Relations and Functions revision notes for JEE

01Key Concepts & Definitions

Ordered Pair

A pair of elements grouped together in a specific, sequential order, denoted as

(p, q)

where

p \in P

and

q \in Q

. The order is crucial;

(a, b)

is not equal to

(b, a)

unless

a=b

.

Equality of Ordered Pairs: Two ordered pairs are equal if and only if their corresponding first elements are equal and their corresponding second elements are equal. JEE Tip This is used to solve linear equations equating coordinate components.

Cartesian Product (

P \times Q

)

Given two non-empty sets

P

and

Q

, their Cartesian product is the set of all possible ordered pairs where the first element is from

P

and the second is from

Q

. Defined as

P \times Q = \{(p,q) : p \in P, q \in Q\}

.

If either $P$ or $Q$ is the null (empty) set, then $P \times Q$ will also be an empty set ( $P \times Q = \phi$ ).
If $A$ and $B$ are non-empty and at least one is an infinite set, $A \times B$ is an infinite set.

Ordered Triplet

The product

A \times A \times A = \{(a, b, c) : a, b, c \in A\}

is a set of ordered triplets. Example:

\mathbb{R} \times \mathbb{R} \times \mathbb{R}

represents the coordinates of all points in three-dimensional space.

Relation (

R

)

A relation from a non-empty set

A

to a non-empty set

B

is a subset of the Cartesian product

A \times B

. It is derived by establishing a relationship between the first and second elements of the ordered pairs.

Arrow Diagram: A visual representation of a relation.
Relation on A: A relation from a set $A$ to the same set $A$ .

Image & Preimage

If

(a, b) \in f

, then the second element

b

is called the image of

a

under

f

. Conversely, the first element

a

is called the preimage of

b

under

f

.

Domain of a Relation/Function

The set of all first elements (

x

) of the ordered pairs in a relation or function.

Range of a Relation/Function

The set of all second elements (

images

,

y

) of the ordered pairs. It is the actual set of outputs produced.

Codomain

The entire destination set

B

in a relation from

A

to

B

. The Range is always a subset of the Codomain (

\text{Range} \subseteq \text{Codomain}

).

Function (

f

)

A relation from a set

A

to a set

B

is said to be a function if every element of set

A

has one and only one image in set

B

.

No two distinct ordered pairs in a function can have the same first element.
Denoted as $f: A \rightarrow B$ .

Real-Valued Function

A function which has either

\mathbb{R}

or one of its subsets as its range.

Real Function

A function where both the domain and range are either the set of real numbers

\mathbb{R}

or subsets of

\mathbb{R}

.

Historical Note

The term "function" first appeared in a 1673 Latin manuscript by Gottfried Wilhelm Leibnitz to describe the "mathematical job" of a curve. Johan Bernoulli assigned it the analytical sense we use today in 1698.

02Formulae, Equations & Set Operations

Cardinality of Cartesian Product: If set $A$ has $p$ elements ( $n(A) = p$ ) and set $B$ has $q$ elements ( $n(B) = q$ ), then the Cartesian product $A \times B$ has exactly $pq$ elements: $n(A \times B) = n(A) \times n(B) = pq$ .
Total Number of Relations: Because every relation from $A$ to $B$ is a subset of $A \times B$ , and a set with $pq$ elements has $2^{pq}$ subsets, the total number of possible relations from $A$ to $B$ is $2^{pq}$ . JEE Tip This formula is extremely frequent in Permutations & Combinations/Sets mixed MCQ questions.

Distributive Properties of Cartesian Products

$A \times (B \cap C) = (A \times B) \cap (A \times C)$ JEE Tip Very useful to split complex Cartesian operations.
$A \times (B \cup C) = (A \times B) \cup (A \times C)$

03Standard Real Functions & Important Graphs

Identity Function

$y = f(x) = x$ .

Domain: $\mathbb{R}$ . Range: $\mathbb{R}$ .

Graph: A straight line passing directly through the origin (0,0) making a 45° angle with the x-axis.

Constant Function

$y = f(x) = c$ (where $c$ is a constant real number).

Domain: $\mathbb{R}$ . Range: $\{c\}$ .
Graph: A straight horizontal line parallel to the x-axis.

Linear Function: $f(x) = mx + c$ (where $m, c$ are constants). Its graph is a straight line.

Polynomial Function

$f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$ .

Condition: $n$ is a non-negative integer and $a_0, a_1, \dots, a_n \in \mathbb{R}$ .
Domain: Generally $\mathbb{R}$ .

Rational Function

$f(x) = \frac{g(x)}{h(x)}$ , where $g(x)$ and $h(x)$ are polynomial functions.

Condition: Defined in a domain where $h(x) \neq 0$ .

Modulus Function

$f(x) = |x|$ .

Definition: $f(x) = x$ if $x \geq 0$ , and $f(x) = -x$ if $x < 0$ .
Domain: $\mathbb{R}$ . Range: Non-negative real numbers.
Graph: V-shaped, symmetric about the y-axis, vertex at the origin.

Signum Function

Definition: $f(x) = 1$ if $x > 0$ ; $f(x) = 0$ if $x = 0$ ; $f(x) = -1$ if $x < 0$ .
Domain: $\mathbb{R}$ . Range: $\{-1, 0, 1\}$ .
Graph: Two horizontal rays (at $y = -1$ and $y = 1$ ) and a distinct point at the origin $(0,0)$ . JEE Tip The Signum function heavily tests continuity and differentiability boundaries at $x=0$ .

Greatest Integer Function (Step Function)

$f(x) = [x]$ .

