Based on your results in (i) and (ii), determine whether f is invertible. If yes, find f⁻¹(x) and hence find f⁻¹(13). [2]

A school library manages its book collection using a classification system. Two books are said to be 'related' if they belong to the same subject category (e.g., both are Mathematics books, or both are Science books, etc.). Formally, let B be the set of all books in the library, and define a relation R on B as:

R = {(b₁, b₂) : b₁ and b₂ belong to the same subject category, b₁, b₂ ∈ B}

The librarian also defines a function f : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6} where f(n) represents the shelf number assigned to book-ID n, given by:

f = {(1, 2), (2, 4), (3, 6), (4, 1), (5, 3), (6, 5)}

A school library manages its book collection using a classification system. Two books are said to be 'related' if they belong to the same subject category (e.g., both are Mathematics books, or both are Science books, etc.). Formally, let B be the set of all books in the library, and define a relation R on B as:

R = {(b₁, b₂) : b₁ and b₂ belong to the same subject category, b₁, b₂ ∈ B}

The librarian also defines a function f : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6} where f(n) represents the shelf number assigned to book-ID n, given by:

f = {(1, 2), (2, 4), (3, 6), (4, 1), (5, 3), (6, 5)}

(i) Show that the relation R defined on the set B is reflexive. [1]
(ii) Show that the relation R defined on the set B is symmetric. [1]
(iii) A student claims that f is a bijective function, and therefore the library can create a perfect 'return-mapping' (i.e., f⁻¹ exists). Verify whether f is one-one and onto, and hence determine whether f⁻¹ exists. [2]

A school library has a collection of books categorised into 4 genres: Mystery (M), Science Fiction (F), Biography (B), and Romance (R), forming the set G = {M, F, B, R}. The librarian defines a relation T on G as:
T = {(x, y) : x and y belong to the same broad category, where Mystery and Science Fiction are grouped as 'Fiction', and Biography and Romance are grouped as 'Non-Fiction'; x, y ∈ G}.

A school library has a collection of books categorised into 4 genres: Mystery (M), Science Fiction (F), Biography (B), and Romance (R), forming the set G = {M, F, B, R}. The librarian defines a relation T on G as:
T = {(x, y) : x and y belong to the same broad category, where Mystery and Science Fiction are grouped as 'Fiction', and Biography and Romance are grouped as 'Non-Fiction'; x, y ∈ G}.

Based on the above, answer the following:
(i) List all the elements of the relation T. [1 mark]
(ii) Check whether T is reflexive. Give a reason. [1 mark]
(iii) A student claims that T is an equivalence relation. Verify whether the student's claim is correct by checking symmetry and transitivity as well. [2 marks]

A school library uses a digital cataloguing system. Every book in the library is assigned a unique ID from the set A = {1, 2, 3, 4, 5}. The librarian defines a function f : A → B that maps each book ID to a 'shelf code' using the rule f(x) = 3x − 1, where B is the range of f.

A school library uses a digital cataloguing system. Every book in the library is assigned a unique ID from the set A = {1, 2, 3, 4, 5}. The librarian defines a function f : A → B that maps each book ID to a 'shelf code' using the rule f(x) = 3x − 1, where B is the range of f.

(i) Write down the set B (range of f). [1 mark]
(ii) Is f a one-one function? Justify your answer. [1 mark]
(iii) The librarian now considers a NEW function g : A → ℕ defined by the same rule g(x) = 3x − 1, where ℕ is the set of all natural numbers. Determine whether g is onto. Justify your answer. [2 marks]

A school library maintains a record of books issued to students. The librarian defines a relation R on the set S = {1, 2, 3, 4, 5, 6}, where S represents student ID numbers, as follows:

R = {(a, b) : a and b have issued a book from the same genre}

After checking the records:
• Students 1, 2, 3 → 'Science Fiction'
• Students 4, 5 → 'Mystery'
• Student 6 → 'Biography'

A school library maintains a record of books issued to students. The librarian defines a relation R on the set S = {1, 2, 3, 4, 5, 6}, where S represents student ID numbers, as follows:

R = {(a, b) : a and b have issued a book from the same genre}

After checking the records, the librarian finds:
• Students 1, 2, 3 have issued books from the genre 'Science Fiction'
• Students 4, 5 have issued books from the genre 'Mystery'
• Student 6 has issued a book from the genre 'Biography'

Based on this situation, answer the following:

(i) List any four ordered pairs that belong to R.

