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Mathematics Grade XI

Grade 11 Mathematics

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Chapter 2: Relations and Functions

Class 11 • Mathematics • Chapter 2

Relations and Functions

How mathematics connects the elements of one set with another in a clear, ordered way.

CBSE / NCERT Aligned

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Chapter Roadmap

Ordered Pairs • Cartesian Product • Relations • Domain, Codomain & Range • Functions • Types of Functions • Algebra of Real Functions • Real-Life Applications

Why Relations and Functions Matter

Think about how your school stores data. Every student is tied to a unique roll number, every roll number is tied to a single set of marks, and every subject is tied to a teacher. Each of these is really a pairing between two groups of objects, and that simple idea of pairing is what this chapter formalises.

In Class 11 you already know what a set is. This chapter takes the next step: it studies how the elements of one set can be matched with the elements of another. When the matching follows one special rule, that every input gets exactly one output, we call it a function, the single most important object in all of higher mathematics. Functions describe how distance depends on time, how an OTT app maps each viewer to recommendations, and how a calculator turns one number into another.

Big idea

A relation is any set of ordered pairs linking two sets. A function is a relation where every element of the first set is linked to exactly one element of the second. Every function is a relation, but not every relation is a function.

Key Terms You Must Know

Term

Meaning

Example

Ordered pair

Two elements written in a fixed order, (a, b); order matters so (a, b) ≠ (b, a) unless a = b.

(2, 5) is different from (5, 2)

Cartesian product A × B

The set of all ordered pairs whose first element is from A and second from B.

A = {1, 2}, B = {x} → A × B = {(1, x), (2, x)}

Relation R

Any subset of A × B; it links some elements of A to elements of B.

R = {(1, x)} on the sets above

Domain

The set of all first elements that actually appear in the relation.

Domain of {(1, x)} is {1}

Codomain

The whole set B from which outputs are allowed.

B = {x, y}

Range

The set of second elements that actually appear; a subset of the codomain.

Range of {(1, x)} is {x}

Function

A relation where each input has exactly one output.

f(x) = x + 3

Core Concepts, Step by Step

1. Ordered Pairs and Their Equality

An ordered pair (a, b) records two objects in a definite order. The first slot is the first coordinate, the second slot the second coordinate. Two ordered pairs are equal only when both coordinates match: (a, b) = (c, d) means a = c and b = d. So (3, 7) and (7, 3) are different points, just as the seat (Row 3, Seat 7) is not the same as (Row 7, Seat 3).

2. The Cartesian Product A × B

Given two non-empty sets A and B, the Cartesian product A × B is the collection of every ordered pair (a, b) with a ∈ A and b ∈ B. If A has m elements and B has n elements, then A × B has exactly m × n ordered pairs, written n(A × B) = n(A) · n(B). Note that A × B and B × A are generally different sets.

3. Relations: Picking Out the Pairs You Care About

A relation R from A to B is simply a subset of A × B. You choose the pairs that satisfy some rule, such as “is double of” or “is less than”. The domain is the set of all first coordinates used, the range is the set of all second coordinates used, and the codomain is the full set B. If A × B has m · n pairs, then the total number of possible relations from A to B is 2 raised to the power (m · n), because each pair is either in R or out of it.

4. Functions: The Special Relations

A relation f from A to B is a function if every element of A is paired with one and only one element of B. No input may be left out, and no input may point to two different outputs. We write f : A → B and f(x) = y, meaning y is the image of x. The set A is the domain, B is the codomain, and the set of all actual images is the range. The mapping diagram below shows the idea at a glance.

A function as a mapping: each input has exactly one arrow out

Function as a mapping: each input has exactly one arrow out

5. Standard Real Functions

NCERT highlights several real-valued functions you must recognise:
• Identity f(x) = x, output equals input.
• Constant f(x) = c, same output for every input.
• Polynomial f(x) = a₀ + a₁x + a₂x² + … with whole-number powers.
• Rational f(x) = p(x) / q(x), defined wherever q(x) ≠ 0.
• Modulus f(x) = |x|, which returns the non-negative size of x.
• Signum f(x) = 1, 0, or −1 according to whether x is positive, zero, or negative.
• Greatest integer f(x) = [x], the largest integer not exceeding x, so [2.7] = 2 and [−1.3] = −2.

Graph of the square function f(x) = x²

Parabola, graph of y = x squared

6. Algebra of Real Functions

If f and g are real functions with overlapping domains, you can combine them: (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), and (fg)(x) = f(x) · g(x). You can also scale a function by a real number k to get (kf)(x) = k · f(x). The quotient (f / g)(x) = f(x) / g(x) is defined only where g(x) ≠ 0. The domain of a combined function is the intersection of the two domains (with zeros of g removed for division).

