MathsCBSE Class 11 · Chapter 2

Relations and Functions Class 11: Notes, Graphs and Solved Questions

Q: What is the difference between a relation and a function?

Every function is a relation, but a function is stricter: each element of the first set must have exactly one image. A relation may leave elements without images or give one element several images.

Q: Is composition of functions in the 2026-27 exam?

CBSE lists composition of functions as taught but assessed only formatively (class activities and tests), so it is not part of the annual exam paper.

Q: Are exponential and logarithmic functions in this chapter?

Yes. The CBSE syllabus names them among the functions whose graphs students should know, even though the rationalised NCERT chapter does not draw them. Learn their domain, range, the points (0, 1) and (1, 0), and the shape of each graph.

Q: Why is range a subset of codomain and not always equal to it?

The codomain is the set you aim into; the range is what you actually hit. For f(x) = x^2 from R to R the codomain is R but the range is only [0, \infty).

Q: How is the Class 11 Maths exam set?

It is a school annual exam of 80 marks (3 hours) plus 20 marks of internal assessment. Unit I, Sets and Functions, carries 23 marks. CBSE gives no Class 11 sample paper, so schools usually mix MCQs, assertion-reason, short and long answers and case-based questions.

Updated 14 October 2026 · Fact-checked

Relations and Functions is Chapter 2 of CBSE Class 11 Maths. It covers ordered pairs and Cartesian products of sets, relations with their domain, codomain and range, functions as special relations, real functions such as identity, constant, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions with graphs, and the algebra of real functions.

What Relations and Functions is about

Mathematics keeps asking how one quantity is linked to another: the cost of a taxi ride to the distance, the area of a square to its side. This chapter gives that link a precise meaning.

It starts with ordered pairs $(a, b)$ , where order matters, and the Cartesian product $A \times B$ , the set of all ordered pairs with first entry from A and second from B. If $n (A) = p$ and $n (B) = q$ , then $n (A \times B) = pq$ . The plane you use for graphs is simply $R \times R$ .

A relation from A to B is any subset of $A \times B$ ; it has a domain (first entries), a range (second entries) and a codomain (the whole of B). A function is a relation in which every element of A is paired with exactly one element of B. Most of the chapter is about real functions: finding their domain and range, drawing the graphs of standard functions (identity, constant, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer), and combining functions by adding, subtracting, multiplying and dividing them. For 2026-27, composition of functions is taught but assessed only formatively.

Weightage: This chapter is part of Unit I, Sets and Functions, which the syllabus gives 23 of the 80 theory marks in the school annual exam (CBSE adds that there is no fixed chapter-wise weightage).Source: CBSE Mathematics (041) syllabus 2026-27, Class XI course structure, Unit I Sets and Functions

Key concepts

Ordered pairs and the Cartesian product

NCERT section 2.2

An ordered pair $(a, b)$ is a pair in which the order of the entries matters: $(2, 5) \neq = (5, 2)$ .

Equality: $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ .

For non-empty sets A and B, the Cartesian product is
$A \times B = {(a, b) : a \in A, b \in B}$

If $n (A) = p$ and $n (B) = q$ , then $n (A \times B) = pq$ .
If A or B is empty, $A \times B = ϕ$ .
In general $A \times B \neq = B \times A$ , though both have the same number of elements.
If A and B are non-empty and one of them is infinite, $A \times B$ is infinite.
$A \times A \times A = {(a, b, c) : a, b, c \in A}$ ; $(a, b, c)$ is an ordered triplet.
$R \times R$ is the set of all points of the coordinate plane, and $R \times R \times R$ is the set of all points of three-dimensional space.

To recover A and B from $A \times B$ : A is the set of first entries and B the set of second entries.

Worked example

If $(2 x - 1, y + 3) = (5, - 1)$ , find x and y. Also, if A = {1, 2, 3} and B = {a, b}, write $A \times B$ and state $n (A \times B)$ . [2 marks]

Try it first, then show the solutionSolution

Equal ordered pairs have equal first and second entries: $2 x - 1 = 5 \Rightarrow x = 3$ ; $y + 3 = - 1 \Rightarrow y = - 4$
$A \times B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}$ , so $n (A \times B) = 3 \times 2 = 6$

Answer: x = 3, y = −4; six ordered pairs as listed.

Relations: domain, codomain and range

NCERT section 2.3

Take two non-empty sets A and B. Any subset R of $A \times B$ is called a relation from A to B; usually it is picked out by some rule linking the first entry to the second. If $(x, y) \in R$ we say y is the image of x.

Domain of R: the set of first entries of the pairs in R.
Range of R: the set of second entries.
Codomain: the whole set B. Always, range ⊂ codomain.

A relation can be written in roster form, in set-builder form, or drawn as an arrow diagram (two ovals with arrows from x to y).

A relation from A to A is called a relation on A.

Counting relations: every subset of $A \times B$ is a relation, so if $n (A) = p$ and $n (B) = q$ there are $2^{pq}$ relations from A to B (including the empty relation).

