CBSE Class 12 Maths: Relations and Functions — Notes 2026

Types of Relations

• Empty relation: no element relates to any other (R = ∅); Universal relation: all elements relate to all (R = A×A)
• Reflexive: (a,a) ∈ R ∀a ∈ A; Symmetric: (a,b) ∈ R → (b,a) ∈ R; Transitive: (a,b) and (b,c) → (a,c) ∈ R
• Equivalence relation: reflexive + symmetric + transitive; congruence modulo n is classic equivalence relation

Functions: Basics

• Function f: A → B; every element of A maps to exactly one element of B; domain A, codomain B, range = f(A)
• Not a function: one element of domain maps to two elements in codomain; or element of domain unmapped
• Vertical line test: if vertical line intersects graph more than once → not a function

Injective (One-One) Functions

• f: A→B injective: f(x₁) = f(x₂) → x₁ = x₂; distinct inputs → distinct outputs
• Test: horizontally, no two different x-values give same f(x); strictly increasing/decreasing functions are injective
• Example: f(x) = 2x+1 is injective; f(x) = x² is not injective (f(2) = f(−2) = 4)

Surjective (Onto) Functions

• f: A→B surjective: every y ∈ B has at least one x ∈ A with f(x) = y; range = codomain
• Test: solve f(x) = y for all y ∈ B; if always solvable, function is onto
• Example: f: ℝ→ℝ, f(x) = 2x+3 is onto; f: ℝ→[0,∞), f(x) = x² is not onto (negative reals unreachable)

Bijective Functions and Inverse

• Bijective: both injective and surjective; one-one and onto; perfect pairing between A and B
• Only bijective functions have inverses; f⁻¹ exists iff f is bijective
• Example: f(x) = 3x−2 on ℝ→ℝ is bijective; inverse: f⁻¹(y) = (y+2)/3

Composition of Functions

• (f∘g)(x) = f(g(x)); apply g first then f; order matters: f∘g ≠ g∘f in general
• Composition is associative: f∘(g∘h) = (f∘g)∘h
• Identity function: f∘I = I∘f = f; inverse: f∘f⁻¹ = f⁻¹∘f = I

CBSE Board Focus

• Relations and Functions: 4–6 marks; equivalence relation proof, bijective function, inverse function
• Proof technique for equivalence: show all 3 properties (R, S, T) separately and systematically
• Composition question: find (f∘g)(x) and (g∘f)(x) — note they are usually different