Summary & JEE Exam Tips: Relations & Functions

Cartesian Product: The set of all possible ordered pairs $(a, b)$ from two sets A and B, denoted $A \times B$ . If $n(A)=p$ and $n(B)=q$ , then $n(A \times B)=pq$ .
Relation: A relation R from A to B is a subset of $A \times B$ . The Domain is the set of all first elements, and the Range is the set of all second elements. The entire set B is the Codomain.
Function: A function is a special relation where every element in the domain (set A) has one and only one image in the codomain (set B).
Standard Real Functions: Key functions include Identity ( $f(x)=x$ ), Constant ( $f(x)=c$ ), Modulus ( $f(x)=|x|$ ), Signum ( $f(x)=sgn(x)$ ), and Greatest Integer ( $f(x)=[x]$ ).
Algebra of Functions: Real functions can be added, subtracted, multiplied, and divided like algebraic expressions, e.g., $(f+g)(x) = f(x)+g(x)$ .

Domain is Everything: Questions on finding the domain are extremely common. Master the core rules: denominator $\ne 0$ , expression inside a square root $\ge 0$ , and argument of a logarithm $> 0$ . Be prepared for combinations of these.
Know Your Graphs: Do not just memorize the definitions of standard functions; learn their graphs by heart. Questions about range, monotonicity, and the number of solutions to an equation are often solved instantly by sketching the graph.
Equivalence Relations: For JEE Advanced, be very clear on how to prove if a relation is reflexive, symmetric, and transitive. This is a common topic for proof-based questions.
Types of Functions (Injectivity/Surjectivity): Understand the definitions of one-one (injective) and onto (surjective) functions. The horizontal line test is a quick check for injectivity on a graph. Problems testing this are frequent in JEE Advanced.
Piecewise & Composite Functions: Be comfortable working with functions defined in pieces (like modulus and greatest integer). Advanced problems often involve finding the domain, range, or continuity of composite functions like $f(g(x))$ .
Functional Equations: This is a classic JEE topic where you must determine a function based on an equation it satisfies, like $f(x+y)=f(x)+f(y)$ .

Relations and Functions