Worked example

Let A = {1, 2, 3, 4, 5, 6} and define R on A by $R = {(x, y) : y = 2 x}$ . Write R in roster form and find its domain, range and codomain. How many relations are there on A? [3 marks]

Try it first, then show the solutionSolution

y = 2x must also lie in A: x = 1, 2, 3 give y = 2, 4, 6. So R = {(1, 2), (2, 4), (3, 6)}
Domain = {1, 2, 3}; range = {2, 4, 6}; codomain = A
$n (A \times A) = 36$ , so the number of relations on A is $2^{36}$

Answer: R = {(1, 2), (2, 4), (3, 6)}; domain {1, 2, 3}, range {2, 4, 6}, codomain A; $2^{36}$ relations.

Functions as special relations

NCERT section 2.4

A relation f from A to B is a function (or mapping) if each element of A is paired with exactly one element of B. Two conditions must both hold:
1. No element of A is left out: the domain of f is the whole of A.
2. No element of A has two images: no two different pairs in f share a first entry.

We write $f : A \to B$ and $f (a) = b$ ; b is the image of a and a is a pre-image of b.

Two elements of A may share the same image; that is allowed.
Some elements of B may have no pre-image; that is also allowed. The range is the set of images, a subset of the codomain B.

Arrow-diagram test: every element on the left must send out exactly one arrow. Graph test (for real functions): a vertical line meets the graph at most once.

Worked example

Which of these relations from A = {1, 2, 3, 4} to B = {a, b, c} are functions? (i) {(1, a), (2, a), (3, b), (4, c)} (ii) {(1, a), (2, b), (3, c)} (iii) {(1, a), (1, b), (2, c), (3, a), (4, b)} [2 marks]

Try it first, then show the solutionSolution

(i) Every element of A has exactly one image (1 and 2 share the image a, which is fine): function
(ii) 4 has no image: not a function. (iii) 1 has two images, a and b: not a function

Answer: Only (i) is a function.

Real functions: finding domain and range

NCERT section 2.4

A real-valued function has its range inside R; a real function also has its domain inside R. When a real function is given only by a formula, its domain is taken to be the largest set of real x for which the formula gives a real value.

Finding the domain: remove the values that cause trouble.

Denominator zero: remove them. $\frac{1}{x - 3}$ has domain $R - {3}$ .
Even root of a negative number: require the inside to be $\geq 0$ . $x - 1$ needs $x \geq 1$ .
Logarithm: require the inside to be $> 0$ .

Finding the range: either reason about the expression (a square is never negative, a square root is never negative), or write $y = f (x)$ , solve for x in terms of y, and see which y give a real x in the domain.

Worked example

Find the domain and range of $f (x) = 16 - x^{2}$ . [3 marks]

Try it first, then show the solutionSolution

Need $16 - x^{2} \geq 0$ , i.e. $x^{2} \leq 16$ , i.e. $- 4 \leq x \leq 4$ . Domain = [−4, 4]
On this domain, $16 - x^{2}$ takes every value from 0 (at x = ±4) to 16 (at x = 0)
So $16 - x^{2}$ takes every value from 0 to 4. Range = [0, 4]

Answer: Domain [−4, 4]; range [0, 4].

Identity, constant, polynomial and rational functions

NCERT section 2.4.1

Identity function $f (x) = x$ on R: domain and range R; graph is the straight line through the origin at 45° to the axes.
Constant function $f (x) = c$ : domain R, range {c}; graph is a horizontal line through (0, c).
Linear function $f (x) = m x + c$ : graph is a straight line.
Polynomial function $f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$ , with n a non-negative integer and real coefficients. $f (x) = x^{2}$ has range $[0, \infty)$ and a U-shaped graph (parabola); $f (x) = x^{3}$ has range R. Note $x^{2/3} + 2 x$ is not a polynomial, since the power $\frac{2}{3}$ is not a whole number.
Rational function $\frac{f ( x )}{g ( x )}$ with f and g polynomials and $g (x) \neq = 0$ . Example: $f (x) = \frac{1}{x}$ has domain and range $R - {0}$ ; its graph has two branches, in the first and third quadrants.

Worked example

Find the domain of $f (x) = \frac{x ^{2} + 3 x + 2}{x ^{2} - 7 x + 10}$ . [2 marks]

Try it first, then show the solutionSolution

The denominator factorises: $x^{2} - 7 x + 10 = (x - 2) (x - 5)$
It is zero at x = 2 and x = 5, so these must be removed

Answer: Domain = $R - {2, 5}$ .

