An ordered pair consists of two elements written in a specific order: $(a, b)$, where $a$ is the first element and $b$ is the second element. Note that $(a, b) \ne (b, a)$ unless $a = b$.
Equality of Ordered Pairs
Two ordered pairs are equal if and only if their corresponding first and second elements are equal:
Classic Exam Question: If $(x + 1, y - 2) = (3, 1)$, find $x$ and $y$.
Comparing coordinates: $x + 1 = 3 \implies x = 2$; and $y - 2 = 1 \implies y = 3$.
Definition: Cartesian Product $A \times B$
Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$.
$$A \times B = \{(a, b) : a \in A, b \in B\}$$
Crucial Properties of Cartesian Products
Cardinality Rule: If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = p \cdot q$.
Non-Commutative: In general, $A \times B \ne B \times A$ unless $A = B$.
Empty Set Product: If either $A = \phi$ or $B = \phi$, then $A \times B = \phi$.
Ordered Triplet: $A \times A \times A = \{(a, b, c) : a, b, c \in A\}$. $(a, b, c)$ is called an ordered triplet.
Geometric Meaning: $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$ represents the coordinates of all points in the 2D Cartesian plane. $\mathbb{R} \times \mathbb{R} \times \mathbb{R} = \mathbb{R}^3$ represents 3D space!
2. Relations: Connecting Two Sets
Intuition: What is a Relation?
In English, a relation is a connection between people (e.g., "is brother of", "is taller than"). In mathematics, a relation between two sets $A$ and $B$ is simply a rule or condition that pairs certain elements of $A$ with elements of $B$.
Formal Definition of Relation
A relation $R$ from set $A$ to set $B$ is simply any subset of the Cartesian product $A \times B$. That is, $R \subseteq A \times B$.
How Many Relations Can Exist?
Since any relation $R$ from $A$ to $B$ is a subset of $A \times B$, and $A \times B$ has $p \cdot q$ elements:
$$\text{Total number of relations from } A \text{ to } B = 2^{n(A) \cdot n(B)} = 2^{pq}$$
Example: If $n(A) = 2$ and $n(B) = 3$, there are $2^{2 \times 3} = 2^6 = 64$ possible relations!
Domain of $R$
The set of all first elements of the ordered pairs in $R$.
$$\text{Domain} = \{a \in A : (a, b) \in R\}$$
Range of $R$
The set of all second elements actually used in $R$.
$$\text{Range} = \{b \in B : (a, b) \in R\}$$
Codomain of $R$
The entire target set $B$, whether all elements are matched or not. Always: $\mathbf{\text{Range} \subseteq \text{Codomain}}$.
Arrow Diagram
Visual depiction: Draw two ellipses for $A$ and $B$, write elements inside, and draw directed arrows from $a$ to $b$ whenever $(a, b) \in R$.
3. Functions: The Special Relation
The Vending Machine Metaphor
Think of a function like a reliable vending machine:
Every valid button you press (every input in the Domain) must produce a snack (an output).
When you press button "A1", you must get exactly one predictable item — never two different items simultaneously. If pressing button 1 gives a soda today and chips tomorrow at random, the machine is broken!
Formal Definition of a Function
A relation $f$ from set $A$ to set $B$ is called a function if:
Every element of $A$ has an image in $B$. (No orphan inputs).
Each element of $A$ has one and only one image in $B$. (No ambiguous multi-outputs).
We write $f: A \to B$, and if $(x, y) \in f$, we write $y = f(x)$, where $y$ is the image of $x$ under $f$, and $x$ is the pre-image of $y$.
Summary: Is It a Function or Not?
Many inputs to one output: ALLOWED! (e.g. $(-2)^2 = 4$ and $(2)^2 = 4$; two inputs can share the same image).
One input to multiple outputs: FORBIDDEN! If an input $x$ points to both $y_1$ and $y_2$ ($y_1 \ne y_2$), it is NOT a function.
Leftover elements in set $A$: FORBIDDEN! Every element in $A$ must be mapped.
Leftover elements in set $B$: ALLOWED! Codomain can have elements that aren't mapped to anything.
Vertical Line Test: If any vertical line intersects the graph of a curve at more than one point, the curve is NOT a function of $x$.
4. Standard Real Functions & Their Graphs
Function Name
Formula $f(x)$
Domain
Range
Key Characteristics
Identity Function
$f(x) = x$
$\mathbb{R}$
$\mathbb{R}$
Straight line through origin at $45^\circ$ angle ($y = x$).
Constant Function
$f(x) = c$
$\mathbb{R}$
$\{c\}$
Horizontal line parallel to the x-axis at height $y = c$.
Square Function
$f(x) = x^2$
$\mathbb{R}$
$[0, \infty)$
Parabola opening upwards with vertex at $(0, 0)$.