Definition: Assumes the value of the greatest integer less than or equal to $x$ . Example: $[x] = -1$ for $-1 \leq x < 0$ and $[x] = 0$ for $0 \leq x < 1$ .
Domain: $\mathbb{R}$ . Range: $\mathbb{Z}$ (All Integers).
Graph: "Staircase" or step-like shape.

04Algebra of Real Functions

For two real functions $f: X \rightarrow \mathbb{R}$ and $g: X \rightarrow \mathbb{R}$ (where $X \subset \mathbb{R}$ ):

Addition: $(f + g)(x) = f(x) + g(x)$ for all $x \in X$ .
Subtraction: $(f - g)(x) = f(x) - g(x)$ for all $x \in X$ .
Multiplication by a Scalar: $(kf)(x) = k \cdot f(x)$ where $k$ is a real number (scalar).
Multiplication (Pointwise): $(fg)(x) = f(x)g(x)$ for all $x \in X$ .
Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ , valid for all $x \in X$ provided $g(x) \neq 0$ .

05Conditions, Edge Cases & Limitations

(Adapted for mathematical boundaries & algebraic sign conventions)

Polynomial Integer Limit: The exponents in a polynomial function must be non-negative integers. Edge Case: $f(x) = x^{2/3} + 2x$ is strictly NOT a polynomial.
Division by Zero Boundary: For rational functions $f(x)/g(x)$ , you must strictly evaluate the roots where $g(x) = 0$ and remove them from the domain. Example: For $f(x) = \frac{x^2 - 5x + 4}{x^2 - 5x + 4}$ , domain is $\mathbb{R} - \{1, 4\}$ .
Square Root Convention: The radicand of a square root must be non-negative. $f(x) = \sqrt{x}$ is defined over non-negative reals.
Modulus Sign Convention: For $x < 0$ , the modulus outputs $-x$ . Since $x$ is already negative, $-x$ becomes positive. Do not assume $-x$ implies a negative output.

06Previous Year JEE Topics & Advanced Extensions

Domain & Range Calculations of Composite Expressions: This chapter establishes the base rules. JEE Advanced heavily combines domains. To find the domain of $(f/g)(x)$ , students must compute $Domain(f) \cap Domain(g)$ AND exclude points where $g(x) = 0$ .
Set Theory + Cartesian Products: Advanced problems frequently ask for intersections of Cartesian spaces. The distributive laws $A \times (B \cap C) = (A \times B) \cap (A \times C)$ are critical shortcuts.
Piecewise Function continuity prep: Thorough understanding of the Modulus ( $|x|$ ), Signum, and Greatest Integer ( $[x]$ ) graphs built here is strictly required for evaluating limits and derivatives in Calculus. Graphical transformations (shifting the vertex of the V-shape of a modulus) is a staple of Advanced MCQs.

07JEE Traps

Based on the foundational topics in this chapter, here are the top 10 JEE traps.

Relation vs. Function Mapping

✕Misconception

Any set of ordered pairs mapping $A$ to $B$ is a function.

✓Reality

A relation is ONLY a function if every element in $A$ maps to one and only one element in $B$ . If $x=2$ maps to both $y=2$ and $y=4$ , it is a relation, not a function.

Codomain vs. Range

✕Misconception

The Codomain and the Range are the exact same thing.

✓Reality

The Codomain is the entire destination set $B$ . The Range is only the actual images produced by the mapping. Range $\subseteq$ Codomain.

Simplifying Before Finding Domain

✕Misconception

You should simplify rational expressions before finding their domain.

✓Reality

Find the domain before canceling common factors. If $f(x) = \frac{x^2-1}{x-1}$ , the domain strictly excludes $x=1$ , even though it simplifies to $x+1$ .

Greatest Integer Function on Negatives

✕Misconception

The Greatest Integer Function just truncates the decimal (e.g., $[-1.5] = -1$ ).

✓Reality

The GIF evaluates to the greatest integer less than or equal to $x$ . For negative numbers, you step down to the left on the number line: $[-1.5] = -2$ .

Cartesian Product Commutativity

✕Misconception

Cartesian products are commutative ( $A \times B = B \times A$ ).

✓Reality

In general, $A \times B \neq B \times A$ . They only equal if Set A = Set B.

Cartesian Product of Empty Set

✕Misconception

$A \times B$ can contain elements even if one set is empty.

✓Reality

If either $A$ or $B$ is an empty set ( $\phi$ ), then $A \times B$ is strictly the empty set $\phi$ .

Counting Relations vs. Elements

✕Misconception

The number of relations from $A$ to $B$ is $n(A) \times n(B)$ .

✓Reality

$n(A \times B) = pq$ is the number of elements in the Cartesian product. The total number of relations is $2^{pq}$ , which is the total number of possible subsets.

Domain of

x^2

✕Misconception

For $f(x) = x^2$ , the domain is only positive real numbers.

✓Reality

The domain of $f(x) = x^2$ is all real numbers $\mathbb{R}$ . Its range is non-negative real numbers.

Fractional Powers Are Not Polynomials

✕Misconception

$f(x) = x^{1/2} + 2$ is a polynomial function of degree $1/2$ .

✓Reality

It is not a polynomial at all. Polynomials require non-negative integer powers.

Piecewise Boundary Values

✕Misconception

For piecewise functions, the boundaries (like $x=0$ ) can map to two different outputs.

✓Reality

To be a valid function, boundary values must uniquely map. If $f(x) = 1 - x$ for $x < 0$ and $f(x) = x + 1$ for $x > 0$ , the value at exactly $x = 0$ must be explicitly defined (e.g., $f(0) = 1$ ) to avoid undefined gaps or function violations.