Modulus, signum and greatest integer functions

NCERT section 2.4.1

Modulus function $f (x) = ∣ x ∣$ : equals x when $x \geq 0$ and $- x$ when $x < 0$ . Domain R, range $[0, \infty)$ . Graph: a V with its corner at the origin.
Signum function $f (x) = \frac{∣ x ∣}{x}$ for $x \neq = 0$ , and $f (0) = 0$ . So $f (x) = 1$ for $x > 0$ , $0$ at $x = 0$ , $- 1$ for $x < 0$ . Domain R, range {−1, 0, 1}. Graph: two horizontal half-lines and the point (0, 0).
Greatest integer function $f (x) = [x]$ : the greatest integer less than or equal to x. So $[2.7] = 2$ , $[5] = 5$ , $[- 2.3] = - 3$ (not −2). Domain R, range Z. Graph: a staircase; on each interval $[n, n + 1)$ it equals n, with a filled dot at the left end and an open circle at the right.

Worked example

Find $[- 2.3] + [2.7] + ∣ - 4∣ + sgn (- 4)$ , where sgn is the signum function. [1 mark]

Try it first, then show the solutionSolution

$[- 2.3] = - 3$ (the greatest integer not exceeding −2.3); $[2.7] = 2$
$∣ - 4∣ = 4$ ; $sgn (- 4) = - 1$ . Sum $= - 3 + 2 + 4 - 1 = 2$

Answer: 2

Exponential and logarithmic functions

NCERT section Syllabus topic (not in the rationalised NCERT chapter)

The 2026-27 syllabus lists these two among the functions whose graphs you should know.

Exponential function $f (x) = a^{x}$ , with $a > 0$ and $a \neq = 1$ . Domain R, range $(0, \infty)$ : $a^{x}$ is always positive. Every graph passes through (0, 1). If $a > 1$ (e.g. $2^{x}$ ) the graph rises from left to right and hugs the x-axis on the left; if $0 < a < 1$ it falls.
Logarithmic function $f (x) = lo g_{a} x$ , with $a > 0$ , $a \neq = 1$ . It undoes the exponential: $y = lo g_{a} x$ means $a^{y} = x$ . Domain $(0, \infty)$ , range R. Every graph passes through (1, 0). For $a > 1$ it rises slowly and drops steeply near the y-axis.

The two graphs are mirror images of each other in the line $y = x$ , because each function reverses the other.

Worked example

For $f (x) = 2^{x}$ and $g (x) = lo g_{2} x$ , find $f (3)$ , $f (- 1)$ and $g (32)$ , and state the domain of g. [2 marks]

Try it first, then show the solutionSolution

$f (3) = 2^{3} = 8$ ; $f (- 1) = 2^{- 1} = \frac{1}{2}$
$g (32) = 5$ because $2^{5} = 32$ ; the logarithm needs a positive input, so the domain of g is $(0, \infty)$

Answer: 8, $\frac{1}{2}$ , 5; domain $(0, \infty)$ .

Algebra of real functions

NCERT section 2.4.2

For real functions $f, g : X \to R$ with $X \subset R$ , define new functions point by point:

Sum: $(f + g) (x) = f (x) + g (x)$
Difference: $(f - g) (x) = f (x) - g (x)$
Scalar multiple: $(α f) (x) = α f (x)$ for a real number α
Product: $(f g) (x) = f (x) g (x)$ (pointwise multiplication)
Quotient: $(\frac{f}{g}) (x) = \frac{f ( x )}{g ( x )}$ , only where $g (x) \neq = 0$

The quotient is the one that changes the domain: you must remove the zeros of g. If f and g have different domains, $f \pm g$ and $f g$ live on the common part of the domains.

(Composition $f \circ g$ is in the syllabus but assessed only formatively in 2026-27.)

Worked example

Let $f (x) = x^{2}$ and $g (x) = x + 2$ . Find $(f + g) (x)$ , $(f - g) (x)$ , $(f g) (x)$ and $(\frac{f}{g}) (x)$ with its domain. [3 marks]

Try it first, then show the solutionSolution

$(f + g) (x) = x^{2} + x + 2$ ; $(f - g) (x) = x^{2} - x - 2$
$(f g) (x) = x^{2} (x + 2) = x^{3} + 2 x^{2}$
$(\frac{f}{g}) (x) = \frac{x ^{2}}{x + 2}$ , defined for $x \neq = - 2$

Answer: $x^{2} + x + 2$ ; $x^{2} - x - 2$ ; $x^{3} + 2 x^{2}$ ; $\frac{x ^{2}}{x + 2}$ on $R - {- 2}$ .

Maths in real life: Relations and Functions

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Cartesian product

Suppose Riya packs 4 shirts and 3 pairs of trousers for a school trip.

How many different outfits (one shirt with one pair of trousers) can she make? Which set describes them?

Try it first, then show the solutionSolution

Let S be the set of shirts and T the set of trousers; an outfit is an ordered pair (shirt, trousers), i.e. an element of $S \times T$
$n (S \times T) = 4 \times 3 = 12$

Answer: 12 outfits, the elements of $S \times T$

Memory hook: Pairs from two sets: multiply the counts.

Greatest integer function

Imagine a courier that charges ₹40 for a parcel plus ₹20 for each complete kilogram of its weight.

Write the charge C as a function of the weight w kg, and find the charge for parcels of 3.7 kg and 4 kg.

