Topics Covered
8 key topics in this chapter
Study Resources
Key Formulae
Essential mathematical expressions for this chapter — understand derivations, not just results.
Exam-Ready Insights
Important points to remember — curated from CBSE Board question patterns.
CBSE always asks you to check whether a given relation is reflexive, symmetric, and/or transitive — do each property separately.
To prove a function is one-one: assume f(x₁)=f(x₂) and show x₁=x₂.
To prove onto: for every y in codomain, find an x in domain with f(x)=y.
Range ⊆ Codomain; equality makes the function onto — a frequent MCQ trap.
Composition is NOT commutative (f∘g ≠ g∘f in general) but IS associative.
Competitive Exam Strategy
Targeted tips for JEE Main, JEE Advanced, NEET, BITSAT, and KVPY.
Number of functions from A(m elements) to B(n elements) = nᵐ. Number of bijections = m! (when m=n). These are quick-solve formulae.
Piecewise-defined functions and their domains appear in JEE Advanced; always check continuity at the boundary points.
BITSAT loves "which of the following is a function?" questions with arrow diagrams — check every element of the domain has exactly one image.
KVPY tests surjective/injective with abstract sets — practice proofs, not just definitions.
Common Mistakes to Avoid
Assuming every function has an inverse — it must be bijective for the inverse to exist.
Confusing domain (where f is defined) with range (actual outputs).
Composition order: f∘g means "g first, f second" — many students reverse this.
A relation from A to B is NOT the same as a relation from B to A.
Key Takeaways
A relation is a subset of A×B; a function is a special relation where every input has exactly one output.
Domain = set of all valid inputs; Codomain = intended output set; Range = actual output set.
Bijection = one-one + onto; only bijections have inverses.
Graph of a function passes the Vertical Line Test.
Even functions: f(−x)=f(x); Odd functions: f(−x)=−f(x).