NCERT Plus One (Class 11) Mathematics • Chapter 2

Relations and Functions

Ordered pairs, Cartesian products, domain-range arrows, vertical line test, toolkit function graphs & function algebra.

📄 Official NCERT Textbook PDF

Topic 2.2: Ordered Pairs & Cartesian Product A × B

Imagine booking movie tickets in Kochi: Seat (Row 2, Seat 5) is near the front, while (Row 5, Seat 2) is way back on the side! Even though both pairs use the same numbers 2 and 5, the ORDER changes your seat completely. When you pair every shirt in closet A with every mundu in drawer B, all ordered combos form the Cartesian Product A × B!
ORDERED PAIR(a, b) = (c, d) only when a = c AND b = d. Usually (a, b) ≠ (b, a)!
CARTESIAN PRODUCTA × B = {(a, b) : a ∈ A, b ∈ B}. Always put A first, B second!
COUNTING RULEIf n(A) = p and n(B) = q, then n(A × B) = p · q
🧠 Memory Trick: A set {1, 2} is a bag where order doesn't matter; an ordered pair (1, 2) is a two-slot address where slot 1 comes from the first set!
💡 Simple Kid-Check: Think of a restaurant combo menu: A = {Puttu, Appam} (main) and B = {Kadala, Stew, Egg} (curry). Total combos = 2 × 3 = 6 pairs like (Puttu, Kadala)!
  • 1
    Empty Set Rule: If either A = ∅ or B = ∅, you can't form any pair, so A × B = ∅.
  • 2
    Infinite Rule: If either A or B is infinite (and the other is non-empty), A × B is infinite too.
  • 3
    Ordered Triplets: A × A × A = {(a, b, c) : a, b, c ∈ A} describes 3D points like (x, y, z)!
🧒 Super-Easy Example: If (x + 1, y − 2) = (3, 1), find x and y.
Match first slots: x + 1 = 3 ⇒ x = 2. Match second slots: y − 2 = 1 ⇒ y = 3!
Students often write A × B = B × A! That is FALSE whenever A ≠ B (and both are non-empty), because (1, a) ≠ (a, 1). Also remember: round brackets (1, 2) mean ordered pair, NOT curly set braces {1, 2}!

Topic 2.3: Relations: Domain, Co-domain, Range & Arrow Diagrams

Suppose A = {1, 2, 3} are students and B = {2, 4, 6, 8} are locker numbers. If we connect each student x to locker y = 2x, arrows go to 2, 4, 6 — leaving locker 8 untouched! Every arrow rule between A and B is a Relation, and locker 8 proves why Co-domain can be bigger than Range!
RELATION RULEAny subset R ⊆ A × B is a relation from A to B
DOMAIN / RANGEDomain: all 1st elements in R
Range: all 2nd elements used!
TOTAL RELATIONSIf n(A) = p, n(B) = q, total relations = 2ᵖᑫ
🧠 Memory Trick: Co-domain = the ENTIRE target room B; Range = ONLY the chairs in B where an arrow actually lands! So always Range ⊆ Co-domain!
💡 Simple Kid-Check: Out of all possible pairs in A × B, pick ONLY the pairs that obey a secret rule (like "y is double of x"). That picked subset is your Relation R!
  • 1
    Domain of R: Collect every starting value x from the chosen pairs (x, y) ∈ R.
  • 2
    Range of R: Collect every landing value y that actually receives an arrow in R.
  • 3
    Co-domain of R: Always the complete second set B, whether every item is used or not!
🎯 Quick Example: Let A = {1, 2, 3}, B = {2, 4, 6, 8}, and R = {(x, y) : y = 2x}.
Pairs: R = {(1, 2), (2, 4), (3, 6)}. So Domain = {1, 2, 3}, Range = {2, 4, 6}, and Co-domain = {2, 4, 6, 8}!
Top Kerala DHSE trap: Writing the whole set B as the Range! Remember, B is the Co-domain, while Range contains ONLY the elements of B that appear as second coordinates in R. Also remember total relations from A to B is 2^(n(A)·n(B)), NOT n(A)·n(B)!

