Relations and Functions: Class 11 Maths · Chapter 2

Mathematics · Companion notes, simulations and practice · Official NCERT chapter PDF ↗

Published 18 Oct 2026 · Updated 18 Oct 2026

85 min
1 simulation
42 practice questions
21 MCQs

What you will learn in Relations and Functions

Form the Cartesian product of two or three finite sets, count its elements and use the equality of ordered pairs.
Describe a relation as a subset of A × B in roster form, set-builder form and as an arrow diagram, and find its domain, codomain and range.
Count the relations from one finite set to another.
Decide whether a relation is a function, and find the domain and range of a real function.
Sketch and read the graphs of the identity, constant, polynomial, rational, modulus, signum and greatest integer functions.
Add, subtract, multiply and divide real functions and state the domain of each result.
Find a linear function from its values and graph functions defined in pieces.

Original companion text aligned to NCERT Mathematics Class 11, Chapter 2, pp. 24–42. Not affiliated with or endorsed by NCERT.

2.1 Introduction

A railway reservation chart matches every berth with a passenger, a marks register matches every roll number with a score, and your phone matches every contact name with a number. Each of these is really a collection of pairs, and in each pair it matters which item is written first. Mathematics is full of the same pattern: '12 is a multiple of 4' links two numbers, and 'triangle $A B C$ is congruent to triangle $P Q R$ ' links two triangles.

This chapter builds the idea in three layers. First we form every possible pair from two sets. Next we keep only the pairs that obey a rule: a relation. Finally we single out relations in which each starting element has exactly one partner: the functions, the working language of the rest of mathematics, from Chapter 3's trigonometric functions to Chapter 12's derivatives.

2.2 Cartesian Products of Sets

A juice-and-tea stall offers three drinks, $D = {chai, coffee, lassi}$ , each in two cup sizes, $C = {S, L}$ . How many different orders can a customer place? Choose the drink first and the size second. Each of the 3 drinks goes with each of the 2 sizes, so there are $3 \times 2 = 6$ orders, which Fig. 2.1 lays out as a table. Every order is an ordered pair (क्रमित युग्म) such as $(lassi, S)$ , whose first entry is a drink from $D$ and whose second entry is a size from $C$ .

2/4Each cell is an ordered pair: drink first, size second.

Fig. 2.1Every drink pairs with every size: the table holds

D \times C

.NCERT Fig. 2.1 · p. 24

A table whose three rows are labelled with the drinks chai, coffee and lassi (the set D) and whose two columns are labelled with the cup sizes S and L (the set C). In the first step the six cells are empty dashed boxes. In the second step each cell fills with its ordered pair, from (chai, S) in the top left to (lassi, L) in the bottom right. In the third step a line under the table counts them: 3 rows times 2 columns gives 6 pairs. In the fourth step the cell (lassi, S) is highlighted and the line under the table reads (lassi, S) is not equal to (S, lassi).

Order matters. $(lassi, S)$ is a small lassi, while $(S, lassi)$ puts a size where a drink should be. Coordinates behave the same way: on graph paper, $(2, 5)$ and $(5, 2)$ are different points. That is why two ordered pairs count as equal only when their first entries agree and their second entries agree.

Definition

Cartesian product

Given sets $A$ and $B$ , their Cartesian product $A \times B$ (read ' $A$ cross $B$ ') collects every ordered pair that takes its first entry from $A$ and its second entry from $B$ : $A \times B = {(a, b) : a \in A, b \in B}$ . When either set is empty, so is $A \times B$ .

Example ${1, 2} \times {x} = {(1, x), (2, x)}$ .

A cinema hall uses the same idea for its seats. A ticket such as G-14 names a row and then a seat number. If the rows run from A to J and every row has 18 seats, the seat labels are exactly the elements of ${A, B, \dots, J} \times {1, 2, \dots, 18}$ , so the hall has $10 \times 18 = 180$ seats. A label written as 14-G is not a seat at all, because the first entry has to be a row.

Four facts follow directly from the definition of the Cartesian product (कार्तीय गुणन).

Equality. $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ .
Counting. When $A$ has $p$ elements and $B$ has $q$ elements, $A \times B$ has $p q$ elements, because each of the $p$ first entries pairs with each of the $q$ second entries. This is (2.1).
Infinite sets. If $A$ and $B$ are non-empty and at least one of them is infinite, then $A \times B$ is infinite as well.
Triples. Three sets combine in the same way: $A \times B \times C$ consists of all $(a, b, c)$ with $a$ from $A$ , $b$ from $B$ and $c$ from $C$ . Such elements are called ordered triplets.

size of a product

n (A \times B) = p q

(2.1)

where

$p, q$: $n (A)$ and $n (B)$

When both sets contain real numbers, an ordered pair is simply a point of the coordinate plane. Fig. 2.2 plots $A \times B$ for $A = {1, 2, 4}$ and $B = {1, 3}$ , and then $B \times A$ . The two products are mirror images of each other in the line $y = x$ and share only the point $(1, 1)$ . Taking all real numbers, $𝐑 \times 𝐑$ is the whole plane, and $𝐑 \times 𝐑 \times 𝐑$ is three-dimensional space, which returns in Chapter 11.

2/3 $B \times A$ is the mirror image in $y = x$ ; $(4, 3)$ and $(3, 4)$ are different points.

Fig. 2.2A product of number sets is a set of points.NCERT Fig. 2.3 · p. 25

A coordinate grid from 0 to 5 on both axes. In the first step six round dots mark A × B for A = {1, 2, 4} and B = {1, 3}: the points (1, 1), (1, 3), (2, 1), (2, 3), (4, 1) and (4, 3). In the second step six square markers appear for B × A, at (1, 1), (1, 2), (1, 4), (3, 1), (3, 2) and (3, 4), the round dots fade, a dashed mirror line y = x is drawn, and rings mark (4, 3) and (3, 4) to show that swapping the entries gives a different point. In the third step the whole grid is shaded and labelled R × R, the whole plane.

Example 2.1NCERT Example 1 · p. 26

Equal pairs

Find $a$ and $b$ if $(3 a - 2, \frac{b}{2} + 1) = (7, 4)$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.2NCERT Example 2 · p. 26

Swapping the factors

Let $A = {p, q}$ and $B = {1, 2, 3}$ . List $A \times B$ and $B \times A$ . Are the two sets equal? Do they have the same number of elements?

Try it yourself first. Then reveal the solution one step at a time.

Example 2.3NCERT Example 3 · p. 26

Products with intersections and unions

Let $A = {a, b}$ , $B = {1, 2, 3}$ and $C = {2, 3, 4}$ . Compare (i) $A \times (B \cap C)$ with $(A \times B) \cap (A \times C)$ , and (ii) $A \times (B \cup C)$ with $(A \times B) \cup (A \times C)$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.4NCERT Example 4 · p. 27

Three coin tosses

A coin is tossed three times and each result is recorded as $H$ or $T$ . Write the set of all possible records as a Cartesian product and list its elements.

Try it yourself first. Then reveal the solution one step at a time.

