Relations and Functions — Mathematics Class 11 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Relations and Functions (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 4 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Relations and Functions (CBSE & HBSE)
Builds on sets to introduce relations, functions, domain/range, and special real-valued functions. CBSE expects 6–8 marks: ordered pairs, function types, and graph sketching. HBSE focuses on domain/range computation and identifying one-one / onto functions.
Cartesian Product & Relations
Ordered Pair
(a, b) is different from {a, b}: order matters and (a, b) = (c, d) iff a = c and b = d.
Cartesian Product
For sets A and B: A × B = {(a, b) : a ∈ A, b ∈ B}.
Properties:
- A × B ≠ B × A in general.
- n(A × B) = n(A) × n(B).
- A × ∅ = ∅; A × (B ∪ C) = (A × B) ∪ (A × C).
- A × A × A = {(a, b, c) : a, b, c ∈ A} — used for 3-D coordinate sets.
Example: A = {1, 2}, B = {x, y}. A × B = {(1, x), (1, y), (2, x), (2, y)}; n = 4.
Relation
A relation R from A to B is any subset of A × B.
- Domain of R = set of first elements of (a, b) ∈ R.
- Range of R = set of second elements.
- Codomain = B (the target set; range ⊆ codomain).
Number of relations from A to B with n(A) = m, n(B) = n is 2^(mn) (every subset of A × B is a relation).
Representation of a Relation
| Method | Example |
|---|---|
| Roster | R = {(1, 4), (2, 5)} |
| Set-builder | R = {(x, y) : y = x + 3, x ∈ A} |
| Arrow diagram | arrows from A to B |
| Graph | plot ordered pairs |
Inverse Relation
R⁻¹ = {(b, a) : (a, b) ∈ R} — swap each pair. Domain(R⁻¹) = Range(R) and vice-versa.
CBSE Tip: When listing a relation by rule, always state the source set explicitly: "R from A to B such that …".
Functions — Definition & Types
Function (Mapping)
A relation f : A → B is a function iff every element of A is associated with exactly one element of B.
- A = domain, B = codomain.
- f(x) = image of x; range = {f(x) : x ∈ A} ⊆ B.
Equality of Functions
f = g iff they have the same domain & codomain AND f(x) = g(x) for every x.
Types of Functions
| Type | Condition |
|---|---|
| One-one (Injective) | f(x₁) = f(x₂) ⇒ x₁ = x₂ |
| Onto (Surjective) | Range = Codomain (every y ∈ B has a pre-image) |
| Bijective | both one-one and onto |
| Many-one | not one-one |
| Into | not onto (range ⊊ codomain) |
To test one-one: solve f(x₁) = f(x₂) and check if it forces x₁ = x₂. To test onto: for arbitrary y ∈ B, solve f(x) = y for x ∈ A; if solvable, onto.
Real-Valued Functions
f : ℝ → ℝ (or subset of ℝ). Domain and range are real-number sets.
Identifying Domain / Range
Common rules:
- Polynomial — domain = ℝ.
- Rational p(x)/q(x) — domain = ℝ minus zeros of q(x).
- Square root √(g(x)) — domain = {x : g(x) ≥ 0}.
- Logarithm log(g(x)) — domain = {x : g(x) > 0}.
Vertical Line Test
A curve in the xy-plane represents a function iff every vertical line intersects it at most once.
HBSE Tip: When finding the range, express x in terms of y and find the set of y for which x is defined.
Standard Real Functions
Identity Function
f(x) = x. Graph is the line y = x; domain & range = ℝ.
Constant Function
f(x) = c (fixed real number). Graph is horizontal line y = c. Range = {c}.
Polynomial Function
f(x) = aₙxⁿ + … + a₁x + a₀ (aₙ ≠ 0). Domain = ℝ.
Rational Function
f(x) = p(x)/q(x). Domain = ℝ − {zeros of q}.
Modulus (Absolute Value) Function
f(x) = |x| = x if x ≥ 0; = −x if x < 0. Graph: V-shape with vertex at origin. Domain = ℝ; range = [0, ∞).
Useful properties:
- |x| ≥ 0, equality iff x = 0
- |xy| = |x||y|, |x/y| = |x|/|y| (y ≠ 0)
- |x + y| ≤ |x| + |y| (triangle inequality)
Signum Function
sgn(x) = 1 if x > 0; = 0 if x = 0; = −1 if x < 0. Range = {−1, 0, 1}.
Greatest Integer Function (Floor)
[x] = greatest integer ≤ x. Examples: [2.3] = 2, [−1.7] = −2, [5] = 5. Graph: step function; range = ℤ.
Even & Odd Functions
- Even: f(−x) = f(x). Graph symmetric about y-axis. Example: x², cos x.
- Odd: f(−x) = −f(x). Graph symmetric about origin. Example: x, x³, sin x.
- A general function may be neither.
CBSE Tip: Modulus, signum, and floor functions are tested via graph sketching — practice drawing them on graph paper to scale.
Algebra of Real Functions
Combining Functions
For real-valued f and g with common domain D:
| Operation | Definition |
|---|---|
| (f + g)(x) | f(x) + g(x) |
| (f − g)(x) | f(x) − g(x) |
| (f · g)(x) | f(x) · g(x) |
| (f / g)(x) | f(x) / g(x), g(x) ≠ 0 |
| (αf)(x) | α · f(x), α ∈ ℝ |
Domain of f + g, f − g, f · g = D. Domain of f / g = D − {x : g(x) = 0}.
Composition (Class 12 preview)
(g ∘ f)(x) = g(f(x)). Domain: those x in domain of f such that f(x) is in domain of g.
Graph Transformations
| Transformation | New graph |
|---|---|
| f(x) + k | shift up by k |
| f(x − h) | shift right by h |
| −f(x) | reflect about x-axis |
| f(−x) | reflect about y-axis |
| a · f(x) | vertical stretch by a |
| f(ax) | horizontal compression by a |
Useful Identities for Function Algebra
- f(x) = even part + odd part = ½[f(x) + f(−x)] + ½[f(x) − f(−x)].
- Sum/product of two even functions is even.
- Sum/product of two odd functions: sum is odd, product is even.
Cartesian Form vs Mapping Form
Given f : A → B by rule y = expression in x, the graph in A × B is the set {(x, f(x)) : x ∈ A}.
HBSE Tip: When combining functions and asked for domain, always intersect individual domains and remove zeros of any denominator.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Relations and Functions notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Relations and Functions chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Relations and Functions.