Cube Function
$f(x) = x^3$
$\mathbb{R}$
$\mathbb{R}$
S-shaped curve passing through origin.
Reciprocal Function
$f(x) = 1/x$
$\mathbb{R} \setminus \{0\}$
$\mathbb{R} \setminus \{0\}$
Rectangular hyperbola in 1st and 3rd quadrants; never touches axes.
Modulus Function
$f(x) = |x|$
$\mathbb{R}$
$[0, \infty)$
V-shaped graph: $x$ for $x \ge 0$, $-x$ for $x < 0$. Always non-negative.
Signum Function
$\text{sgn}(x)$
$\mathbb{R}$
$\{-1, 0, 1\}$
Outputs $1$ if $x > 0$, $0$ if $x = 0$, and $-1$ if $x < 0$. Discontinuous at $0$.
Greatest Integer Function (GIF)
$f(x) = [x]$
$\mathbb{R}$
$\mathbb{Z}$ (Integers)
Step function ("Floor"): outputs greatest integer $\le x$. E.g. $[2.9]=2, [-1.2]=-2$.
Deep Dive: Greatest Integer Function $[x]$
The greatest integer function rounds any real number DOWN to the nearest integer on the number line:
$[3.7] = 3$
$[4.0] = 4$
$[0.85] = 0$
$[-0.4] = -1$ (Careful! Round down to the left on the number line!)
$[-2.8] = -3$
5. Algebra of Real Functions
Operations on Real Functions
Let $f: X \to \mathbb{R}$ and $g: X \to \mathbb{R}$ be two real functions. Their domain for basic operations is the intersection of their domains: $D_f \cap D_g$.
Example 2: Elements of Cartesian Product & Number of RelationsEasy
Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Write $A \times B$. How many relations are there from $A$ to $B$? Write three such relations.
Step 1: Form all ordered pairs:
$$A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$$
Step 2: Find total relations:
Since $n(A \times B) = 4$, the number of subsets of $A \times B$ is $2^4 = 16$. Hence, there are 16 relations.
Step 3: Three examples of relations:
$R_1 = \phi$ (empty relation),
$R_2 = \{(1, 3), (2, 4)\}$,
$R_3 = A \times B$ (universal relation).
Example 3: Ordered Triplet $P \times P \times P$Medium
If $P = \{-1, 1\}$, form the set $P \times P \times P$.
Step 1: First compute $P \times P$:
$$P \times P = \{(-1, -1), (-1, 1), (1, -1), (1, 1)\}$$
Step 2: Combine each pair with each element of $P$:
$$P \times P \times P = \{(-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), (1,-1,-1), (1,-1,1), (1,1,-1), (1,1,1)\}$$
Total elements: $2 \times 2 \times 2 = 8$.
Example 4: Relation Domain, Range, and Arrow DiagramMedium
Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : y = x + 1\}$.
(i) Write $R$ in roster form. (ii) Write down its domain, codomain, and range.
Step 1: Substitute values of $x \in A$ such that $y = x + 1 \in A$:
Example 5: Testing if a Piecewise Relation is a FunctionMedium
The relation $f$ is defined by:
$$f(x) = \begin{cases} x^2, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases}$$
The relation $g$ is defined by:
$$g(x) = \begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases}$$
Show that $f$ is a function and $g$ is not a function.
For relation $f$: The overlap occurs at $x = 3$.
From first piece: $f(3) = 3^2 = 9$.
From second piece: $f(3) = 3(3) = 9$.
Both definitions yield the exact same value $9$ at the boundary. For every other $x \in [0, 10]$, $x$ belongs to only one piece and has a unique output. Thus, each input has exactly one image. $f$ is a function!
For relation $g$: The overlap occurs at $x = 2$.
From first piece: $g(2) = 2^2 = 4$.
From second piece: $g(2) = 3(2) = 6$.
Here, the single input $x = 2$ gives two distinct outputs ($4$ and $6$). This violates the uniqueness condition of a function. $g$ is NOT a function!
Example 6: Domain and Range of Radical Function $\sqrt{9 - x^2}$Hard
Find the domain and range of the real-valued function $f(x) = \sqrt{9 - x^2}$.
Step 1: Domain
For $f(x)$ to be a real number, the term under the square root must be non-negative:
$$9 - x^2 \ge 0 \implies x^2 \le 9 \implies -3 \le x \le 3$$
$$\mathbf{\text{Domain} = [-3, 3]}$$
Step 2: Range
Let $y = \sqrt{9 - x^2}$.
Since the square root symbol represents the principal (positive) square root: $y \ge 0$.
Squaring both sides: $y^2 = 9 - x^2 \implies x^2 = 9 - y^2$.