Example 2.5NCERT Example 5 · p. 27

Products of number sets

Describe geometrically (i) $𝐙 \times 𝐙$ , (ii) $𝐑 \times 𝐑$ and (iii) $𝐑 \times 𝐑 \times 𝐑$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.6NCERT Example 6 · p. 27

Recovering the sets

If $A \times B = {(1, u), (1, v), (1, w), (4, u), (4, v), (4, w)},$ find $A$ and $B$ .

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 4

n (A) = 5

and

n (B) = 4

, how many elements does

A \times B

have?

Show answer

Answer 20

By (2.1),

n (A \times B) = 5 \times 4 = 20

. Adding the sizes (9) is the common slip: every element of

A

pairs with every element of

B

Practice · 10 questions

Practice 2.1: questions with answers

Short answerEasyNCERT Exercise 2.1, Q1
Find $x$ and $y$ if $(\frac{x}{4} + 2, \frac{1}{3} - y) = (\frac{11}{4}, - \frac{2}{3})$ .
Show solution
Answer $x = 3$ , $y = 1$
Equate first entries: $\frac{x}{4} + 2 = \frac{11}{4}$ , so $\frac{x}{4} = \frac{3}{4}$ and $x = 3$ .
Equate second entries: $\frac{1}{3} - y = - \frac{2}{3}$ , so $y = \frac{1}{3} + \frac{2}{3} = 1$ .
Quick answerEasyNCERT Exercise 2.1, Q2
Set $A$ has 4 elements and $B = {a, e, i, o, u}$ . How many elements does $A \times B$ have?
Show answer
Answer 20
$n (A \times B) = 4 \times 5 = 20$ .
Short answerEasyNCERT Exercise 2.1, Q3
Let $S = {p, q}$ and $T = {0, 1, 9}$ . Write $S \times T$ and $T \times S$ .
Show solution
$S \times T = {(p, 0), (p, 1), (p, 9), (q, 0), (q, 1), (q, 9)} .$
$T \times S = {(0, p), (0, q), (1, p), (1, q), (9, p), (9, q)} .$
Each has 6 elements, but the sets are different.
4
Short answerEasyNCERT Exercise 2.1, Q4
State whether each statement is true or false. Correct each false statement.
1. If $A = {1}$ and $B = {2, 3}$ , then $A \times B = B \times A$ .
  TrueFalse
  Show solution
  False. $A \times B = {(1, 2), (1, 3)}$ but $B \times A = {(2, 1), (3, 1)}$ . Correct version: $A \times B$ and $B \times A$ have the same number of elements but are different sets.
2. For finite sets $A$ and $B$ , $n (A \times B) = n (B \times A)$ .
  TrueFalse
  Show solution
  True: both equal $n (A) \cdot n (B)$ .
3. If $A = {4, 5}$ , then $A \times \emptyset$ has two elements.
  TrueFalse
  Show solution
  False. There is no second entry to pair with, so $A \times \emptyset = \emptyset$ , which has no elements.
Short answerEasyNCERT Exercise 2.1, Q5
If $A = {0, 1}$ , list $A \times A \times A$ . How many of its triplets have entries that add up to 2?
Show solution
Answer8 triplets; 3 of them add up to 2.
$A \times A \times A$ has $2^{3} = 8$ triplets: $(0, 0, 0)$ , $(0, 0, 1)$ , $(0, 1, 0)$ , $(0, 1, 1)$ , $(1, 0, 0)$ , $(1, 0, 1)$ , $(1, 1, 0)$ , $(1, 1, 1)$ .
Entries add up to 2 when exactly two entries are 1: $(0, 1, 1)$ , $(1, 0, 1)$ and $(1, 1, 0)$ , so 3 triplets.
Short answerEasyNCERT Exercise 2.1, Q6
If $A \times B = {(2, m), (2, n), (5, m), (5, n), (7, m), (7, n)},$ find $A$ and $B$ .
Show solution
Answer $A = {2, 5, 7}$ , $B = {m, n}$
First entries: $A = {2, 5, 7}$ . Second entries: $B = {m, n}$ . Check: $3 \times 2 = 6$ pairs.
7
Short answerMediumNCERT Exercise 2.1, Q7
Let $A = {p, q}$ , $B = {1, 3, 5}$ and $C = {3, 5, 7}$ .
1. Verify that $A \times (B \cap C) = (A \times B) \cap (A \times C)$ .
  Show solution
  $B \cap C = {3, 5}$ , so the left side is ${(p, 3), (p, 5), (q, 3), (q, 5)} .$
  A pair is in both $A \times B$ and $A \times C$ exactly when its second entry is in $B$ and in $C$ , i.e. is 3 or 5. That gives the same four pairs, so the two sides are equal.
2. With $E = {p, q, r}$ and $D = {1, 3, 5, 7, 9}$ , show that $A \times C \subset E \times D$ .
  Show solution
  Take any $(x, y) \in A \times C$ . Then $x \in A \subset E$ and $y \in C \subset D$ , so $(x, y) \in E \times D$ . Hence $A \times C \subset E \times D$ ; for example $(r, 1) \in E \times D$ is not in $A \times C$ , so the inclusion is strict.
Short answerEasyNCERT Exercise 2.1, Q8
For $A = {a}$ and $B = {1, 2, 3}$ , list the elements of $A \times B$ , say how many subsets it has, and write them all out.
Show solution
$A \times B = {(a, 1), (a, 2), (a, 3)}$ , a set with 3 elements, so it has $2^{3} = 8$ subsets:
$\emptyset$ , ${(a, 1)}$ , ${(a, 2)}$ , ${(a, 3)}$ , ${(a, 1), (a, 2)}$ , ${(a, 1), (a, 3)}$ , ${(a, 2), (a, 3)}$ and $A \times B$ itself.
Short answerMediumNCERT Exercise 2.1, Q9
Sets $A$ and $B$ have $n (A) = 2$ and $n (B) = 3$ . The pairs $(p, u)$ , $(q, v)$ and $(p, w)$ belong to $A \times B$ , where $p \neq q$ and $u$ , $v$ , $w$ are distinct. Find $A$ and $B$ .
Show solution
Answer $A = {p, q}$ , $B = {u, v, w}$
Every first entry lies in $A$ , so $p, q \in A$ . They are distinct and $n (A) = 2$ , so $A = {p, q}$ .
Every second entry lies in $B$ , so $u, v, w \in B$ . They are distinct and $n (B) = 3$ , so $B = {u, v, w}$ .
Short answerMediumNCERT Exercise 2.1, Q10
Two of the nine elements of $A \times A$ are $(2, - 1)$ and $(5, 2)$ . Determine $A$ , then list the rest of $A \times A$ .
Show solution
$n (A)^{2} = 9$ , so $n (A) = 3$ . The entries $2$ , $- 1$ and $5$ must all belong to $A$ , and that is already three elements, so $A = {- 1, 2, 5}$ .
The other seven elements are $(- 1, - 1)$ , $(- 1, 2)$ , $(- 1, 5)$ , $(2, 2)$ , $(2, 5)$ , $(5, - 1)$ and $(5, 5)$ .

2.3 Relations

Four friends, Asha, Ravi, Meena and Kabir, and four games, cricket, kabaddi, chess and hockey, give $4 \times 4 = 16$ possible (student, game) pairs. Usually only some of these pairs interest us, for example the pairs in which the student actually plays the game. Picking out pairs by a rule like this produces a relation (संबंध). Fig. 2.3 draws one as an arrow diagram, with an arrow from each student to every game that student plays.

