NCERT Solutions for Class 11 Maths Chapter 2 – Relations and Functions
If Chapter 1 gave you the language of sets, Chapter 2 teaches you what happens when two sets start talking to each other. Relations and Functions is one of the most practically rich chapters in Class 11 Mathematics — the ideas here form the mathematical foundation behind everything from coordinate geometry to calculus and even computer programming logic.
A relation is simply a rule that pairs elements from one set with elements of another. But not every relation is a function. A function demands something stricter: every input must produce exactly one output. This single constraint is what makes functions so powerful and so widely used across science, engineering, and economics. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
NCERT's Chapter 2 walks you from the basic idea of a Cartesian product all the way to graphing real-valued functions and understanding their domains and ranges. The chapter has three exercises and a miscellaneous section, each building on the previous. Students often find this chapter approachable once they stop thinking about it abstractly and start drawing arrow diagrams and graphs — both of which are expected in CBSE answers. These solutions are written to match the way a well-prepared student writes answers in a board exam: each step justified, notation consistent with NCERT, and no logical jumps. Whether you are stuck on Exercise 2.1's Cartesian products or trying to decode a composite function problem in the miscellaneous exercise, these solutions lay it out clearly.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 2.1 | Cartesian Product of Sets | 10 Questions |
| Exercise 2.2 | Relations – Definition and Representation | 9 Questions |
| Exercise 2.3 | Functions – Types and Properties | 5 Questions |
| Miscellaneous | Mixed Problems on Relations and Functions | 12 Questions |
Chapter 2 – Relations and Functions: Concepts, Explanation and Key Tables
The Cartesian Product: Where It All Starts
Before you can define a relation, you need the Cartesian product. If A and B are two non-empty sets, their Cartesian product A × B is the set of all ordered pairs (a, b) where a belongs to A and b belongs to B. The key word here is ordered — (a, b) and (b, a) are not the same unless a = b.
For example, if A = {1, 2} and B = {x, y}, then A × B = {(1,x), (1,y), (2,x), (2,y)}. Notice that the order of the pair matters and the number of elements in A × B is always n(A) × n(B). This is tested directly in Exercise 2.1.
What is a Relation?
A relation R from set A to set B is any subset of the Cartesian product A × B. It is a way of expressing which elements of A are connected to which elements of B under some rule. You can represent a relation in three ways — as a set of ordered pairs, as a table, or as an arrow diagram. CBSE expects students to be comfortable with all three.
The Jump from Relation to Function
A function is a special type of relation. Specifically, f: A → B is a function if and only if every element in A (the domain) is associated with exactly one element in B (the codomain). Two things are non-negotiable for a function: every element of A must be mapped (no element left out), and no element of A can map to two different elements of B.
This is why checking whether a graph represents a function using the Vertical Line Test is such a reliable method — any vertical line should intersect the graph at most once.
Types of Functions
| Type of Function | What Makes It Unique | Example |
|---|---|---|
| One-One (Injective) | No two different inputs give the same output | f(x) = 2x + 1 |
| Onto (Surjective) | Every element in the codomain has at least one pre-image | f: ℝ → ℝ, f(x) = x³ |
| Bijective (One-One + Onto) | Both injective and surjective; perfect pairing | f(x) = x (identity function) |
| Many-One | Two or more inputs map to the same output | f(x) = x² (both 2 and –2 map to 4) |
| Into | At least one element in codomain has no pre-image | f: ℝ → ℝ, f(x) = x² (negatives unreached) |
| Constant Function | Every input gives the same output | f(x) = 7 |
| Identity Function | Every input maps to itself | f(x) = x |
| Polynomial Function | Defined by a polynomial expression | f(x) = 3x² – 5x + 2 |
| Rational Function | Ratio of two polynomials | f(x) = (x+1)/(x–2), x ≠ 2 |
| Modulus Function | Gives the absolute value of input | f(x) = |
| Signum Function | Returns –1, 0, or 1 based on sign of x | f(x) = x/ |
| Greatest Integer Function | Maps x to the largest integer ≤ x | f(x) = ⌊x⌋; f(2.7) = 2 |
Domain, Codomain, and Range
These three terms are fundamental and frequently confused. The domain is the complete set of valid inputs. The codomain is the set the function maps into (it is declared, not calculated). The range is the actual set of outputs — it is always a subset of the codomain, and sometimes equal to it.