Try it first, then show the solutionSolution

The number of complete kilograms is $[w]$ , so $C (w) = 40 + 20 [w]$
$C (3.7) = 40 + 20 \times 3 = 100$ ; $C (4) = 40 + 20 \times 4 = 120$

Answer: $C (w) = 40 + 20 [w]$ ; ₹100 and ₹120

Memory hook: 'Complete units' means round down: the greatest integer function.

Linear function and its inverse use

Suppose a prepaid phone plan costs ₹150 a month plus ₹0.50 for each minute of calls beyond the free limit.

Write the monthly bill B as a function of the extra minutes m. Find the bill for 120 extra minutes, and the extra minutes used if the bill is ₹270.

Try it first, then show the solutionSolution

$B (m) = 150 + 0.5 m$ , a linear function with domain the non-negative numbers
$B (120) = 150 + 60 = 210$
$150 + 0.5 m = 270 \Rightarrow 0.5 m = 120 \Rightarrow m = 240$

Answer: ₹210; 240 extra minutes

Memory hook: Fixed charge + rate × quantity = a straight-line function.

Topics in this chapter

Relations and Functions formula sheet

Cartesian product: $A \times B = {(a, b) : a \in A, b \in B}, n (A \times B) = n (A) n (B)$
Equal ordered pairs: $(a, b) = (c, d) ⟺ a = c and b = d$
Number of relations: $n (A) = p, n (B) = q \Rightarrow number of relations from A to B = 2^{pq}$
Modulus and signum: $∣ x ∣ = {x, - x, x \geq 0 x < 0, sgn (x) = ⎩ ⎨ ⎧ 1, 0, - 1, x > 0 x = 0 x < 0$
Greatest integer function: $[x] = n for n \leq x < n + 1, n \in Z$; Range Z; [−2.3] = −3.
Exponential and logarithm: $y = lo g_{a} x ⟺ a^{y} = x (a > 0, a \neq = 1, x > 0)$; $a^{x}$ : domain R, range (0, ∞). $lo g_{a} x$ : domain (0, ∞), range R.
Algebra of functions: $(f \pm g) (x) = f (x) \pm g (x), (f g) (x) = f (x) g (x), (\frac{f}{g}) (x) = \frac{f ( x )}{g ( x )}, g (x) \neq = 0$

How the school exam asks Relations and Functions

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Ordered pairs and Cartesian products	Equal ordered pairs; write $A \times B$ ; find A and B from $A \times B$ ; count elements; verify $A \times (B \cap C) = (A \times B) \cap (A \times C)$ .	Match entries; list systematically, first entry fixed while the second runs through B.
Relations	Write a relation in roster form from a rule; find domain, range and codomain; read an arrow diagram; count relations.	Test each element of A against the rule; use $2^{pq}$ for counting.
Is it a function?	Decide from a set of pairs, an arrow diagram, a graph or a piecewise rule whether a relation is a function.	Check both conditions: every element has an image, and no element has two.
Domain and range of real functions	Rational functions, square roots, modulus, $x^{2} + c$ , $\frac{x ^{2}}{1 + x ^{2}}$ and similar.	Domain: remove zeros of denominators, keep square-root inputs ≥ 0. Range: reason about the expression or solve for x.
Graphs of standard functions	Sketch modulus, signum, greatest integer, $x^{2}$ , $x^{3}$ , $\frac{1}{x}$ , $a^{x}$ , $lo g_{a} x$ or a piecewise function.	Make a short table of values; mark included end points with filled dots and excluded ones with open circles.
Algebra of functions	Find $f + g$ , $f - g$ , $f g$ , $\frac{f}{g}$ and their values; find a linear function from given pairs.	Work point by point; for the quotient, state the excluded points.

Tricks to remember Relations and Functions

Short memory aids. Say each one aloud once, then use it on a question the same day.

Domain first, codomain whole, range what is hit
Domain = inputs used; codomain = the target set; range = the targets actually hit.
Use it when: Writing domain, codomain and range of a relation.
One arrow out of every dot
For a function, every element of A sends out exactly one arrow. Many arrows may land on the same target.
Use it when: Function or not.
2 to the power of the pairs
Relations = subsets of $A \times B$ , so there are $2^{pq}$ of them.
Use it when: Counting relations.
Greatest integer goes down the number line
[x] moves left to the nearest integer: [3.9] = 3, [−3.1] = −4.
Use it when: Negative values of [x].
Bottom zero, root negative: throw out
Domain = R minus zeros of the denominator, minus x making an even root's inside negative.
Use it when: Domains of real functions.

Relations and Functions: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Counting a product · 1 mark · MCQExam level

If A has 3 elements and B = {3, 4, 5}, the number of elements in $A \times B$ is:
(a) 6
(b) 8
(c) 9
(d) 27

Try it first, then show the solutionSolution

$n (A \times B) = 3 \times 3 = 9$ .