3/3The range is the set of games that are hit: hockey is in the codomain but not in the range.

Fig. 2.3One relation, three sets to read from it.NCERT Fig. 2.4 · p. 28

Two ovals: the left one, set A, holds the students Asha, Ravi, Meena and Kabir; the right one, set B, holds the games cricket, kabaddi, chess and hockey. Four arrows go from Asha to cricket, Asha to chess, Ravi to kabaddi and Kabir to cricket. In the second step Asha, Ravi and Kabir are boxed as the domain, while Meena fades with the note that she has no arrow. In the third step cricket, kabaddi and chess are boxed as the range, while hockey fades with the note that no arrow reaches it, though it is still in the codomain B.

Definition

Relation

Given non-empty sets $A$ and $B$ , any subset $R$ of $A \times B$ is called a relation from $A$ to $B$ . Usually a rule connecting the two entries decides which pairs belong to $R$ . When $(a, b) \in R$ , we call $b$ an image of $a$ .

ExampleFrom ${1, 2, 3}$ to ${1, 4, 9}$ , the rule 'the second entry is the square of the first' gives $R = {(1, 1), (2, 4), (3, 9)}$ .

Definition

Domain

The domain of a relation $R$ from $A$ to $B$ is the set of all first entries of the pairs in $R$ . It is a subset of $A$ .

Definition

Codomain

The codomain of a relation from $A$ to $B$ is the whole set $B$ , whether or not every element of $B$ is used.

Definition

Range

The range of $R$ collects the second entries (the images) of its pairs. It is always a subset of the codomain.

Read these off Fig. 2.3. The relation is ${(Asha, cricket), (Asha, chess), (Ravi, kabaddi), (Kabir, cricket)} .$ Its domain (प्रांत) is ${Asha, Ravi, Kabir}$ : Meena plays none of these games, so she is never a first entry. Its codomain (सहप्रांत) is the set of all four games, but its range (परिसर) is only ${cricket, kabaddi, chess}$ , since nobody plays hockey. Asha has two images (प्रतिबिंब), cricket and chess.

There are three standard ways to present a relation. Roster form lists the pairs. Set-builder form states the rule, as in $R = {(x, y) : y = x^{2}, x \in A, y \in B}$ . An arrow diagram draws the pairs. A relation from a set $A$ to the same set $A$ is called a relation on $A$ .

Example 2.7NCERT Example 7 · p. 28

Doubling inside one set

Let $A = {1, 2, 3, \dots, 9}$ and let $R = {(x, y) : y = 2 x, x, y \in A}$ be a relation on $A$ . (i) Write $R$ in roster form and describe its arrow diagram. (ii) Find the domain, codomain and range of $R$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.8NCERT Example 8 · p. 29

Reading a rule from arrows

An arrow diagram from $P = {6, 8, 9}$ to $Q = {2, 3, 4, 5}$ has exactly these arrows: $6 \to 2$ , $6 \to 3$ , $8 \to 2$ , $8 \to 4$ and $9 \to 3$ . Write the relation in set-builder form and in roster form, and find its domain and range.

Try it yourself first. Then reveal the solution one step at a time.

How many relations can be defined from $A$ to $B$ ? A relation is just a subset of $A \times B$ , so we are counting subsets. With $p$ elements in $A$ and $q$ in $B$ , the product $A \times B$ has $p q$ pairs, and each pair is either in the relation or out of it: two choices per pair, made independently.

number of relations

N = 2^{p q}

(2.2)

where

$N$: the count of all subsets of $A \times B$
$p, q$: $n (A)$ and $n (B)$

The count includes the two extreme cases: the empty relation $\emptyset$ , in which nothing is related, and the whole of $A \times B$ , in which everything is.

Example 2.9NCERT Example 9 · p. 29

Counting relations

Let $A = {x, y}$ and $B = {1, 2, 3}$ . (i) How many relations are there from $A$ to $B$ ? (ii) How many of them contain the pair $(x, 1)$ ?

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 4

A = {1, 2, \dots, 8}

, let

R = {(x, y) : x + y = 10}

. What is the range of

R

{1, 2, \dots, 8}

{5}

{2, 3, \dots, 8}

{10}

Show answer

Answer (C) ${2, 3, \dots, 8}$

The pairs are

(2, 8), (3, 7), \dots, (8, 2)

, so the second entries run from 2 to 8. The element 1 is not in the range: its partner would be 9, which is not in

A

. The codomain is all of

A

; the range is smaller.

Practice · 9 questions

Practice 2.2: questions with answers

Short answerEasyNCERT Exercise 2.2, Q1
Let $A = {1, 2, 3, \dots, 20}$ and $R = {(x, y) : y = 4 x, x, y \in A}$ . Find the domain, codomain and range of $R$ .
Show solution
$4 x \leq 20$ only for $x \leq 5$ , so $R = {(1, 4), (2, 8), (3, 12), (4, 16), (5, 20)} .$
Domain $= {1, 2, 3, 4, 5}$ ; codomain $= A$ ; range $= {4, 8, 12, 16, 20}$ .
Short answerEasyNCERT Exercise 2.2, Q2
On $𝐍$ , take $R = {(x, 3 x - 1) : x \in 𝐍, x \leq 4}$ . List $R$ and give its domain and range.
Show solution
$x = 1, 2, 3, 4$ give $y = 2, 5, 8, 11$ . So $R = {(1, 2), (2, 5), (3, 8), (4, 11)}$ , with domain ${1, 2, 3, 4}$ and range ${2, 5, 8, 11}$ .
Short answerMediumNCERT Exercise 2.2, Q3
Let $A = {2, 3, 4, 7}$ and $B = {5, 6, 10}$ . Write in roster form the relation $R$ from $A$ to $B$ defined by ' $x + y$ is a multiple of 3'.
Show solution
Check all 12 sums. The multiples of 3 are $2 + 10 = 12$ , $3 + 6 = 9$ , $4 + 5 = 9$ and $7 + 5 = 12$ ; the other eight sums (7, 8, 8, 13, 10, 14, 13, 17) are not.
So $R = {(2, 10), (3, 6), (4, 5), (7, 5)}$ .
Short answerEasyNCERT Exercise 2.2, Q4
An arrow diagram from $P = {10, 12, 14}$ to $Q = {5, 6, 7, 8}$ has the arrows $10 \to 5$ , $12 \to 6$ and $14 \to 7$ . Write the relation in set-builder form and in roster form, and find its domain and range.
Show solution
Each arrow halves its starting number, so the rule is $y = \frac{x}{2}$ .
Set-builder form: $R = {(x, y) : y = \frac{x}{2}, x \in P, y \in Q}$ . Roster form: $R = {(10, 5), (12, 6), (14, 7)}$ .
Domain $= {10, 12, 14}$ ; range $= {5, 6, 7}$ (8 is in $Q$ but receives no arrow).
5
Short answerMediumNCERT Exercise 2.2, Q5
On $A = {2, 3, 4, 6, 9}$ , let $R = {(a, b) : a \neq b and b is a multiple of a}$ .
1. Write $R$ in roster form.
  Show solution
  Multiples inside $A$ : of 2 are 4 and 6; of 3 are 6 and 9; 4, 6 and 9 have none other than themselves. So $R = {(2, 4), (2, 6), (3, 6), (3, 9)}$ .
2. Find the domain of $R$ .
  Show solution
  Domain $= {2, 3}$ .
3. Find the range of $R$ .
  Show solution
  Range $= {4, 6, 9}$ .
Short answerEasyNCERT Exercise 2.2, Q6
Find the domain and range of $R = {(x, x^{2} - 1) : x \in {- 2, - 1, 0, 1, 2}}$ .
Show solution
The pairs are $(- 2, 3)$ , $(- 1, 0)$ , $(0, - 1)$ , $(1, 0)$ and $(2, 3)$ .
Domain $= {- 2, - 1, 0, 1, 2}$ ; range $= {- 1, 0, 3}$ (repeated values are listed once).
Short answerEasyNCERT Exercise 2.2, Q7
The relation $R = {(p, 2^{p})}$ pairs each prime $p$ below 8 with a power of 2. Write $R$ in roster form.
Show solution
The primes below 8 are 2, 3, 5 and 7, so $R = {(2, 4), (3, 8), (5, 32), (7, 128)}$ .
Quick answerEasyNCERT Exercise 2.2, Q8
How many relations can be defined from ${p, q, r}$ to ${1, 2, 3}$ ?
Show answer
Answer 512
$A \times B$ has 9 pairs, so there are $2^{9} = 512$ relations.
Short answerMediumNCERT Exercise 2.2, Q9
Let $R = {(a, b) : a, b \in 𝐍 and b = a + 2}$ . Find the domain and range of $R$ .
Show solution
Every natural number $a$ has the partner $a + 2 \in 𝐍$ , so the domain is $𝐍$ .
The second entries are $3, 4, 5, \dots$ The numbers 1 and 2 never appear, because they would need $a = - 1$ or $a = 0$ . So the range is ${3, 4, 5, \dots} = 𝐍 - {1, 2}$ .

