Summary and Quick Revision

The Chapter in One Idea

Everything here is one construction refined twice:

Pair up two sets ($A \times B$), select pairs by a rule (a relation), then demand exactly one partner per input (a function).

Cartesian product → relation → function: each is a special case of the one before, and each keyword (domain, codomain, range, image) keeps its meaning down the chain.

Exam weight at a glance: JEE Main draws from this chapter nearly every session — counting relations/functions, domain-range of algebraic functions, and greatest-integer/fractional-part values. In Boards, expect ordered-pair equations, roster/domain/range of a given relation, function evaluation, and the algebra of functions. And the entire calculus sequence — limits, continuity, derivatives — is built on the function vocabulary fixed here.

Cartesian Product and Relations — Formula Card

• Ordered pair: (a, b) with order significant; $(a, b) = (x, y) \iff a = x$ and $b = y$.
• Cartesian product: $A \times B = \{(a, b) : a \in A, b \in B\}$; $A \times \phi = \phi$; in general $A \times B \neq B \times A$ (equal only when A = B for non-empty sets).
• Counting: $n(A) = p$, $n(B) = q$ $\Rightarrow$ $n(A \times B) = pq$; if either factor is infinite (both non-empty), so is the product.
• Distributivity: $A \times (B \cap C) = (A \times B) \cap (A \times C)$; same for $\cup$. Also $(A \times B) \cap (B \times A) = (A \cap B) \times (A \cap B)$.
• Geometry: $\mathbb{R} \times \mathbb{R}$ = the plane; $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ = space; (a, b, c) is an ordered triplet.
• Relation from A to B: ANY subset of $A \times B$. Domain = first elements used; range = second elements used; codomain = all of B; range $\subset$ codomain.
• Counting relations: $2^{pq}$ (subsets of the product). Extremes: the empty relation and the universal relation $A \times B$.
• Representations: roster, set-builder, arrow diagram.

Functions — Formula Card

• Function test: every element of A has ONE and ONLY ONE image. Same output for two inputs: fine. Two outputs for one input, or an input with no image: fails.
• Notation: $f : A \to B$, f(a) = b; a is a preimage, b the image. Range = set of images $\subset$ codomain B.
• Real function: domain and range inside $\mathbb{R}$.
• Standard functions (domain; range):

• Polynomial: non-negative integer powers only ($x^{2/3}$ disqualifies). Rational: quotient of polynomials, denominator non-zero. Linear: f(x) = mx + c; for $m \neq 0$, domain and range both $\mathbb{R}$.
• Greatest integer: $[x] = n \iff n \leq x < n + 1$; $[-2.1] = -3$. Piecewise functions: check the boundary points where formulas meet.

Algebra of Functions and the Domain Playbook — Formula Card

Pointwise operations (f, g real functions):

• $(f \pm g)(x) = f(x) \pm g(x)$; $(fg)(x) = f(x)g(x)$; $(\alpha f)(x) = \alpha f(x)$; $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$.
• Domains: sum/difference/product on dom f $\cap$ dom g; quotient additionally deletes zeros of g.

Domain playbook:

1. polynomial → $\mathbb{R}$; 2. $\frac{1}{g}$ → exclude roots of g; 3. $\sqrt{g}$ → $g \geq 0$; 4. $\frac{1}{\sqrt{g}}$ → $g > 0$ strictly.

• Quadratic signs: upward parabola with roots $\alpha < \beta$: $\geq 0$ outside $[\alpha, \beta]$, $\leq 0$ inside.
• Modulus: $|x| < a \iff -a < x < a$; $|x| > a \iff x < -a$ or $x > a$; $|x - c| \leq a \iff x \in [c - a, c + a]$.

Range toolkit: solve for x in terms of y (legal y = range); complete the square for quadratics; chain bounds for composites; $\frac{ax + b}{cx + d}$ has range $\mathbb{R} - \left\{\frac{a}{c}\right\}$.

JEE Counting and Bracket Results — Quick Card

Counting ($n(A) = m$, $n(B) = n$):

• Relations from A to B: $2^{mn}$; relations on A: $2^{m^2}$.
• Functions from A to B: $n^m$ (domain size in the exponent!).
• With k inputs' values fixed: $n^{m-k}$. One-one functions ($m \leq n$): $n(n-1)\cdots(n-m+1)$; bijections of an m-set: $m!$.
• Not one-one = total $-$ one-one.

Greatest integer and fractional part:

• $x = [x] + \{x\}$, $[x] \in \mathbb{Z}$, $\{x\} \in [0, 1)$; $\{x\} = 0 \iff x \in \mathbb{Z}$.
• $[x + n] = [x] + n$ and $\{x + n\} = \{x\}$ for integer n.
• $[x] + [-x]$ = 0 (integer x) or $-1$ (otherwise); $\{x\} + \{-x\}$ = 0 or 1 correspondingly.
• Solve bracket equations by x = n + t (n integer, $t \in [0, 1)$) — the unknowns decouple.
• Range of $x - [x]$: $[0, 1)$; graph of $\{x\}$: sawtooth, period 1.

One-one / onto preview: one-one = distinct inputs, distinct images; onto = range equals codomain. f(x) = mx + c ($m \neq 0$) on $\mathbb{R}$: both; $x^2$ on $\mathbb{R} \to \mathbb{R}$: neither.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by thousands of students every year:

1. (a, b) is not {a, b} — order matters in pairs; equality means BOTH coordinates match.
2. $A \times B \neq B \times A$ in general — same count, mirrored pairs.
3. Images must stay in the target set — for y = x + 1 on {1, …, 6}, the input 6 has no image; drop it from the domain.
4. Repeated first element (different partners) kills a function; repeated second element never does.
5. Piecewise overlap points: both formulas must agree there, else not a function.
6. $[-2.1] = -3$, not $-2$; and $\{-2.3\} = 0.7$, never negative.
7. Exponent placement: functions $n^m$ (domain up), relations $2^{mn}$ — never swap.
8. Quotient domains: delete zeros of the denominator AFTER intersecting; a root in a denominator makes $\geq$ into $>$.
9. Cancelling changes domains: $\frac{x^2 - 1}{x - 1}$ is NOT the function x + 1 — the point x = 1 stays excluded.
10. Open vs closed range endpoints: check whether the extreme value is actually attained (x² + 2 attains 2; $\frac{x^2}{1 + x^2}$ never attains 1).

Done revising? Take the Section 7 mock drill under exam timing — that is the real test of readiness.