Answer: (c) 9

Example 2: Greatest integer function · 1 mark · Assertion-reasonHarder exam level

Assertion (A): The range of $f (x) = [x]$ , $x \in R$ , is Z.
Reason (R): For each integer n, $[x] = n$ for every x in the interval $[n, n + 1)$ .
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

Try it first, then show the solutionSolution

R is true by the definition of [x]. The intervals $[n, n + 1)$ cover the whole real line, so [x] is always an integer, and each integer n is reached; hence the range is exactly Z and R explains A.

Answer: (a)

Example 3: Finding a linear function · 2 marks · Very short answerHarder exam level

A linear function f from Z to Z, $f (x) = a x + b$ , contains the pairs (1, 4) and (3, 10). Find a and b, and f(−2). [2 marks]

Try it first, then show the solutionSolution

$a + b = 4$ and $3 a + b = 10$ ; subtracting, $2 a = 6$ , so a = 3 and b = 1
$f (x) = 3 x + 1$ , so $f (- 2) = - 5$

Answer: a = 3, b = 1; f(−2) = −5

Example 4: Domain and range with a square root · 3 marks · Short answerHarder exam level

Find the domain and range of $f (x) = x + 3$ and of $g (x) = ∣ x - 2∣ + 1$ . [3 marks]

Try it first, then show the solutionSolution

f: need $x + 3 \geq 0$ , so domain $[- 3, \infty)$ ; $x + 3$ takes every value $\geq 0$ , so range $[0, \infty)$
g: defined for all real x, so domain R
$∣ x - 2∣ \geq 0$ with equality at x = 2, so $g (x) \geq 1$ and every value $\geq 1$ is reached: range $[1, \infty)$

Answer: f: $[- 3, \infty)$ and $[0, \infty)$ ; g: R and $[1, \infty)$

Example 5: Is f a function? · 3 marks · Short answerHarder exam level

f is defined by $f (x) = x + 2$ for $0 \leq x \leq 2$ and $f (x) = 2 x$ for $2 \leq x \leq 6$ . The relation g is defined by $g (x) = x + 2$ for $0 \leq x \leq 3$ and $g (x) = 2 x$ for $3 \leq x \leq 6$ . Show that f is a function and g is not. [3 marks]

Try it first, then show the solutionSolution

The only point covered by both rules of f is x = 2: $2 + 2 = 4$ and $2 \times 2 = 4$ agree, so x = 2 has one image. Every other x has one rule: f is a function
For g the shared point is x = 3: $3 + 2 = 5$ but $2 \times 3 = 6$
So 3 has two images, 5 and 6: g is not a function

Answer: f is a function; g is not, since g(3) would be both 5 and 6.

Example 6: Range by solving for x · 3 marks · Short answerHarder exam level

Find the domain and range of $f (x) = \frac{x - 2}{3 - x}$ . [3 marks]

Try it first, then show the solutionSolution

Denominator zero at x = 3, so domain $R - {3}$
Let $y = \frac{x - 2}{3 - x}$ : $3 y - x y = x - 2 \Rightarrow x (1 + y) = 3 y + 2 \Rightarrow x = \frac{3 y + 2}{1 + y}$
This gives a real x for every $y \neq = - 1$ (and x is never 3, since that would need 3 + 3y = 3y + 2). Range $R - {- 1}$

Answer: Domain $R - {3}$ ; range $R - {- 1}$

Example 7: Counting functions · 3 marks · Short answerHOTS / competency

Let A = {x, y, z} and B = {1, 2}. (a) How many relations are there from A to B? (b) How many of them are functions from A to B? (c) Why is the second number much smaller? [3 marks]

Try it first, then show the solutionSolution

(a) $n (A \times B) = 6$ , so there are $2^{6} = 64$ relations
(b) A function picks exactly one image for each of x, y, z, with 2 choices each: $2 \times 2 \times 2 = 8$ functions
(c) A function must give every element exactly one image; most subsets of $A \times B$ leave an element out or give it two images

Answer: 64 relations; 8 functions.

Example 8: A relation that is not a function · 2 marks · Very short answerHOTS / competency

Let $f = {(a^{2}, a^{3}) : a \in Z}$ , a subset of $Z \times Z$ . Is f a function from Z to Z? Give two separate reasons. [2 marks]

Try it first, then show the solutionSolution

Take a = 1: the pair (1, 1) is in f. Take a = −1: the pair (1, −1) is in f. So 1 has two images
Also 2 is not a perfect square, so 2 has no image at all. Either fault means f is not a function from Z to Z

Answer: No: 1 is paired with both 1 and −1, and 2 has no image.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Seating a quiz final

Suppose three schools, P, Q and R, reach a quiz final and the hall has two tables, T1 and T2. Let S = {P, Q, R} and T = {T1, T2}. The organiser records each seating plan as a set of pairs (school, table).