2.4 Functions

Some relations have a special property. In India every SIM card is registered to exactly one person, although one person may hold two or three SIMs. So the relation 'SIM → registered owner' sends each SIM to exactly one person, while 'person → SIM' can send one person to several numbers. Relations of the first kind behave like a machine that turns each input into one definite output. They are called functions (फलन); the words map and mapping mean the same thing.

Definition

Function

Let $A$ and $B$ be non-empty sets. A relation $f \subseteq A \times B$ is a function from $A$ to $B$ when each element of $A$ is the first entry of exactly one pair in $f$ . We then write $f : A \to B$ , and $f (a) = b$ for the pair $(a, b)$ .

Example $f : {1, 2, 3} \to {0, 1}$ with $f (1) = 1$ , $f (2) = 0$ , $f (3) = 1$ .

We read $f (a) = b$ as ' $f$ of $a$ equals $b$ ': the output $b$ is the image of $a$ , while $a$ is a preimage of $b$ (पूर्व प्रतिबिंब). Two separate conditions are packed into the definition, and Fig. 2.4 tests each one. First, no element of $A$ may be left out, so the domain of $f$ is the whole of $A$ . Second, no element of $A$ may have two images, so no two pairs of $f$ share a first entry. Two different inputs may share an output, and some elements of $B$ may receive no arrow at all: neither breaks the rule.

2/31 now has two images, $a$ and $b$ : not a function.

Fig. 2.4One arrow out of every element of

A

, no more and no less.

Two ovals, A holding 1, 2 and 3 and B holding a, b, c and d. In the first step the arrows are 1 to a, 2 to c and 3 to c, and a tick label says this is a function: 2 and 3 share the image c and d receives nothing, which is allowed. In the second step an extra dashed arrow from 1 to b appears, 1 is ringed, and the label reads that 1 has two images, so this is not a function. In the third step the extra arrow is gone but the arrow from 3 is removed, 3 is ringed, and the label reads that 3 has no image, so this is not a function either.

Test the relations of Section 2.3 against this. In Example 2.7 the numbers 5 to 9 have no image, so that relation is not a function on $A$ . In Example 2.8 the number 6 has two images, 2 and 3. Of the 64 relations in Example 2.9, only those that contain exactly one pair starting with $x$ and exactly one pair starting with $y$ are functions, and there are $3 \times 3 = 9$ of them.

When $A$ and $B$ are sets of real numbers, a relation is a set of points in the plane and the test becomes visual. A function never puts two points on the same vertical line, and it has a point above (or below) every $x$ in its domain. Try this in the simulation.

SimulationIs it still a function?

A coordinate grid from 0 to 8 on both axes with four draggable points A (1, 3), B (2, 5), C (4, 5) and D (6, 2), each showing its coordinates and snapping to whole numbers. Together they form the relation {(1, 3), (2, 5), (4, 5), (6, 2)}. Dragging a point changes one pair, so you can create two pairs with the same first entry and see when the relation stops being a function.

Fig. 2.5Each point is one pair

(x, y)

of a relation. Drag the points and watch for two of them on one vertical line.

Try this

Drag $D$ to $(4, 1)$ , directly below $C$ . Which input now has two images?
$B$ and $C$ share the $y$ -coordinate 5. Does that stop the relation being a function?
Arrange the points so that the domain is ${1, 2, 3, 4}$ and the relation is still a function.

Example 2.10NCERT Example 10 · p. 31

Squares of natural numbers

Let $R = {(x, y) : y = x^{2}, x, y \in 𝐍}$ . Find the domain, codomain and range of $R$ . Is $R$ a function from $𝐍$ to $𝐍$ ?

Try it yourself first. Then reveal the solution one step at a time.

Example 2.11NCERT Example 11 · p. 31

Which lists are functions?

Decide whether each relation is a function from ${1, 2, 3, 4}$ to $𝐍$ , giving a reason. (i) ${(1, 5), (2, 5), (3, 5), (4, 5)}$ (ii) ${(1, 2), (2, 3), (3, 4)}$ (iii) ${(1, 1), (2, 4), (3, 9), (2, 6), (4, 16)}$

Try it yourself first. Then reveal the solution one step at a time.

Definition

Real function

A real function is one whose domain and range both lie inside $𝐑$ . If only the range is required to lie in $𝐑$ , the function is called real-valued.

Example $f (x) = \sqrt{x}$ with domain $[0, \infty)$ .

Most functions in this book are real functions (वास्तविक फलन) given by a formula. When only the formula is given, we take its domain to be the largest set of real numbers for which the formula makes sense. For example, $\frac{1}{x - 2}$ has domain $𝐑 - {2}$ and $\sqrt{x + 1}$ has domain $[- 1, \infty)$ .

Example 2.12NCERT Example 12 · p. 31

A table of auto fares

In one town an auto-rickshaw charges a fixed ₹25 plus ₹15 for every kilometre, so the fare for $x$ km is $f (x) = 15 x + 25$ rupees, where $x \in 𝐍$ . Make a table of $f (x)$ for $x = 1$ to $6$ .

Try it yourself first. Then reveal the solution one step at a time.

2.4.1 Some functions and their graphs

The graph of a real function $f$ is the set of points $(x, f (x))$ , a subset of $𝐑 \times 𝐑$ . A sketch is often the fastest way to see a function's domain, its range and how it behaves. Here are the seven families you will keep using for the rest of school mathematics.