Key Results & Proofs

This chapter is built mostly on definitions, but two counting results are genuinely proved. Here they are, reasoned line by line; everything else in the chapter follows from the definitions of pair, relation and function.

Result 1: Size of the Cartesian Product

Statement. If n(A) = m and n(B) = n, then n(A × B) = m × n.

Proof

Counting ordered pairs is a two-step choice.

Build a pair (a, b) by choosing its two coordinates one after the other.

Choose the first coordinate a from A: there are m ways.

For each such a, choose the second coordinate b from B: there are n ways.

n(A × B)

=

m × n

the two independent choices multiply

Every one of the m first-coordinates pairs with all n second-coordinates, so there are m × n ordered pairs in all.

Result 2: Number of Relations

Statement. If n(A) = m and n(B) = n, the number of possible relations from A to B is 2^(m·n).

Proof

A relation is any subset of A × B, so we count subsets.

n(A × B)

=

m · n

from Result 1

A relation keeps some of these pairs and drops the rest, so for each pair we make one yes/no choice.

There are m·n pairs, each with 2 independent choices, and the choices multiply.

number of relations

=

2 × 2 × … (m·n times)

two choices per pair

=

2^(m·n)

Each distinct pattern of keep/drop choices is a different relation, so there are exactly 2^(m·n) of them, including the empty relation and the whole set A × B.

Worked Examples

Example 1

Question: If A = {1, 2} and B = {a, b}, write A × B and find n(A × B).

▶ Show full working

A × B is every pairing with the first element from A and the second from B.

A × B

=

{(1,a), (1,b), (2,a), (2,b)}

pair each element of A with each of B

n(A × B)

=

n(A) × n(B) = 2 × 2

multiply the sizes

=

4

count

Answer: A × B = {(1, a), (1, b), (2, a), (2, b)}; n(A × B) = 4.

Example 2

Question: If n(A) = 3 and n(B) = 4, how many relations are possible from A to B?

▶ Show full working

Count all the pairs first; then each pair is either kept or dropped.

n(A × B)

=

3 × 4 = 12

total pairs

number of relations

=

2¹²

2 choices (keep/drop) for each pair

=

4096

evaluate

Answer: 4096 relations.

Example 3

Question: Given (x + 1, 5) = (3, 2y − 1), find x and y.

▶ Show full working

Equal ordered pairs mean equal first parts and equal second parts.

x + 1

=

3

match the first coordinates

x

=

2

subtract 1 from both sides

2y − 1

=

5

match the second coordinates

2y

=

6

add 1 to both sides

y

=

3

divide by 2

Answer: x = 2, y = 3.

Example 4

Question: Let A = {1, 2, 3, 4} and R = {(a, b) : b = a + 1}. Write R, its domain and range.

▶ Show full working

Test each value of a; the partner b must also belong to A.

a = 1 → b = 2 ✓ a = 2 → b = 3 ✓ a = 3 → b = 4 ✓ a = 4 → b = 5 ✗ (5 not in A)

R

=

{(1,2), (2,3), (3,4)}

keep only the valid pairs

domain

=

{1, 2, 3}

the first coordinates used

range

=

{2, 3, 4}

the second coordinates used

Answer: R = {(1, 2), (2, 3), (3, 4)}; domain {1, 2, 3}, range {2, 3, 4}.

Example 5

Question: Is R = {(1, 2), (1, 3), (2, 4)} a function from {1, 2} to {2, 3, 4}?

▶ Show full working

A function allows each input exactly one output, so check every input.

Input 1 appears in (1, 2) and in (1, 3), so it points to both 2 and 3.

One input with two different outputs breaks the function rule.

Answer: No, R is not a function.

Example 6

Question: If f(x) = x² − 3x + 2, find f(0), f(2) and f(−1).

▶ Show full working

‘f(0)’ means replace every x in the formula with 0, and so on.

f(0)

=

0² − 3(0) + 2 = 2

put x = 0

f(2)

=

2² − 3(2) + 2 = 4 − 6 + 2 = 0

put x = 2

f(−1)

=

(−1)² − 3(−1) + 2 = 1 + 3 + 2 = 6

put x = −1 (mind the signs)

Answer: f(0) = 2, f(2) = 0, f(−1) = 6.

Example 7

Question: Find the domain and range of f(x) = |x − 2|.

▶ Show full working

The modulus measures size, so it accepts any input and never returns a negative.

domain

=

R

every real number is allowed

The smallest output is 0, when x − 2 = 0 (that is x = 2); after that it grows without limit.

range

=

[0, ∞)

outputs from 0 upward

Answer: Domain R; range [0, ∞).

Example 8

Question: Evaluate the signum function at x = 4, x = 0 and x = −7.

▶ Show full working

Signum reports only the sign of a number: +1, 0 or −1.

sgn(4)

=

1

4 is positive

sgn(0)

=

0

0 is neither positive nor negative

sgn(−7)

=

−1

−7 is negative

Answer: 1, 0 and −1 respectively.