(i) · 1 mark
How many elements does $S \times T$ have?
Answer
Answer: 6
$3 \times 2 = 6$
(ii) · 1 mark
How many relations from S to T are possible?
Answer
Answer: 64
$2^{6} = 64$
(iii) · 1 mark
A valid plan seats every school at exactly one table. How many valid plans are there?
Answer
Answer: 8
A valid plan is a function from S to T: $2^{3} = 8$ .
(iv) · 1 mark
Is {(P, T1), (Q, T1), (R, T2), (P, T2)} a valid plan?
Answer
Answer: No
P is at two tables, so it is a relation but not a function.

Parking charges

Suppose a mall charges ₹30 to enter its car park plus ₹20 for every complete hour a car stays. A car parked for t hours (t > 0) pays $P (t) = 30 + 20 [t]$ .

(i) · 1 mark
Find the charge for 2.5 hours.
Answer
Answer: ₹70
$30 + 20 \times 2 = 70$
(ii) · 1 mark
Find the charge for 45 minutes.
Answer
Answer: ₹30
[0.75] = 0, so $P = 30$ .
(iii) · 1 mark
For which times is the charge exactly ₹90?
Answer
Answer: $3 \leq t < 4$ hours
$30 + 20 [t] = 90 \Rightarrow [t] = 3$ .
(iv) · 1 mark
What kind of graph does P have?
Answer
Answer: A staircase
It is constant on each interval [n, n + 1) and jumps by ₹20 at each whole hour.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

If A = {−1, 1} and B = {0, 2}, write $A \times B$ and $B \times A$ , and state whether they are equal. [2 marks] (2 marks)

Step	Marks
$A \times B = {(- 1, 0), (- 1, 2), (1, 0), (1, 2)}$ ; $B \times A = {(0, - 1), (0, 1), (2, - 1), (2, 1)}$	1
For example $(- 1, 0) \in A \times B$ but $(- 1, 0) \in / B \times A$ , so $A \times B \neq = B \times A$ (both have 4 elements)	1

Find the domain and range of the real function $f (x) = \frac{3}{2 + x ^{2}}$ . [3 marks] (3 marks)

Step	Marks
$2 + x^{2} \geq 2 > 0$ for all real x, so the domain is R	1
Let $y = \frac{3}{2 + x ^{2}}$ . Then $y > 0$ and $2 + x^{2} = \frac{3}{y} \Rightarrow x^{2} = \frac{3}{y} - 2$	1
We need $x^{2} \geq 0$ : $\frac{3}{y} \geq 2 \Rightarrow y \leq \frac{3}{2}$ (y > 0). Range = $(0, \frac{3}{2}]$ , the top value at x = 0	1

Let $f (x) = 2 - x$ for $x < 0$ , $f (0) = 2$ , and $f (x) = x + 2$ for $x > 0$ . (a) Show that f is a function on R. (b) Find f(−3), f(0) and f(4). (c) Sketch the graph. (d) Find the range of f. (e) Show that f(x) = 2 + |x| for every real x. [5 marks] (5 marks)

Step	Marks
(a) Every real x falls in exactly one of x < 0, x = 0, x > 0, so each x has exactly one image: f is a function	1
(b) f(−3) = 2 − (−3) = 5; f(0) = 2; f(4) = 6	1
(c) Graph: the line y = 2 − x to the left of the y-axis and y = x + 2 to the right, meeting at (0, 2): a V shape with its corner at (0, 2)	1
(d) For x ≠ 0 the value is greater than 2, and f(0) = 2, so the range is $[2, \infty)$	1
(e) For x < 0, \|x\| = −x so 2 + \|x\| = 2 − x; for x > 0, 2 + \|x\| = 2 + x; at 0 both are 2 ✓	1

Where students lose marks in Relations and Functions

The slips examiners see most often, and what to write instead.

Mistake: Treating (a, b) and (b, a) as the same
Thinking of pairs like sets.
Wrong working
${(1, a)} = {(a, 1)}$ , so $A \times B = B \times A$
Right working
$(1, a) \neq = (a, 1)$ , so in general $A \times B \neq = B \times A$
Do this: Order matters in an ordered pair.
Usually costs: 1 mark.
Mistake: Giving the codomain as the range
Mixing up the target set with the images actually reached.
Do this: Range = set of second entries actually used; codomain = the whole of B.
Usually costs: 1 mark.
Mistake: Calling a relation a function when some element has no image
Checking only that no element has two images.
Do this: Check both conditions: every element of A has an image, and only one.
Usually costs: 1-2 marks.
Mistake: Writing [−2.3] = −2
Dropping the decimal part instead of going down.
Wrong working
[−2.3] = −2
Right working
−3 ≤ −2.3 < −2, so [−2.3] = −3
Do this: [x] is the greatest integer that does not exceed x.
Usually costs: 1 mark.
Mistake: Forgetting to exclude zeros of g in $\frac{f}{g}$
Simplifying first and losing the restriction.
Do this: State 'x ≠ …' for every value where the original denominator is zero.
Usually costs: 1 mark.
Mistake: Wrong range for $x^{2} + c$ or functions
Writing R by habit.
Do this: A square is never negative and a square root is never negative: $x^{2} + 2$ has range $[2, \infty)$ .
Usually costs: 1 mark.