(i) The identity function. The function $f : 𝐑 \to 𝐑$ , $f (x) = x$ , returns every input unchanged (तत्समक फलन). Its graph is the line $y = x$ , which passes through the origin and makes an angle of $45^{\circ}$ with each axis. Its domain and range are both $𝐑$ .

(ii) Constant functions. For a fixed number $c$ , the function $f : 𝐑 \to 𝐑$ , $f (x) = c$ , gives the same output to every input (अचर फलन). Its graph is a horizontal line at height $c$ . The domain is $𝐑$ , but the range is the single-element set ${c}$ . Fig. 2.6 shows both graphs.

2/2 $y = 2$ : every input gives 2, so the range is ${2}$ .

Fig. 2.6Identity and constant functions are both straight lines.NCERT Fig. 2.8 · p. 32

Axes from minus 4 to 4. In the first step the line y = x rises through the origin at 45 degrees, with marked points (−3, −3) and (2, 2); labels give its domain and range as all real numbers. In the second step the identity line fades and the horizontal line y = 2 appears, with marked points at x = −3 and x = 1, both at height 2; the labels now give the domain as all real numbers and the range as the single value 2.

(iii) Polynomial functions. These are built from whole-number powers of $x$ (बहुपद फलन).

Definition

Polynomial function

A function $f : 𝐑 \to 𝐑$ is a polynomial function if $f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$ for some non-negative integer $n$ and real numbers $a_{0}, a_{1}, \dots, a_{n}$ .

Example $x^{5} - 3 x + 7$ and $\frac{2}{3} x^{2}$ .

polynomial function

\begin{aligned} f (x) & = a_{0} + a_{1} x \\ + a_{2} x^{2} + \dots \\ + a_{n} x^{n} \end{aligned}

(2.3)

The coefficients may be any real numbers, but the powers must be $0, 1, 2, \dots$ So $g (x) = π x^{3} - 5 x + \frac{1}{2}$ is a polynomial function, while $h (x) = x^{1 / 2} + 1$ and $k (x) = x + \frac{1}{x}$ are not, because they contain the powers $\frac{1}{2}$ and $- 1$ .

Example 2.13NCERT Example 13 · p. 33

The squaring function

For $f : 𝐑 \to 𝐑$ , $f (x) = x^{2}$ , find $f (x)$ at $x = - 2$ , $- 1.5$ , $- 1$ , $- 0.5$ , $0$ , $0.5$ , $1$ , $1.5$ , $2$ . Sketch the graph and state the domain and range.

Try it yourself first. Then reveal the solution one step at a time.

2/3A smooth curve through them is the parabola $y = x^{2}$ .

Fig. 2.7Two power functions with different shapes.NCERT Fig. 2.10 · p. 33

Axes with x from minus 3 to 3 and y from minus 8 to 8 (the vertical scale is compressed). In the first step nine dots mark the table values of x squared from x = −2 to 2: 4, 2.25, 1, 0.25, 0, 0.25, 1, 2.25, 4. In the second step a smooth U-shaped parabola passes through the dots; labels note that its range is from 0 upwards and that it is its own mirror image in the y-axis. In the third step the parabola fades and the cubic curve y = x cubed appears, falling to −8 at x = −2, flattening through the origin and rising to 8 at x = 2; labels note that its range is all real numbers and that it rises everywhere.

Example 2.14NCERT Example 14 · p. 34

The cubing function

Sketch the graph of $f : 𝐑 \to 𝐑$ , $f (x) = x^{3}$ , using the same inputs as in Example 2.13, and find its range.

Try it yourself first. Then reveal the solution one step at a time.

(iv) Rational functions. Dividing one polynomial function by another gives a rational function (परिमेय फलन).

Definition

Rational function

A function of the form $\frac{p (x)}{q (x)}$ , where $p$ and $q$ are polynomial functions. It is defined wherever $q (x) \neq 0$ .

Example $\frac{x + 1}{x - 2}$ , defined for every $x \neq 2$ .

Example 2.15NCERT Example 15 · p. 34

The reciprocal function

Let $f (x) = \frac{1}{x}$ for $x \in 𝐑 - {0}$ . Find $f (x)$ at $x = - 4$ , $- 2$ , $- 1$ , $- 0.5$ , $- 0.25$ , $0.25$ , $0.5$ , $1$ , $2$ , $4$ , sketch the graph and find the domain and range.

Try it yourself first. Then reveal the solution one step at a time.

2/3Join them: one branch for $x > 0$ , one for $x < 0$ .

Fig. 2.8The reciprocal function has two branches.NCERT Fig. 2.12 · p. 35

Axes from minus 4 to 4. In the first step ten dots mark the table values of 1/x, from (−4, −0.25) to (4, 0.25), with points far from the axes at (0.25, 4) and (−0.25, −4). In the second step two smooth branches join the dots: one in the first quadrant and one in the third, each falling as x moves right. In the third step both axes are drawn dashed and labelled: x is never 0 and y is never 0, so the graph never meets either axis.

(v) The modulus function. The function $f (x) = ∣ x ∣$ gives the distance of $x$ from 0 on the number line, so it is never negative (मापांक फलन). In two pieces:

modulus function

∣ x ∣ = {\begin{cases} x, & x \geq 0 \\ - x, & x < 0 \end{cases}

(2.4)

To draw the modulus function, start from the line $y = x$ and flip the part below the $x$ -axis upwards (Fig. 2.9). The result is a V with its corner at the origin. The domain is $𝐑$ and the range is $[0, \infty)$ .

3/3Opposite inputs have equal outputs: $∣ - 3 ∣ = ∣ 3 ∣ = 3$ .

Fig. 2.9Folding

y = x

gives

y = ∣ x ∣

.NCERT Fig. 2.13 · p. 35

Axes from minus 4 to 4. In the first step the whole line y = x is drawn. In the second step the half of the line to the left of the y-axis flips upwards about the x-axis, leaving a faint dashed copy where it was; the solid graph is now a V with its corner at the origin, labelled y = |x|. In the third step dots at (−3, 3) and (3, 3) are joined by a dashed level line, and labels note that |−3| = |3| = 3 and that the range is from 0 upwards.

(vi) The signum function. Signum is Latin for 'sign', and the signum function reports only the sign of its input (चिह्न फलन):

signum function

sgn x = {\begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ - 1, & x < 0 \end{cases}

(2.5)

Its graph (Fig. 2.10) consists of two horizontal rays at heights 1 and $- 1$ , each with a hollow end on the $y$ -axis, and the single point $(0, 0)$ . The domain is $𝐑$ and the range is ${- 1, 0, 1}$ . For $x \neq 0$ you can also write $sgn x = \frac{∣ x ∣}{x}$ .

3/3 $sgn 0 = 0$ : the filled dot at the origin completes the graph.

Fig. 2.10The signum function keeps only the sign.NCERT Fig. 2.14 · p. 36

Axes with x from minus 4 to 4 and y from minus 3 to 3. In the first step a horizontal ray at height 1 runs to the right from a hollow circle at (0, 1), labelled: x positive gives 1. In the second step a ray at height −1 runs to the left from a hollow circle at (0, −1), labelled: x negative gives −1. In the third step a filled dot is placed at the origin, labelled: x = 0 gives 0, and the range is listed as −1, 0 and 1.