Example 9

Question: If [x] is the greatest integer function, find [3.6], [−2.4] and [5].

▶ Show full working

[x] rounds down to the integer at or below x, never up.

[3.6]

=

3

largest integer not above 3.6

[−2.4]

=

−3

step down (more negative), not −2

[5]

=

5

5 is already an integer

Answer: 3, −3 and 5.

Example 10

Question: If f(x) = x + 1 and g(x) = 2x, find (f + g)(x), (fg)(x) and (f / g)(3).

▶ Show full working

Add or multiply the formulas; for division, work each out at 3 then divide.

(f + g)(x)

=

(x + 1) + 2x = 3x + 1

add the formulas

(fg)(x)

=

(x + 1)(2x) = 2x² + 2x

multiply the formulas

(f / g)(3)

=

f(3) / g(3) = 4 / 6

g(3) = 6 ≠ 0, so division is allowed

=

2/3

simplify

Answer: 3x + 1; 2x² + 2x; (f / g)(3) = 2/3.

Example 11

Question: Find the domain of f(x) = 1 / (x − 5).

▶ Show full working

We may never divide by zero, so keep the denominator non-zero.

x − 5

=

0

find where the bottom is zero

x

=

5

this single value is banned

Answer: Domain = R − {5}.

Example 12

Question: Find the domain of f(x) = √(x − 3).

▶ Show full working

A square root needs the inside to be zero or positive.

Require x − 3 ≥ 0. – square-root rule

So x ≥ 3. – add 3 to both sides

Answer: Domain = [3, ∞).

Where You Meet This in Real Life

Databases and spreadsheets

Every lookup table, whether roll number to marks or product code to price, is a function: one key, one value. Relational databases are literally built on the idea of relations between sets of data.

Maps and GPS

A location is an ordered pair of coordinates (latitude, longitude). The order matters, exactly like ordered pairs, which is why swapping the two values lands you in a different place.

Recommendation engines

Streaming and shopping apps define a function from each viewer to a list of suggestions, recomputed as your inputs change.

Currency and unit conversion

Converting rupees to dirhams, or kilometres to miles, is a function: one amount in gives one amount out through a fixed rule.

Computer programming

Every program function takes inputs and returns one defined output, the exact mathematical idea of a function, used billions of times a day.

Practice Sets A-D

Practice Set A – Basics

A1. If A = {p, q} and B = {1, 2, 3}, write A × B and state n(A × B).

▶ Reveal full working

Pair each element of A with each element of B.

A × B

=

{(p,1),(p,2),(p,3),(q,1),(q,2),(q,3)}

all pairings

n(A × B)

=

2 × 3 = 6

multiply the sizes

Answer: A × B as listed; n(A × B) = 6.

A2. Find x and y if (2x, y + 1) = (6, 4).

▶ Reveal full working

Match the first and second coordinates separately.

2x

=

6

first coordinates

x

=

3

divide by 2

y + 1

=

4

second coordinates

y

=

3

subtract 1

Answer: x = 3, y = 3.

A3. State the domain and range of {(1, 4), (2, 5), (3, 6)}.

▶ Reveal full working

Domain = the first numbers; range = the second numbers.

domain

=

{1, 2, 3}

first coordinates

range

=

{4, 5, 6}

second coordinates

Answer: Domain {1, 2, 3}; range {4, 5, 6}.

A4. If f(x) = 3x − 2, find f(1) and f(−2).

▶ Reveal full working

Replace x by the given number each time.

f(1)

=

3(1) − 2 = 1

put x = 1

f(−2)

=

3(−2) − 2 = −8

put x = −2

Answer: f(1) = 1, f(−2) = −8.

Practice Set B – Conceptual

B1. Is every relation a function? Justify with an example.

▶ Reveal full working

A function needs each input to have exactly one output.

Take {(1, 2), (1, 3)}: input 1 points to both 2 and 3.

That breaks the rule, so it is a relation but not a function.

Answer: No; only relations with unique outputs are functions.

B2. How many relations are possible from a set with 2 elements to a set with 3 elements?

▶ Reveal full working

Count the pairs, then use 2 choices per pair.

n(A × B)

=

2 × 3 = 6

total pairs

number of relations

=

2⁶ = 64

keep or drop each pair

Answer: 64.

B3. Classify f(x) = 7 and g(x) = x as standard functions.

▶ Reveal full working

Look at what each function returns.

f(x) = 7 returns 7 for every input → constant function.

g(x) = x returns the input unchanged → identity function.

Answer: f is constant; g is identity.

B4. Why is the greatest integer of −0.5 equal to −1 and not 0?

▶ Reveal full working

[x] rounds down to the integer at or below x.

Integers not greater than −0.5 are −1, −2, −3, …

The largest of these is −1; 0 is greater than −0.5, so it is excluded.