Topic 2.4.1: Functions as Special Relations & Vertical Line Intuition

Think of a Kerala juice vending machine: press ONE button (input x), and ONE exact drink (output y) must drop out! If pressing "Lime" drops nothing, OR randomly drops both Lime and Coffee, that machine is faulty — NOT a function! Two different buttons CAN drop the same ₹20 price tag, though — that's still a totally valid Function!
FUNCTION RULEEvery element of A has one and only one image in B
TWO TESTS1) No orphan left in A
2) No input in A splits to 2 outputs!
VERTICAL LINE TESTAny vertical line x = c cuts a function graph at most once!
🧠 Memory Trick: Many-to-One is OK! One-to-Many is BANNED! Never repeat the SAME first coordinate with two different outputs!
💡 Simple Kid-Check: Check the list of pairs (x, y). If you see the SAME x twice with different y-values — like (2, 5) and (2, 9) — shout "NOT A FUNCTION!"
Relation Pairs / GraphFunction?Why?
{(1, 4), (2, 4), (3, 7)}YESInputs 1, 2, 3 each have 1 unique output (many-to-one is valid!)
{(2, 1), (2, 3), (4, 5)}NOInput 2 splits to two outputs (1 and 3)
y = x² (U-parabola)YESEvery vertical line hits the curve only once
x = y² or Circle x² + y² = 9NOA vertical line hits upper & lower branches (2 outputs!)
Watch out for {(1, 5), (2, 5), (3, 5)}! Many students mistakenly say "Not a function because 5 repeats!" NO — outputs are completely allowed to repeat! Only repeated inputs (first numbers) like (1, 5) and (1, 6) disqualify a function!

Topic 2.4.1: Identity, Constant & Polynomial Function Graphs

Picture three friends cycling along Alappuzha canal road! Appu is parked at the tea stall at milestone y = 3 forever (Constant). Meera matches her distance to her watch minutes y = x (Identity). Meanwhile, a cricket ball's height or accelerating rocket bends into smooth Polynomial curves like x² and x³!
IDENTITY f(x) = xDomain = ℝ, Range = ℝ. Straight 45° line through (0, 0)!
CONSTANT f(x) = cDomain = ℝ, Range = {c}. Horizontal line parallel to X-axis!
SQUARE & CUBEx² U-curve: Range [0, ∞)
x³ S-curve: Range ℝ
🧠 Memory Trick: Even powers (x², x⁴) never dip below zero — both negative and positive inputs turn non-negative [0, ∞)!
💡 Simple Kid-Check: An Identity function is like an echo mirror — whatever number you shout in (x), it echoes right back (x). A Constant function is a stubborn parrot — no matter what x you ask, it always repeats c!
FunctionEquationDomainRangeGraph Shape
Identityf(x) = xℝℝLine at 45° through origin
Constantf(x) = cℝ{c}Flat horizontal line at height c
Squaref(x) = x²ℝ[0, ∞)Symmetric U-shaped parabola
Cubef(x) = x³ℝℝS-curve rising from Q3 to Q1
Common 1-mark mistake: Writing the Range of Constant Function f(x) = 5 as ℝ or 5! A range MUST be written as a set: {5}! And for f(x) = x², remember Range is [0, ∞) (bracket closed at 0!), NOT ℝ!

Topic 2.4.1: Rational, Modulus |x|, Signum & Greatest Integer [x] Functions

Four famous exam guests! Distance from home never cares whether you went left or right — it stays positive (Modulus |x|). A railway signal only shouts −1 (reverse), 0 (stop), or +1 (go) — that's Signum! And rounding ₹49.90 down to completed rupees gives 49 — that's the staircase Greatest Integer [x]!
MODULUS |x|Sharp V-shape at (0, 0). D = ℝ, R = [0, ∞)
SIGNUM sgn(x)+1 (x > 0), 0 (x = 0), −1 (x < 0). Range = {−1, 0, 1}!
GREATEST INT [x]Largest integer ≤ x (step left!). D = ℝ, R = ℤ
🧠 Memory Trick: For [x], picture standing on a number line and looking LEFT to the nearest integer step! So [2.9] = 2, AND [−2.1] = −3!
💡 Simple Kid-Check: For Rational Function f(x) = 1/x, dividing by zero explodes! So 0 is banned from BOTH Domain and Range: D = ℝ − {0}, R = ℝ − {0}.
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    Modulus |x|: Equals x if x ≥ 0, and −x if x < 0. Never negative!
  • 2
    Signum f(x): Written piecewise as |x|/x when x ≠ 0 and 0 at x = 0. Only 3 possible outputs: {−1, 0, 1}!
  • 3
    Greatest Integer [x]: Creates a staircase of horizontal steps with closed dot on the LEFT and open circle on the RIGHT!
⚠️ Negative Step Check: What is [−1.4]? Look to the LEFT of −1.4 on the number line → you hit −2 (because −2 ≤ −1.4 < −1)!
The #1 DHSE trap in this chapter: writing [−2.3] = −2! WRONG — −2 is GREATER than −2.3! Greatest integer means largest integer less than or equal to x, so [−2.3] = −3! Also never forget: Range of Signum is ONLY {−1, 0, 1}, and Domain of 1/x is ℝ − {0}.