(vii) The greatest integer function. The greatest integer function $f (x) = [x]$ returns the greatest integer that is less than or equal to $x$ (महत्तम पूर्णांक फलन). So $[3.7] = 3$ , $[5] = 5$ and $[- 0.4] = - 1$ . For negative inputs, rounding down moves away from zero.

greatest integer function

\begin{array}{c} [x] = n \\ when n \leq x < n + 1 \end{array}

(2.6)

where

$n$: an integer

On each interval $[n, n + 1)$ the value stays at $n$ , and at the next integer it jumps up by 1. The graph is a staircase (Fig. 2.11): each step includes its left end (filled dot) but not its right end (hollow dot). The domain is $𝐑$ and the range is the set of integers $𝐙$ .

2/2Each step covers $[n, n + 1)$ : filled at the left end, hollow at the right.

Fig. 2.11The greatest integer function is a staircase.NCERT Fig. 2.15 · p. 36

Axes from minus 3 to 4. In the first step two inputs are marked on the x-axis, 2.6 and −1.3, with dashed arrows from each down or across to its output level: 2.6 goes to 2 and −1.3 goes to −2; labels stress that the function always rounds down. In the second step the full staircase appears: horizontal steps at heights −3 to 3, each starting with a filled dot at its left end and ending with a hollow circle at its right end; labels give the range as the integers and note that the filled end belongs to the step.

InteractiveFunction lab

Two modes. In Graphs mode, buttons choose the identity, constant, square, cube, reciprocal, modulus, signum or greatest integer function; the graph is drawn on a grid from −5 to 5, a slider moves an input x and marks the point (x, f(x)), and a panel states the domain and range (with a slider for the constant c). In Algebra mode, two menus choose f and g from x, x², x + 1, 2x − 3, |x|, √x and 1/x, and buttons choose f + g, f − g, f · g, f ÷ g or k · f (with a slider for k). The graphs of f and g are drawn faintly and the result boldly, with hollow circles at removed points, and the panel gives f(x), g(x), the result and the domain of the result, including the zeros of g that a quotient must leave out.

Fig. 2.12Graph the seven families, move an input along the

x

-axis, and combine two functions by adding, subtracting, multiplying or dividing them.

Quick check

1 of 4

Match each function with its range.

Tap an item on the left, then its match on the right.

Show answer

Answer

$x^{2}$ : $[0, \infty)$
$x^{3}$ : $𝐑$
$sgn x$ : ${- 1, 0, 1}$
$\frac{1}{x}$ : $𝐑 - {0}$

Squares are never negative; cubes take every real value; signum takes only three values;

\frac{1}{x}

takes every value except 0.

2.4.2 Algebra of real functions

Real functions can be combined much as numbers are, one input at a time. Take two real functions $f$ and $g$ with the same domain $X \subseteq 𝐑$ . To find $(f + g) (x)$ , work out $f (x)$ and $g (x)$ and add them. The same pointwise idea defines the difference, the product, the quotient and a scalar multiple $α f$ , where the scalar $α$ is a real number.

Table 2.1The algebra of real functions

f, g : X \to 𝐑

, for each

x \in X

(

α

is a real number).

Combination	Value at $x$
sum	$(f + g) (x) = f (x) + g (x)$
difference	$(f - g) (x) = f (x) - g (x)$
scalar multiple	$(α f) (x) = α f (x)$
product	$(f g) (x) = f (x) g (x)$
quotient	$(\frac{f}{g}) (x) = \frac{f (x)}{g (x)}$ , where $g (x) \neq 0$

The quotient needs an extra condition: $(\frac{f}{g}) (x)$ exists only where $g (x) \neq 0$ , so its domain is $X$ with the zeros of $g$ removed. If $f$ and $g$ start with different domains, the combinations are defined on the part they share. The product $f g$ is called pointwise multiplication, to keep it apart from composition, the chaining of one function after another that you will study in Class 12.

On a graph, adding two functions means stacking heights (Fig. 2.13). At each $x$ , the point of $f + g$ sits a distance $g (x)$ above the point of $f$ , or below it when $g (x)$ is negative.

3/3Doing this at every $x$ traces the graph of $f + g$ .

Fig. 2.13To add functions, add their heights at every

x

Axes with x from minus 4 to 4 and y from minus 2 to 7. In the first step the straight line f(x) = x/2 + 1 and the dashed parabola g(x) = x²/4 are drawn. In the second step two stacked bars appear: at x = 2 a bar of height f(2) = 2 with a bar of height g(2) = 1 on top, reaching 3; at x = −3 a short bar down to f(−3) = −0.5 with a bar of height g(−3) = 2.25 starting from there and reaching 1.75. Dots mark the tops. In the third step the curve of f + g, equal to x²/4 + x/2 + 1, passes through both dots while f and g fade.

Example 2.16NCERT Example 16 · p. 37

Four combinations

Let $f (x) = x^{2} - 1$ and $g (x) = x + 3$ be real functions on $𝐑$ . Write down formulas for $f + g$ , $f - g$ , $f g$ and $\frac{f}{g}$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.17NCERT Example 17 · p. 37

A square root in the mix

Let $f (x) = \sqrt{x}$ and $g (x) = 3 - x$ on $X = [0, \infty)$ . Express $f + g$ , $f - g$ , $f g$ and $\frac{f}{g}$ as formulas, and give the domain of each.

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 3

f (x) = 2 x + 1

and

g (x) = x^{2}

, find

(f - g) (3)

Show answer

Answer -2

f (3) = 7

and

g (3) = 9

, so

(f - g) (3) = 7 - 9 = - 2

. Watch the order:

g - f

would give

+ 2

Practice · 5 questions

Practice 2.3: questions with answers

1
Short answerEasyNCERT Exercise 2.3, Q1
For each set of pairs, say whether it is a function; for those that are, state the domain and range.
1. ${(- 2, 4), (- 1, 1), (0, 0), (1, 1), (2, 4)}$
  Show solution
  A function: no first entry repeats. Domain ${- 2, - 1, 0, 1, 2}$ ; range ${0, 1, 4}$ .
2. ${(4, 2), (4, - 2), (9, 3), (9, - 3)}$
  Show solution
  Not a function: 4 has the two images 2 and $- 2$ (and 9 has two as well).
3. ${(5, 1), (6, 1), (7, 0)}$
  Show solution
  A function. Domain ${5, 6, 7}$ ; range ${0, 1}$ .
2
Short answerMediumNCERT Exercise 2.3, Q2
Find the domain and range of each real function.
1. $f (x) = 3 - ∣ x ∣$
  Show solution
  Every real $x$ is allowed, so the domain is $𝐑$ . Since $∣ x ∣ \geq 0$ , $f (x) \leq 3$ , and $f (x) = 3 - t$ takes every value $\leq 3$ as $t = ∣ x ∣$ runs over $[0, \infty)$ . Range $(- \infty, 3]$ .
2. $f (x) = \sqrt{16 - x^{2}}$
  Show solution
  Need $16 - x^{2} \geq 0$ , i.e. $- 4 \leq x \leq 4$ : domain $[- 4, 4]$ . Then $16 - x^{2}$ runs from 0 (at $x = \pm 4$ ) to 16 (at $x = 0$ ), so the range is $[0, 4]$ .
3
Short answerEasyNCERT Exercise 2.3, Q3
For $g (x) = 7 - 3 x$ , find the values below.
1. $g (0)$
  Show answer
  Answer 7
  $7 - 0 = 7$ .
2. $g (4)$
  Show answer
  Answer -5
  $7 - 12 = - 5$ .
3. $g (- 2)$
  Show answer
  Answer 13
  $7 + 6 = 13$ .
4
Short answerMediumNCERT Exercise 2.3, Q4
The function $s (F) = \frac{5 (F - 32)}{9}$ converts a temperature of $F$ degrees Fahrenheit into degrees Celsius.
1. Find $s (212)$ .
  Show answer
  Answer 100
  $\frac{5 \times 180}{9} = 100$ : water boils at $212^{\circ} F$ .
2. Find $s (98.6)$ .
  Show answer
  Answer 37
  $\frac{5 \times 66.6}{9} = 37$ : normal body temperature.
3. Find $s (- 40)$ .
  Show answer
  Answer -40
  $\frac{5 \times (- 72)}{9} = - 40$ : the two scales agree at $- 40$ .
4. Find $F$ when $s (F) = 25$ .
  Show answer
  Answer 77
  $\frac{5 (F - 32)}{9} = 25$ gives $F - 32 = 45$ , so $F = 77$ .
5
Short answerMediumNCERT Exercise 2.3, Q5
Find the range of each function.
1. $f (x) = 4 x + 1$ , $x \in 𝐑$ , $x < 0$
  Show solution
  $x < 0$ gives $4 x < 0$ , so $f (x) < 1$ ; every $y < 1$ comes from $x = \frac{y - 1}{4} < 0$ . Range $(- \infty, 1)$ .
2. $f (x) = 3 - x^{2}$ , $x \in 𝐑$
  Show solution
  $x^{2} \geq 0$ , so $f (x) \leq 3$ , with every value $\leq 3$ reached. Range $(- \infty, 3]$ .
3. $f (x) = - x$ , $x \in 𝐑$
  Show solution
  Every real $y$ is $f (- y)$ , so the range is $𝐑$ .

Miscellaneous Examples

These examples combine the ideas of the chapter and preview methods you will use again in coordinate geometry and calculus.

Example 2.18NCERT Example 18 · p. 38

Graph of a linear function

Sketch the graph of $f : 𝐑 \to 𝐑$ , $f (x) = 6 - 2 x$ , and find where it meets the axes.

Try it yourself first. Then reveal the solution one step at a time.

2/2One step right, two steps down: slope $- 2$ , $y$ -intercept 6.

Fig. 2.14A linear function has a straight-line graph.NCERT Fig. 2.16 · p. 38

Axes with x from minus 2 to 5 and y from minus 4 to 10. In the first step five dots mark (0, 6), (1, 4), (2, 2), (3, 0) and (4, −2), with the intercepts (0, 6) and (3, 0) labelled. In the second step a straight line passes through all of them, and a dashed right-angled step from (1, 4) to (2, 4) to (2, 2) shows that one unit across means two units down; labels give the slope m = −2 and the y-intercept c = 6.

Linear functions. Any function $f (x) = m x + c$ with constants $m$ and $c$ is called a linear function. Its graph is a straight line with slope $m$ that crosses the $y$ -axis at $(0, c)$ . The identity function ( $m = 1$ , $c = 0$ ) and the constant functions ( $m = 0$ ) are special cases; Chapter 9 studies straight lines in detail.

Example 2.19NCERT Example 19 · p. 39

A relation on the integers

Let $R = {(a, b) : a, b \in 𝐙 and a + b is even}$ . Show that (i) $(a, a) \in R$ for every integer $a$ ; (ii) if $(a, b) \in R$ , then $(b, a) \in R$ ; (iii) if $(a, b) \in R$ and $(b, c) \in R$ , then $(a, c) \in R$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.20NCERT Example 20 · p. 39

Finding a linear rule

A linear function $f (x) = m x + c$ on $𝐙$ takes the values $f (- 2) = 7$ , $f (0) = 3$ , $f (1) = 1$ and $f (3) = - 3$ . Find $f (x)$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.21NCERT Example 21 · p. 39

Domain of a rational function

Find the domain of $f (x) = \frac{2 x + 3}{x^{2} + x - 6}$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 2.22NCERT Example 22 · p. 39

A function in two pieces

A function is given by $f (x) = 2 - x$ for $x < 1$ and $f (x) = x + 1$ for $x \geq 1$ . Draw its graph and find its range.

Try it yourself first. Then reveal the solution one step at a time.

3/3The line $x = 1$ meets the graph once: $f (1) = 2$ .

Fig. 2.15A filled dot belongs to the graph; a hollow dot does not.NCERT Fig. 2.17 · p. 39

Axes with x from minus 3 to 5 and y from minus 1 to 7. In the first step the left piece, the line y = 2 − x, falls from (−3, 5) towards a hollow circle at (1, 1). In the second step the right piece, the line y = x + 1, rises from a filled dot at (1, 2) to (5, 6). In the third step a dashed vertical line at x = 1 is drawn and the filled dot (1, 2) is ringed: the line meets the graph only once, so f(1) = 2 and f is a function.

Quick check

1 of 3

A linear function has

f (0) = - 4

and

f (2) = 2

. What is

f (x)

2 x - 4

3 x - 4

- 4 x + 3

3 x + 2

Show answer

Answer (B) $3 x - 4$

f (0) = c = - 4

, then

2 m - 4 = 2

gives

m = 3

. Option A uses the value 2 as the slope, but the rise of 6 happens over a run of 2.

Recap

Relations and Functions: key points and summary

An ordered pair $(a, b)$ has a first and a second entry, and $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ (Section 2.2).
$A \times B = {(a, b) : a \in A, b \in B}$ , with $n (A \times B) = n (A) \cdot n (B)$ and $A \times \emptyset = \emptyset$ . In general $A \times B \neq B \times A$ .
$𝐑 \times 𝐑$ is the coordinate plane and $𝐑 \times 𝐑 \times 𝐑$ is space.
A relation from $A$ to $B$ is a subset of $A \times B$ , written in roster form, in set-builder form or as an arrow diagram (Section 2.3).
The domain is the set of first entries, the range is the set of images, and the codomain is all of $B$ ; the range is a subset of the codomain.
There are $2^{p q}$ relations from a set of $p$ elements to a set of $q$ elements ((2.2)).
A function $f : A \to B$ is a relation in which every element of $A$ has exactly one image; inputs may share an image (Section 2.4).
A real function has its domain and range inside $𝐑$ ; a formula's domain is the largest set of reals where it makes sense.
Identity ( $y = x$ ), constant ( $y = c$ ), polynomial and rational functions, with the parabola $x^{2}$ , the cubic $x^{3}$ and the two-branch $\frac{1}{x}$ (Section 2.4.1).
$∣ x ∣$ is a V with range $[0, \infty)$ , $sgn x$ has range ${- 1, 0, 1}$ , and $[x]$ is a staircase with range $𝐙$ .
$f \pm g$ , $f g$ , $α f$ and $\frac{f}{g}$ are defined pointwise; $\frac{f}{g}$ needs $g (x) \neq 0$ (Section 2.4.2).
A linear function $m x + c$ has a straight-line graph with slope $m$ ; a piecewise function needs one value at each break point (Miscellaneous Examples).