(ii) Check whether R is reflexive. Give a reason.

(iii) The librarian claims: 'If student a has issued a book from the same genre as student b, and student b has issued a book from the same genre as student c, then student a has also issued a book from the same genre as student c.' Verify whether this claim is mathematically correct by checking if R is transitive.

(iii) OR

A new student (ID = 7) joins and issues a 'Mystery' genre book. The librarian now updates R accordingly. Determine whether the updated relation R (on the new set S' = {1, 2, 3, 4, 5, 6, 7}) is an equivalence relation. Justify your answer by checking all three properties.

A school library system uses a digital cataloguing software. The librarian assigns each book a unique 'Library Code' using a function f : ℕ → ℕ defined by f(n) = 2n + 3, where n ∈ ℕ (ℕ = {1, 2, 3, …}).

A school library system uses a digital cataloguing software. The librarian assigns each book a unique 'Library Code' using a function f : ℕ → ℕ defined by f(n) = 2n + 3, where n ∈ ℕ (ℕ = {1, 2, 3, …}).

Based on the above information, answer the following questions:

(i) Show that f is one-one. [1]
(ii) Determine whether f is onto. Justify your answer. [1]
(iii) The librarian considers a new relation R on ℕ defined as:
R = {(m, n) : m, n ∈ ℕ and |m − n| is even}.
Show that R is an equivalence relation. [2]

A school library system assigns each student a unique Library Card Number. The librarian models this as a function f : S → N, where S = {Aarav, Bela, Chirag, Diya} is the set of students and N = {101, 102, 103, 104} is the set of card numbers, defined by:

f(Aarav) = 101, f(Bela) = 102, f(Chirag) = 103, f(Diya) = 104

A school library system assigns each student a unique Library Card Number. The librarian models this as a function f : S → N, where S = {Aarav, Bela, Chirag, Diya} is the set of students and N = {101, 102, 103, 104} is the set of card numbers, defined by:

f(Aarav) = 101, f(Bela) = 102, f(Chirag) = 103, f(Diya) = 104

(i) Is f one-one? Justify your answer. [1]
(ii) Is f onto? Justify your answer. [1]
(iii) The librarian wants to add a fifth student, Esha, to set S. She considers assigning f(Esha) = 101 (reusing Aarav's card number). Will f remain one-one? Give reason. Also, if instead a brand-new card number 105 is added to N (making N = {101, 102, 103, 104, 105}) but no new student is added to S, will f remain onto? Justify. [2]

A data analytics company assigns each employee a unique 'performance code' using a function f : ℝ − {−1} → ℝ defined by f(x) = (3x − 2)/(x + 1). The HR manager claims that this function is both one-one and onto ℝ, ensuring a perfect one-to-one correspondence between employees and performance codes.

A data analytics company assigns each employee a unique 'performance code' using a function f : ℝ → ℝ defined by f(x) = (3x − 2)/(x + 1), for all x ∈ ℝ − {−1}.

The HR manager claims:
• Assertion (A): The function f is one-one.
• Assertion (B): The function f is onto ℝ.

(i) Verify whether f is one-one. [2 marks]
(ii) Determine the range of f and hence decide whether f is onto ℝ. Justify your answer. [2 marks]

A relation R on the set A = {1, 2, 3} is defined as R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Which of the following is correct?

Let f : ℝ − {2} → ℝ be defined by f(x) = (x − 1)/(x − 2). Is f one-one?

Assertion (A): The function f : ℝ → ℝ defined by f(x) = 2x + 5 is both one-one and onto.

Reason (R): A function f : ℝ → ℝ is onto if and only if for every y ∈ ℝ (Codomain), there exists x ∈ ℝ (Domain) such that f(x) = y.

Let f : {1, 2, 3, 4} → {1, 2, 3, 4} be defined by f(1) = 2, f(2) = 3, f(3) = 4, f(4) = 1. Which of the following is true?

