Relations and Functions revision notes

Key Concepts & Definitions

Ordered Pair

A pair of elements grouped together in a specific, sequential order, denoted as (p,q)(p, q)(p,q) where p∈Pp \in Pp∈P and q∈Qq \in Qq∈Q. The order is crucial; (a,b)(a, b)(a,b) is not equal to (b,a)(b, a)(b,a) unless a=ba=ba=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 TIPThis is used to solve linear equations equating coordinate components.

Cartesian Product (P×QP \times QP×Q)

Given two non-empty sets PPP and QQQ, their Cartesian product is the set of all possible ordered pairs where the first element is from PPP and the second is from QQQ. Defined as P×Q={(p,q):p∈P,q∈Q}P \times Q = \{(p,q) : p \in P, q \in Q\}P×Q={(p,q):p∈P,q∈Q}. If either PPP or QQQ is the null (empty) set, then P×QP \times QP×Q will also be an empty set (P×Q=ϕP \times Q = \phiP×Q=ϕ). If AAA and BBB are non-empty and at least one is an infinite set, A×BA \times BA×B is an infinite set.

Ordered Triplet

The product A×A×A={(a,b,c):a,b,c∈A}A \times A \times A = \{(a, b, c) : a, b, c \in A\}A×A×A={(a,b,c):a,b,c∈A} is a set of ordered triplets. Example: R×R×R\mathbb{R} \times \mathbb{R} \times \mathbb{R}R×R×R represents the coordinates of all points in three-dimensional space.

Relation (RRR)

A relation from a non-empty set AAA to a non-empty set BBB is a subset of the Cartesian product A×BA \times BA×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 AAA to the same set AAA.

Image & Preimage

If (a,b)∈f(a, b) \in f(a,b)∈f, then the second element bbb is called the image of aaa under fff. Conversely, the first element aaa is called the preimage of bbb under fff.

Domain of a Relation/Function

The set of all first elements (xxx) of the ordered pairs in a relation or function.

Range of a Relation/Function

The set of all second elements (imagesimagesimages, yyy) of the ordered pairs. It is the actual set of outputs produced.

Codomain

The entire destination set BBB in a relation from AAA to BBB. The Range is always a subset of the Codomain (Range⊆Codomain\text{Range} \subseteq \text{Codomain}Range⊆Codomain).

Function (fff)

A relation from a set AAA to a set BBB is said to be a function if every element of set AAA has one and only one image in set BBB. No two distinct ordered pairs in a function can have the same first element. Denoted as f:A→Bf: A \rightarrow Bf:A→B.

Real-Valued Function

A function which has either R\mathbb{R}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 R\mathbb{R}R or subsets of R\mathbb{R}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.

Formulae, 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 TIPThis 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 TIPVery useful to split complex Cartesian operations.
  • $A \times (B \cup C) = (A \times B) \cup (A \times C)$

Standard 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 TIPThe 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.

Algebra 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 $g(x) \neq 0$.

Conditions, 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.

Previous 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.