NCERT exercise guide: Relations and Functions

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Exercise 2.1 (ordered pairs and Cartesian products)

What it tests: Equal ordered pairs, writing products, counting elements, finding A and B from a product, verifying identities such as $A \times (B \cap C) = (A \times B) \cap (A \times C)$ .

How to approach it: Equate entries; list pairs in order; first entries give A, second entries give B.

Same kind as Exercise 2.1

The product A × A has 9 elements, two of which are (2, 5) and (5, 7). Find A and the remaining elements of A × A.

Show the solutionSolution

n(A × A) = 9 means n(A) = 3; the pairs show 2, 5 and 7 are in A, so A = {2, 5, 7}
Remaining pairs: (2, 2), (2, 7), (5, 2), (5, 5), (7, 2), (7, 5), (7, 7)

Answer: A = {2, 5, 7}; the seven pairs listed.

Exercise 2.2 (relations)

What it tests: Writing relations in roster form from a rule, domain, codomain and range, arrow diagrams, counting relations.

How to approach it: Run each element of the first set through the rule; collect first and second entries.

Same kind as Exercise 2.2

Let A = {1, 2, 3, …, 20} and R = {(x, y) : y = 4x − 1, x, y ∈ A}. Find the domain, codomain and range of R.

Show the solutionSolution

y = 4x − 1 must be in A, so x = 1, 2, 3, 4, 5 (x = 6 gives 23): R = {(1, 3), (2, 7), (3, 11), (4, 15), (5, 19)}
Domain {1, 2, 3, 4, 5}; range {3, 7, 11, 15, 19}; codomain A

Answer: Domain {1, 2, 3, 4, 5}, range {3, 7, 11, 15, 19}, codomain A.

Exercise 2.3 and Miscellaneous Exercise (functions)

What it tests: Function or not, domain and range of real functions, values of functions, the Celsius-Fahrenheit function, algebra of functions, finding a linear function.

How to approach it: Apply the two-condition test; use the domain rules; substitute carefully.

Same kind as Exercise 2.3

The function $c (F) = \frac{5 ( F - 32 )}{9}$ converts degrees Fahrenheit into degrees Celsius. Find c(212), c(−40) and the value of F for which c(F) = 37.

Show the solutionSolution

$c (212) = \frac{5 \times 180}{9} = 100$ ; $c (- 40) = \frac{5 \times ( - 72 )}{9} = - 40$
$\frac{5 ( F - 32 )}{9} = 37 \Rightarrow F - 32 = 66.6 \Rightarrow F = 98.6$

Answer: 100, −40, F = 98.6.