Answer: [−0.5] = −1.

Practice Set C – Application / Numerical

C1. Find the domain of f(x) = (x + 2) / (x² − 9).

▶ Reveal full working

The bottom of a fraction cannot be zero.

x² − 9

=

0

set the denominator to zero

x

=

±3

solve x² = 9

Ban both x = 3 and x = −3.

Answer: Domain = R − {−3, 3}.

C2. If f(x) = x² and g(x) = x − 1, find (f − g)(x) and (fg)(2).

▶ Reveal full working

Subtract the formulas; for (fg)(2) evaluate each at 2 then multiply.

(f − g)(x)

=

x² − (x − 1)

subtract g

=

x² − x + 1

the minus flips both terms

(fg)(2)

=

f(2) × g(2) = 4 × 1

f(2) = 4, g(2) = 1

=

4

multiply

Answer: x² − x + 1; (fg)(2) = 4.

C3. Find the range of f(x) = x² + 1 for real x.

▶ Reveal full working

Squaring any real number never gives a negative.

x² ≥ 0 for every real x.

Add 1: x² + 1 ≥ 1, with the smallest value at x = 0.

Answer: Range = [1, ∞).

C4. Find the domain of f(x) = √(4 − x).

▶ Reveal full working

The inside of the root must be 0 or positive.

Require 4 − x ≥ 0.

So x ≤ 4.

Answer: Domain = (−∞, 4].

Practice Set D – HOTS / Multi-step

D1. If A × B has 9 elements and two of them are (1, 3) and (2, 5), find possible sets A and B.

▶ Reveal full working

9 = n(A) × n(B); the simplest split is 3 × 3.

9

=

n(A) × n(B) = 3 × 3

choose 3 elements each

First parts 1, 2 belong to A; second parts 3, 5 belong to B.

Complete each to size 3, e.g. A = {1, 2, 3}, B = {3, 5, 7}.

Answer: n(A) = n(B) = 3; e.g. A = {1, 2, 3}, B = {3, 5, 7}.

D2. Find the domain of f(x) = √(x − 1) / (x − 4).

▶ Reveal full working

Two rules apply: the inside of the root ≥ 0, and the bottom ≠ 0.

Square-root rule: x − 1 ≥ 0, so x ≥ 1.

Denominator rule: x − 4 ≠ 0, so x ≠ 4.

Combine: x ≥ 1 with a hole at x = 4.

Answer: Domain = [1, 4) ∪ (4, ∞).

D3. If f(x) = (x − 1)/(x + 1), show that f(1/x) = −f(x) for x ≠ 0, −1.

▶ Reveal full working

Replace x by 1/x, clear the small fractions, then compare with −f(x).

f(1/x)

=

(1/x − 1)/(1/x + 1)

replace every x by 1/x

=

(1 − x)/(1 + x)

multiply top and bottom by x

−f(x)

=

−(x − 1)/(x + 1) = (1 − x)/(1 + x)

flip the sign of f(x)

Both sides equal (1 − x)/(1 + x).

Answer: f(1/x) = −f(x), as required.

D4. A relation R on {1, 2, 3, 4, 5, 6} is defined by (a, b) ∈ R if b is exactly divisible by a. Write the domain and range of R.

▶ Reveal full working

For each a, list the multiples b that lie in the set.

1 → 1,2,3,4,5,6 2 → 2,4,6 3 → 3,6 4 → 4 5 → 5 6 → 6

domain

=

{1, 2, 3, 4, 5, 6}

every a was used

range

=

{1, 2, 3, 4, 5, 6}

every b appeared as a multiple

Answer: Domain = range = {1, 2, 3, 4, 5, 6}.

Chapter Summary

Everything in One Glance

Ordered Pairs

(a, b) keeps order; equality means matching both coordinates.

Cartesian Product

A × B holds every pair; n(A × B) = n(A) · n(B).

Relations

A subset of A × B; has domain, codomain and range; 2^(mn) relations possible.

Functions

Each input has exactly one output; written f : A → B with f(x) = y.

Standard Functions

Identity, constant, polynomial, rational, modulus, signum, greatest integer.

Algebra of Functions

Add, subtract, multiply, scale and divide functions on shared domains.

Are You Exam-Ready?

8-Point Exam Quick-Check

When are two ordered pairs equal?

What is n(A × B) if n(A) = 5 and n(B) = 6?

How many relations are possible from a 3-element set to a 2-element set?

State the difference between codomain and range.

What single test decides whether a relation is a function?

Give the value of [−3.2] and the value of the signum of −9.

Write the domain of f(x) = 1/(x − 7).

If f(x) = x and g(x) = x + 4, what is (fg)(2)?

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Class 11 Mathematics Chapter 2: Relations and Functions, Complete Notes and Practice

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