The function f : ℝ → ℝ defined by f(x) = 3x + 7 is:

Assertion (A): The relation R = {(x, y) : x + y = 10 and x, y ∈ ℕ} is not a reflexive relation.

Reason (R): For reflexivity, (x, x) must belong to R for every x ∈ ℕ, but x + x = 2x ≠ 10 for all x ∈ ℕ.

Assertion (A): The relation R = {(a, b) : a + b is even, a, b ∈ ℕ} is a reflexive relation on ℕ.

Reason (R): For any n ∈ ℕ, n + n = 2n, which is always even.

Let R be a relation on ℝ defined by R = {(a, b) : a ≤ b²}. Check whether R is reflexive.

Assertion (A): The function f : ℕ → ℕ defined by f(x) = 2x + 1, ∀ x ∈ ℕ, is one-one but not onto.

Reason (R): For y = 2 ∈ ℕ (Codomain), there does not exist any x ∈ ℕ (Domain) such that f(x) = 2x + 1 = 2.

Let f : ℝ → ℝ be defined by f(x) = 2x + 7. Which of the following is correct?

Check whether the function f : ℝ → ℝ defined by f(x) = 3x + 5 is one-one (injective) or not.

Check whether the function f : ℝ → ℝ defined by f(x) = 3x + 5 is one-one (injective) or not.

A function f : ℝ → ℝ is defined as f(x) = 5x − 3. Check whether f is one-one.

Check whether the function f : ℝ − {2} → ℝ − {1} defined as f(x) = (x − 1)/(x − 2) is one-one and onto.

Show that f : [0, ∞) → [−3, ∞) defined by f(x) = 9x² + 6x − 3 is both one-one and onto. Also, find q ∈ [0, ∞) such that f(q) = 6.

Let R be a relation on ℕ defined by R = {(a, b) : a, b ∈ ℕ and 3a + b = 30}. Find the domain and range of R. Also check whether R is reflexive, symmetric and transitive. Hence, state whether R is an equivalence relation or not.

A relation R is defined on ℕ × ℕ (where ℕ is the set of natural numbers) as:
(a, b) R (c, d) ⟺ ad = bc
Show that R is an equivalence relation.

Let R be a relation defined on ℤ (the set of all integers) by R = {(a, b) : (a − b) is divisible by 5, a, b ∈ ℤ}. Show that R is an equivalence relation.

Let f : ℝ − {2} → ℝ − {1} be defined by f(x) = (3x − 1)/(3x − 6). Show that f is one-one and onto.

Let A = ℝ − {2} and B = ℝ − {1}. A function f : A → B is defined by f(x) = (x − 1)/(x − 2). Show that f is bijective (one-one and onto).

A school library uses a system to assign unique ID codes to books. The librarian defines a function f : ℕ → ℕ by f(n) = 2n, where ℕ is the set of natural numbers and f(n) represents the ID code assigned to book number n.

A school library uses a system to assign unique ID codes to books. The librarian defines a function f : ℕ → ℕ by f(n) = 2n, where ℕ is the set of natural numbers and f(n) represents the ID code assigned to book number n.

Based on the above information, answer the following questions:
(i) Show that f is one-one. [1]
(ii) Is f onto? Justify your answer. [1]
(iii) The librarian considers a new coding function g : ℕ → ℕ defined by g(n) = n + 1. Check whether g is one-one and onto. Hence determine whether g is bijective. [2]

A school library system assigns each book a unique digital code. The librarian defines a function f : ℕ → ℕ by f(n) = 2n, where ℕ is the set of natural numbers, representing that every book with catalogue number n is assigned the digital code 2n. A second librarian proposes a new coding rule g : ℕ → ℕ defined by g(n) = n + 1.

A school library system assigns each book a unique digital code. The librarian defines a function f : ℕ → ℕ by f(n) = 2n, where ℕ is the set of natural numbers, representing that every book with catalogue number n is assigned the digital code 2n.

(i) Show that f is a one-one function. [1]
(ii) Determine whether f is an onto function. Justify your answer. [1]
(iii) A second librarian proposes a new coding rule g : ℕ → ℕ defined by g(n) = n + 1. Find the composite function (g ∘ f)(n) and determine whether (g ∘ f) is one-one. [2]