Practice questions on Relations and Functions

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
If $(x + 1, y - 2) = (3, 1)$ , then (x, y) is:
(a) (2, 3)
(b) (4, 3)
(c) (2, −1)
(d) (3, 2)
Hint
Equate entries.
Answer
(a) (2, 3)
Q2 · 1 mark · MCQExam level
The range of the signum function is:
(a) R
(b) {−1, 1}
(c) {−1, 0, 1}
(d) $[- 1, 1]$
Hint
Three cases.
Answer
(c) {−1, 0, 1}
Q3 · 1 mark · Assertion-reasonExam level
Assertion (A): {(1, 2), (2, 2), (3, 2)} is a function from {1, 2, 3} to {2, 4}.
Reason (R): In a function, different elements may have the same image.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
A constant function is allowed.
Answer
(a)
Q4 · 1 mark · Very short answerExam level
If $A \times B = {(a, x), (a, y), (b, x), (b, y)}$ , find A and B.
Hint
First and second entries.
Answer
A = {a, b}, B = {x, y}.
Q5 · 2 marks · Very short answerHarder exam level
Write $R = {(x, x^{2} + 1) : x is an odd natural number less than 9}$ in roster form and give its range.
Hint
x = 1, 3, 5, 7.
Answer
R = {(1, 2), (3, 10), (5, 26), (7, 50)}; range {2, 10, 26, 50}.
Q6 · 2 marks · Very short answerHarder exam level
Find the domain of $f (x) = \frac{x + 2}{x - 1}$ .
Hint
Two conditions.
Answer
Need $x + 2 \geq 0$ and $x \neq = 1$ : domain $[- 2, \infty) - {1}$ , i.e. $[- 2, 1) \cup (1, \infty)$ .
Q7 · 2 marks · Very short answerHarder exam level
If $f (x) = x^{3}$ , find $\frac{f ( 2.1 ) - f ( 2 )}{2.1 - 2}$ .
Hint
$2. 1^{3} = 9.261$ .
Answer
$\frac{9.261 - 8}{0.1} = 12.61$ .
Q8 · 3 marks · Short answerHarder exam level
Let A = {2, 3, 4, 6, 8} and R = {(a, b) : a, b ∈ A, a < b and a divides b}. Write R in roster form and find its domain and range.
Hint
a must be strictly smaller than b.
Answer
R = {(2, 4), (2, 6), (2, 8), (3, 6), (4, 8)}; domain {2, 3, 4}; range {4, 6, 8}.
Q9 · 3 marks · Short answerHarder exam level
Let f, g : R → R be $f (x) = 2 x + 1$ and $g (x) = x - 4$ . Find $f + g$ , $f - g$ and $\frac{f}{g}$ , with the domain of the last.
Hint
Zero of g.
Answer
$(f + g) (x) = 3 x - 3$ ; $(f - g) (x) = x + 5$ ; $(\frac{f}{g}) (x) = \frac{2 x + 1}{x - 4}$ , $x \neq = 4$ .
Q10 · 3 marks · Short answerHOTS / competency
Let R = {(a, b) : a, b ∈ N and $b = 2 a$ }. Decide, with an example in each case, whether (i) every natural number a is related to itself, and (ii) whenever a is related to b, b is also related to a.
Hint
Try a = 1 and the pair (3, 6).
Answer
(i) False: (1, 1) ∉ R since $1 \neq = 2 \times 1$ (in fact no a is related to itself). (ii) False: (3, 6) ∈ R since $6 = 2 \times 3$ , but (6, 3) ∉ R since $3 \neq = 12$ .
Q11 · 3 marks · Short answerHOTS / competency
Sketch $f (x) = 2^{x}$ and $g (x) = lo g_{2} x$ on the same axes and state one point on each graph and how the two graphs are related.
Hint
Swap x and y.
Answer
$2^{x}$ passes through (0, 1), (1, 2), (3, 8) and stays above the x-axis; $lo g_{2} x$ passes through (1, 0), (2, 1), (8, 3) and stays to the right of the y-axis. Each point (a, b) on one gives (b, a) on the other, so the graphs are mirror images in the line y = x.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.If n(A) = 3 and n(B) = 4, what is n(A × B)?Show answer
12.
2.How many relations are there from A to B if n(A) = 2 and n(B) = 3?Show answer
$2^{6} = 64$ .
3.What two conditions make a relation a function?Show answer
Every element of the first set has an image, and no element has more than one image.
4.What is the range of the signum function?Show answer
{−1, 0, 1}.
5.What is [−0.5]?Show answer
−1.
6.Domain of $\frac{1}{x - 4}$ ?Show answer
$R - {4}$ .
7.Range of $2^{x}$ ?Show answer
$(0, \infty)$ .
8.What does $R \times R$ represent?Show answer
All points of the coordinate plane.

Best order to study it

Learn ordered pairs, their equality and the Cartesian product.
Practise counting $n (A \times B)$ and recovering A and B from a product.
Learn relations with domain, codomain, range and arrow diagrams; count relations.
Learn the definition of a function and the two conditions to test.
Practise domains and ranges of real functions.
Learn the standard functions and sketch each graph from a small table.
Finish with the algebra of functions and finding a linear function from pairs.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you solve for unknowns from equal ordered pairs?
Can you write $A \times B$ and $A \times A \times A$ for small sets?
Can you find A and B from $A \times B$ ?
Can you write a relation in roster form and give its domain, range and codomain?
Can you count relations from A to B?
Can you decide whether a relation is a function, with reasons?
Can you find the domain and range of a rational or square-root function?
Can you draw the graphs of |x|, sgn x, [x], $x^{2}$ , $x^{3}$ , $\frac{1}{x}$ , $a^{x}$ and $lo g_{a} x$ ?
Can you find $f + g$ , $f - g$ , $f g$ and $\frac{f}{g}$ with the right domain?
Can you find a linear function from two of its pairs?

Relations and Functions: frequently asked questions

What is the difference between a relation and a function?

Every function is a relation, but a function is stricter: each element of the first set must have exactly one image. A relation may leave elements without images or give one element several images.

Is composition of functions in the 2026-27 exam?

CBSE lists composition of functions as taught but assessed only formatively (class activities and tests), so it is not part of the annual exam paper.

Are exponential and logarithmic functions in this chapter?

Yes. The CBSE syllabus names them among the functions whose graphs students should know, even though the rationalised NCERT chapter does not draw them. Learn their domain, range, the points (0, 1) and (1, 0), and the shape of each graph.

Why is range a subset of codomain and not always equal to it?

The codomain is the set you aim into; the range is what you actually hit. For f(x) = x^2 from R to R the codomain is R but the range is only [0, \infty).

How is the Class 11 Maths exam set?

It is a school annual exam of 80 marks (3 hours) plus 20 marks of internal assessment. Unit I, Sets and Functions, carries 23 marks. CBSE gives no Class 11 sample paper, so schools usually mix MCQs, assertion-reason, short and long answers and case-based questions.

Based on: CBSE Mathematics (041) syllabus 2026-27 (Classes XI-XII); NCERT Mathematics, Textbook for Class XI, Chapter 2 (Reprint 2026-27)