Q1. Is f(x) = 2x + 3 one-one?
Assume f(a) = f(b): 2a + 3 = 2b + 3
⇒ 2a = 2b ⇒ a = b
So f is one-one (injective).
Relations and Functions — Class 12
Functions are the backbone of Class 12 maths. Sort out one-one, onto and invertible functions and the equivalence relations.
Types of relationsReflexive (aRa), symmetric (aRb ⇒ bRa), transitive (aRb, bRc ⇒ aRc). All three ⇒ equivalence relation.
One-one (injective)Different inputs give different outputs: f(a) = f(b) ⇒ a = b.
Onto (surjective)Every element of the codomain is an image of some input.
Invertible functionA function is invertible iff it is both one-one and onto (a bijection); then f⁻¹ exists.
Solved examples
Q2. Find (g∘f)(x) if f(x) = x + 1 and g(x) = x².
(g∘f)(x) = g(f(x)) = g(x + 1)
= (x + 1)²
Formula sheet
| Quantity | Formula |
|---|---|
| Composition | (g∘f)(x) = g(f(x)) |
| Inverse condition | f∘f⁻¹ = f⁻¹∘f = identity |
| Bijection | one-one + onto ⇒ invertible |
Common mistakes & exam wins
- A relation is an equivalence relation only if it is reflexive AND symmetric AND transitive.
- One-one + onto = bijection = invertible.
- Composition is generally NOT commutative: g∘f ≠ f∘g.
Frequently asked questions
What is an equivalence relation?
A relation that is reflexive, symmetric and transitive.
When is a function invertible?
When it is both one-one (injective) and onto (surjective), i.e. a bijection.
What is the composition of functions?
(g∘f)(x) = g(f(x)) — apply f first, then g.