When asked to find the domain of a function algebraically, two conditions to check are: the denominator must not be zero, and expressions under even roots must be non-negative.
Key Formulas for Exercise 2.1
| Formula | Meaning |
|---|---|
| n(A × B) = n(A) × n(B) | Number of elements in the Cartesian product |
| A × B ≠ B × A (unless A = B) | Cartesian product is not commutative |
| A × (B ∪ C) = (A × B) ∪ (A × C) | Distributive property |
| A × (B ∩ C) = (A × B) ∩ (A × C) | Distributive property |
| If A ⊆ B, then A × C ⊆ B × C | Subset property |
| n(A × A × A) = [n(A)]³ | For three-set Cartesian product |
Exercise-wise Focus
| Exercise | Core Skill Being Tested | Common Mistakes |
|---|---|---|
| 2.1 | Writing Cartesian products; finding number of elements | Forgetting ordered pairs are direction-sensitive |
| 2.2 | Defining relations; finding domain and range of a relation | Confusing range of relation with codomain |
| 2.3 | Deciding if a mapping is a function; identifying function type | Applying the vertical line test incorrectly on arrow diagrams |
| Miscellaneous | Domain/range of complex functions; algebra of functions | Sign errors when finding domain of rational/root functions |
Algebra of Functions
NCERT introduces operations on functions in this chapter. If f and g are two functions with the same domain, you can define their sum, difference, product, and quotient as new functions. The domain of the quotient function f/g excludes values where g(x) = 0.
Study Tips for Chapter 2
- Draw arrow diagrams for every relation or function question in Exercise 2.2 — it takes 30 seconds and prevents errors.
- When checking if a function is one-one, set f(x₁) = f(x₂) and prove x₁ = x₂. If you can always prove this, it is one-one.
- For the greatest integer function, practise plotting it — its staircase graph appears in CBSE objective questions regularly.
- The range of |x| is [0, ∞) — a surprisingly common slip is writing (–∞, ∞).
FAQs for NCERT Solutions for Class 11 Maths Chapter 2 Relations & Functions
A relation is a connection between elements of two sets and is represented as a collection of ordered pairs. A function is a special type of relation in which each element of the first set is associated with exactly one element of the second set. Every function is a relation, but every relation is not a function. Understanding this difference is essential for solving problems in Chapter 2. NCERT Solutions explain the concepts with examples and diagrams, helping students identify whether a given relation qualifies as a function. Mastering these basics is important because functions are used extensively in higher mathematics and various real-life applications.
NCERT Solutions provide accurate and detailed answers to every exercise question in the chapter. They help students understand concepts such as relations, functions, Cartesian products, domain, codomain, and range through clear explanations. Step-by-step solutions make complex problems easier to solve and improve conceptual clarity. Students can use these solutions for homework, revision, and exam preparation. Regular practice enhances logical reasoning and mathematical accuracy. Since many examination questions are based on NCERT textbook exercises, studying these solutions thoroughly can improve performance and boost confidence. They also provide a strong foundation for advanced mathematical concepts covered in later classes.
The Cartesian product is the set of all possible ordered pairs formed by taking one element from each of two sets. It serves as the foundation for defining relations and functions. Understanding Cartesian products helps students determine the number of possible relations between sets and represent them mathematically. NCERT Solutions include detailed explanations and solved examples that demonstrate how Cartesian products are formed and used. Regular practice of these questions helps students improve their understanding of ordered pairs and set operations. Since Cartesian products are directly related to relations and functions, mastering this concept is essential for success in this chapter.
The domain of a function is the set of all possible input values. The codomain is the set that contains all possible output values, while the range is the actual set of output values obtained from the function. Students often confuse codomain and range, making it important to understand their definitions clearly. NCERT Solutions provide numerous examples that explain these concepts step by step. Learning domain, codomain, and range helps students analyze functions accurately and solve related questions with confidence. These concepts are fundamental to advanced mathematical topics and are frequently tested in school and competitive examinations.