Relations and Functions Class 11 Notes & Important Questions

Q: What is in CBSE Class 11 Maths Chapter 2, Relations and Functions?

The topics are: Ordered pairs and Cartesian products, Relations, Real functions: domain and range, Functions, Algebra of real functions. This page has revision notes for each, and 41 board-style practice questions with step-marked solutions.

Q: How many marks is Relations and Functions worth?

It is in the Sets and Functions unit, which carries 23 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Relations and Functions?

Relation accepted as a function although one element has two images (or none). Fix: check every element of the domain appears exactly once as a first entry. Greatest integer of negative numbers rounded towards zero. Fix: [−2.5] = −3 (the integer just below).

Q: Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Revision notes

Relations and Functions: revision notes

1. Ordered pairs and Cartesian products

$(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ . The Cartesian product $A \times B = {(a, b) : a \in A, b \in B}$ ; $n (A \times B) = n (A) \times n (B)$ . In general $A \times B \neq B \times A$ . $R \times R$ is the plane.

2. Relations

A relation $R$ from $A$ to $B$ is any subset of $A \times B$ . Its domain is the set of first elements, its range the set of second elements, and $B$ is the codomain. If $n (A) = p$ and $n (B) = q$ , there are $2^{p q}$ relations from $A$ to $B$ .

3. Functions

A function $f : A \to B$ is a relation in which every element of $A$ has exactly one image in $B$ . Domain $A$ ; range $= {f (x) : x \in A} \subset B$ .

4. Real functions: domain and range

Domain: all real $x$ for which the formula makes sense: denominators $\neq 0$ , expressions under $\sqrt{}$ $\geq 0$ .
Identity $f (x) = x$ ; constant $f (x) = c$ ; polynomial; rational $\frac{p (x)}{q (x)}$ , $q (x) \neq 0$ .
Modulus $| x |$ : $x$ if $x \geq 0$ , $- x$ if $x < 0$ ; range $[0, \infty)$ .
Signum: $1$ for $x > 0$ , $0$ at $0$ , $- 1$ for $x < 0$ ; range ${- 1, 0, 1}$ .
Greatest integer $[x]$ : the greatest integer $\leq x$ ; $[3.7] = 3$ , $[- 2.5] = - 3$ ; range $Z$ .

5. Algebra of real functions

On the common domain: $(f \pm g) (x) = f (x) \pm g (x)$ , $(f g) (x) = f (x) g (x)$ , $(\frac{f}{g}) (x) = \frac{f (x)}{g (x)}$ where $g (x) \neq 0$ ; $(α f) (x) = α f (x)$ .

Worked example 1

Find the domain of $f (x) = \frac{x}{x^{2} - 9}$ .

$x^{2} - 9 \neq 0$ : domain $R - {- 3, 3}$ .

Worked example 2

Find the range of $f (x) = 3 - x^{2}$ .

$x^{2} \geq 0$ , so $f (x) \leq 3$ : range $(- \infty, 3]$ .

Worked example 3

$A = {1, 2}$ , $B = {x, y, z}$ . How many relations are there from $A$ to $B$ ?

$n (A \times B) = 6$ , so $2^{6} = 64$ relations.

Common errors

Calling a relation a function when some element of the domain has two images, or none.
$[- 2.5] = - 2$ : the greatest integer function goes down, to $- 3$ .
Cancelling $x - 2$ in $\frac{x^{2} - 4}{x - 2}$ and forgetting that $2$ is still not in the domain.
Mixing up codomain and range.

Exam tips

For a range, write $y = f (x)$ , solve for $x$ and find the $y$ for which a valid $x$ exists.
State the domain with every quotient $\frac{f}{g}$ .

Topics in this chapter: Ordered pairs and Cartesian products · Relations · Real functions: domain and range · Functions · Algebra of real functions.

Formulas for this chapter: Chapter 2 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can work with ordered pairs and Cartesian products and recognise a function among relations.

Read first: 1. Ordered pairs and Cartesian products; 2. Relations; 3. Functions 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can find domains and ranges of standard real functions and combine functions.

Read first: 4. Real functions: domain and range; 5. Algebra of real functions; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can analyse modulus, signum, greatest integer and rational functions fully, with their graphs in mind.

Read first: 4. Modulus, signum, greatest integer; rational functions 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can reason about ranges of less obvious functions and about products of sets.

Read first: 4. Domain and range; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 41 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceOrdered pairs and Cartesian products

If $(x + 1, y - 2) = (3, 1)$ , then $(x, y)$ is

(a) $(3, 2)$
(b) $(4, - 1)$
(c) $(2, - 1)$
(d) $(2, 3)$

Q2·1 mark·Multiple choiceRelations

If $n (A) = 3$ and $n (B) = 4$ , the number of relations from $A$ to $B$ is

(a) $12$
(b) $7$
(c) $2^{12}$
(d) $2^{7}$

Q3·1 mark·Multiple choiceReal functions: domain and range

The domain of $f (x) = \frac{1}{x - 3}$ is

(a) $R$
(b) $R - {3}$
(c) $(3, \infty)$
(d) $R - {0}$

Worked solution·3 marks·Short answerAlgebra of real functions

Let $f (x) = \sqrt{x - 1}$ and $g (x) = \sqrt{5 - x}$ . Find the domain of $f + g$ , and write $(f g) (x)$ as a single square root.

Full worked solution Free, no sign-in

$f$ needs $x \geq 1$ ; $g$ needs $x \leq 5$ M1
Domain of $f + g$ : $[1, 5]$ A1
$(f g) (x) = \sqrt{(x - 1) (5 - x)}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original combinatorics problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S59 · Problem S10 · Problem S58 · Problem S09. All 22 →

Knowledge organiser: Relations and Functions on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Relations and Functions

Relation accepted as a function although one element has two images (or none). Fix: check every element of the domain appears exactly once as a first entry.
Greatest integer of negative numbers rounded towards zero. Fix: [−2.5] = −3 (the integer just below).
Domain of a quotient without excluding zeros of the denominator. Fix: solve g(x) = 0 and remove those points.
Range stated as the codomain. Fix: the range is the set of values actually taken.
Cancelled common factor used to change the domain. Fix: the domain comes from the original formula.

Test Relations and Functions against the clock

Want it against the clock? Take a timed 30-mark chapter test on Relations and Functions, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on sets, relations and functions.

Relations and Functions: common questions

What is in CBSE Class 11 Maths Chapter 2, Relations and Functions?