Timeline

How the idea of a function took shape

1637
Descartes
La Géométrie describes curves by pairs of coordinates, the idea behind the Cartesian product.
1673
Leibniz
First uses the word 'function', for quantities attached to a curve.
1718
Johann Bernoulli
Calls a function any quantity built from a variable and constants.
c. 1734
Euler
Introduces the notation f(x).
1748
Euler's Introductio
Makes functions, as formulas, the centre of analysis.
1837
Dirichlet
Allows any rule that gives each x one definite y, formula or not.
1939
Bourbaki
Defines a function as a set of ordered pairs, a subset of a Cartesian product, as in this chapter.

Find your NCERT exercise question

Our practice questions are written for ncert.ai and follow the NCERT exercises. Find the one that matches the question in your textbook; its answer is with it.

In your NCERT textbook	Practice question here
NCERT Exercise 2.1, Q1	Practice 2.1 · Q1
NCERT Exercise 2.1, Q2	Practice 2.1 · Q2
NCERT Exercise 2.1, Q3	Practice 2.1 · Q3
NCERT Exercise 2.1, Q4	Practice 2.1 · Q4
NCERT Exercise 2.1, Q5	Practice 2.1 · Q5
NCERT Exercise 2.1, Q6	Practice 2.1 · Q6
NCERT Exercise 2.1, Q7	Practice 2.1 · Q7
NCERT Exercise 2.1, Q8	Practice 2.1 · Q8
NCERT Exercise 2.1, Q9	Practice 2.1 · Q9
NCERT Exercise 2.1, Q10	Practice 2.1 · Q10
NCERT Exercise 2.2, Q1	Practice 2.2 · Q1
NCERT Exercise 2.2, Q2	Practice 2.2 · Q2
NCERT Exercise 2.2, Q3	Practice 2.2 · Q3
NCERT Exercise 2.2, Q4	Practice 2.2 · Q4
NCERT Exercise 2.2, Q5	Practice 2.2 · Q5
NCERT Exercise 2.2, Q6	Practice 2.2 · Q6
NCERT Exercise 2.2, Q7	Practice 2.2 · Q7
NCERT Exercise 2.2, Q8	Practice 2.2 · Q8
NCERT Exercise 2.2, Q9	Practice 2.2 · Q9
NCERT Exercise 2.3, Q1	Practice 2.3 · Q1
NCERT Exercise 2.3, Q2	Practice 2.3 · Q2
NCERT Exercise 2.3, Q3	Practice 2.3 · Q3
NCERT Exercise 2.3, Q4	Practice 2.3 · Q4
NCERT Exercise 2.3, Q5	Practice 2.3 · Q5
NCERT Miscellaneous Exercise, Q1	Miscellaneous exercises · Q1
NCERT Miscellaneous Exercise, Q2	Miscellaneous exercises · Q2
NCERT Miscellaneous Exercise, Q3	Miscellaneous exercises · Q3
NCERT Miscellaneous Exercise, Q4	Miscellaneous exercises · Q4
NCERT Miscellaneous Exercise, Q5	Miscellaneous exercises · Q5
NCERT Miscellaneous Exercise, Q6	Miscellaneous exercises · Q6
NCERT Miscellaneous Exercise, Q7	Miscellaneous exercises · Q7
NCERT Miscellaneous Exercise, Q8	Miscellaneous exercises · Q8
NCERT Miscellaneous Exercise, Q9	Miscellaneous exercises · Q9
NCERT Miscellaneous Exercise, Q10	Miscellaneous exercises · Q10
NCERT Miscellaneous Exercise, Q11	Miscellaneous exercises · Q11
NCERT Miscellaneous Exercise, Q12	Miscellaneous exercises · Q12

Quick check: MCQs on Relations and Functions

2.2 Cartesian Products of Sets4 questions
2.3 Relations4 questions
2.4 Functions3 questions
2.4.1 Some functions and their graphs4 questions
2.4.2 Algebra of real functions3 questions
Miscellaneous Examples3 questions

Glossary · 14

Key terms (English–Hindi)

Cartesian productकार्तीय गुणनkartiya gunan: $A \times B$ , the set of all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$ .First seen in the chapter →
codomainसहप्रांतsahaprant: The set $B$ for a relation or function from $A$ to $B$ .First seen in the chapter →

domainप्रांतprant: The set of first entries of a relation; for a function $f : A \to B$ , the whole of $A$ .First seen in the chapter →

functionफलनphalan: A relation from $A$ to $B$ in which every element of $A$ has exactly one image.First seen in the chapter →

greatest integer functionमहत्तम पूर्णांक फलनmahattam purnank phalan: $[x]$ , the largest integer not exceeding $x$ ; for example $[2.7] = 2$ and $[- 2.7] = - 3$ .First seen in the chapter →

imageप्रतिबिंबpratibimb: If $(a, b)$ belongs to a relation, $b$ is an image of $a$ ; for a function, $f (a)$ is the image of $a$ .First seen in the chapter →

modulus functionमापांक फलनmapank phalan: $∣ x ∣$ , equal to $x$ for $x \geq 0$ and to $- x$ for $x < 0$ .First seen in the chapter →

ordered pairक्रमित युग्मkramit yugm: A pair $(a, b)$ in which order matters: $(a, b) = (c, d)$ only when $a = c$ and $b = d$ .First seen in the chapter →

polynomial functionबहुपद फलनbahupad phalan: $a_{0} + a_{1} x + \dots + a_{n} x^{n}$ with real coefficients and whole-number powers.First seen in the chapter →

rangeपरिसरparisar: The set of all images; it is always a subset of the codomain.First seen in the chapter →
rational functionपरिमेय फलनparimey phalan: A quotient $\frac{p (x)}{q (x)}$ of polynomial functions, defined where $q (x) \neq 0$ .First seen in the chapter →
real functionवास्तविक फलनvastavik phalan: A function whose domain and range are subsets of $𝐑$ .First seen in the chapter →
relationसंबंधsambandh: A subset of $A \times B$ , usually chosen by a rule linking first and second entries.First seen in the chapter →

signum functionचिह्न फलनchihn phalan: $sgn x$ : 1 for $x > 0$ , 0 at $x = 0$ , $- 1$ for $x < 0$ .First seen in the chapter →

Relations and Functions: Class 11 Maths · Chapter 2

What you will learn in Relations and Functions

2.1 Introduction

2.2 Cartesian Products of Sets

Practice 2.1: questions with answers

2.3 Relations

Practice 2.2: questions with answers

2.4 Functions

2.4.1 Some functions and their graphs

2.4.2 Algebra of real functions

Practice 2.3: questions with answers

Miscellaneous Examples

Relations and Functions: key points and summary

Practice questions with answers (NCERT “Miscellaneous exercises”)

Find your NCERT exercise question

Quick check: MCQs on Relations and Functions

Key terms (English–Hindi)

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