Topic 2.4.2: Algebra of Real Functions (+, −, ·, /)

Suppose two juice stalls on MG Road earn f(x) = x² rupees and g(x) = 2x + 1 rupees on day x. You can add their earnings (f + g)(x), compare differences (f − g)(x), multiply (fg)(x), or take their ratio (f/g)(x) — just watch out for any day when stall g earns zero!
SUM & DIFF(f ± g)(x) = f(x) ± g(x) on shared domain D(f) ∩ D(g)
SCALAR & PRODUCT(αf)(x) = α·f(x)  |  (fg)(x) = f(x)·g(x)
QUOTIENT (f/g)(f/g)(x) = f(x)/g(x), strictly where g(x) ≠ 0!
🧠 Memory Trick: Combining two functions only works where BOTH machines are switched ON (D(f) ∩ D(g)), and for division, also ban every x that makes denominator g(x) = 0!
💡 Simple Kid-Check: Adding, subtracting, or multiplying two functions works term-by-term at each x! Only division needs an extra security guard checking g(x) ≠ 0!
🧮 Worked Example: Given f(x) = x² and g(x) = 2x + 1 (both defined on ℝ):
• (f + g)(x) = x² + 2x + 1 (Domain: ℝ)
• (f − g)(x) = x² − 2x − 1 (Domain: ℝ)
• (fg)(x) = x²(2x + 1) = 2x³ + x² (Domain: ℝ)
• (f/g)(x) = x²2x + 1, Domain: ℝ − {−1/2} (because 2x + 1 = 0 ⇒ x = −1/2!)
Two classic domain traps in Algebra of Functions: (1) Writing Domain of (f/g)(x) as ℝ when denominator vanishes at a root (e.g. for g(x) = 2x + 1, MUST exclude x = −1/2!). (2) When f(x) = √x and g(x) = x, domain of f + g shrinks to [0, ∞) ∩ ℝ = [0, ∞)!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 1 Marks): If (x + 1, y − 2) = (3, 1), find the values of x and y.

Equating corresponding coordinates:
x + 1 = 3 ⇒ x = 2, and y − 2 = 1 ⇒ y = 3.

Q2 (Kerala HSE Say • 2 Marks): If P = {1, 2}, form the set P × P × P.

All 2³ = 8 ordered triplets:
{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)}.

Q3 (Kerala HSE March • 2 Marks): Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

n(A) = 3, n(B) = 2 ⇒ n(A × B) = 3 × 2 = 6.
Number of relations = 2⁶ = 64.

Q4 (Kerala HSE March • 3 Marks): Let A = {1, 2, 3, ..., 14}. Define a relation R from A to A by R = {(x, y) : 3x − y = 0, x, y ∈ A}. Write its domain, co-domain and range.

Rule gives y = 3x. Keeping y ≤ 14: R = {(1,3), (2,6), (3,9), (4,12)}.
• Domain: {1, 2, 3, 4}
• Range: {3, 6, 9, 12}
• Co-domain: A = {1, 2, ..., 14}

Q5 (Kerala HSE Model • 2 Marks): Is R = {(2, 1), (3, 1), (4, 2)} a function? Justify your answer and state its range.

Yes, every first element (2, 3, 4) has a unique second element (many-to-one is allowed!).
Range: {1, 2}.

Q6 (Kerala HSE March • 3 Marks): Draw/describe the graph of f(x) = |x| and find the domain and range of f(x) = −|x|.

For f(x) = −|x|, every value becomes non-positive (inverted V opening downward from origin).
• Domain: ℝ
• Range: (−∞, 0]

Q7 (Kerala HSE Say • 3 Marks): Find the domain and range of the real function f(x) = √(x − 1).

Inside square root must be ≥ 0: x − 1 ≥ 0 ⇒ x ≥ 1.
• Domain: [1, ∞)
• Since square root is always ≥ 0, Range: [0, ∞).

Q8 (NCERT / DHSE Model • 3 Marks): Let f(x) = √x and g(x) = x be two functions defined over non-negative real numbers. Find (f+g)(x), (f−g)(x), (fg)(x), and (f/g)(x).

• (f+g)(x) = √x + x (x ≥ 0)
• (f−g)(x) = √x − x (x ≥ 0)
• (fg)(x) = x^(3/2) (x ≥ 0)
• (f/g)(x) = 1/√x, strictly for x > 0 (0 excluded!).