Since $x \in [-3, 3]$, we know $x^2 \ge 0$, which gives:
$$9 - y^2 \ge 0 \implies y^2 \le 9 \implies -3 \le y \le 3$$
Combining with $y \ge 0$:
$$0 \le y \le 3$$
$$\mathbf{\text{Range} = [0, 3]}$$
Example 7: Domain of a Rational FunctionHard
Find the domain of the function:
$$f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$$
Step 1: The numerator is a polynomial defined for all $x \in \mathbb{R}$.
Step 2: For a rational function, division by zero is undefined. Hence, we must find where the denominator equals zero:
$$x^2 - 8x + 12 = 0 \implies (x - 2)(x - 6) = 0 \implies x = 2 \text{ or } x = 6$$
Let $f(x) = \sqrt{x}$ and $g(x) = x$. Find $(f + g)(x)$, $(f - g)(x)$, $(f \cdot g)(x)$ and $\left(\frac{f}{g}\right)(x)$ along with their domains.
Domains of individual functions:
Domain of $f$ is $[0, \infty)$ because $x$ is under a square root.
Domain of $g$ is $\mathbb{R}$.
Their intersection is $D_f \cap D_g = [0, \infty)$.
Quotient:
$\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}}$.
Since denominator $g(x) = x \ne 0$, we must exclude $0$.
Domain of $\frac{f}{g}$ is $(0, \infty)$.
8. NCERT Key Exercise Questions Solved
NCERT Exercise 2.1 — Question 5 & 7
Q: If $A \times B$ has 6 elements and three of them are $(1, 4), (2, 6), (3, 4)$, find the sets $A$ and $B$, and the remaining elements of $A \times B$.
View Full Solution
First elements belong to $A$: $\{1, 2, 3\} \subseteq A$.
Second elements belong to $B$: $\{4, 6\} \subseteq B$.
Since $n(A \times B) = n(A) \cdot n(B) = 6$, and $|\{1, 2, 3\}| = 3$, $|\{4, 6\}| = 2$, we have $3 \times 2 = 6$.
Therefore, $A = \{1, 2, 3\}$ and $B = \{4, 6\}$.
The full set $A \times B = \{(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6)\}$.
Remaining elements: $\{(1, 6), (2, 4), (3, 6)\}$.
NCERT Exercise 2.2 — Question 2
Q: Define a relation $R$ on the set $\mathbb{N}$ of natural numbers by $R = \{(x, y) : y = x + 5,\; x \text{ is a natural number less than } 4,\; x, y \in \mathbb{N}\}$. Depict this relationship using roster form. Write down the domain and the range.
View Full Solution
Natural numbers less than 4 are $x \in \{1, 2, 3\}$.
For $x = 1 \implies y = 1 + 5 = 6$.
For $x = 2 \implies y = 2 + 5 = 7$.
For $x = 3 \implies y = 3 + 5 = 8$.
Roster form: $R = \{(1, 6), (2, 7), (3, 8)\}$.
Domain: $\{1, 2, 3\}$.
Range: $\{6, 7, 8\}$.
NCERT Exercise 2.3 — Question 2
Q: Find the domain and range of the following real functions:
(i) $f(x) = -|x|$
(ii) $f(x) = \sqrt{9 - x^2}$
View Full Solution
(i) For $f(x) = -|x|$:
Domain: Since $|x|$ is defined for all real numbers, Domain = $\mathbb{R}$.
Range: For all $x \in \mathbb{R}$, $|x| \ge 0$. Multiplying by $-1$ reverses the inequality: $-|x| \le 0$. Hence, Range = $(-\infty, 0]$.
Range: At $x = \pm 3$, $f(x) = 0$. At $x = 0$, $f(x) = \sqrt{9} = 3$. Hence, Range = $[0, 3]$.
NCERT Miscellaneous Exercise — Question 11 & 12
Q: Let $A = \{9, 10, 11, 12, 13\}$ and let $f: A \to \mathbb{N}$ be defined by $f(n) = \text{the highest prime factor of } n$. Find the range of $f$.
View Full Solution
Find prime factorisation of each element in $A$:
$9 = 3^2 \implies$ highest prime factor is $3$.
$10 = 2 \times 5 \implies$ highest prime factor is $5$.
$11 = 11 \implies$ highest prime factor is $11$.
$12 = 2^2 \times 3 \implies$ highest prime factor is $3$.
$13 = 13 \implies$ highest prime factor is $13$.
The set of outputs is $\{3, 5, 11, 3, 13\}$.
Range of $f = \{3, 5, 11, 13\}$.
9. Common Exam Traps & Pitfalls
Trap 1: Confusing Codomain and Range
If $f: A \to B$, the codomain is always the entire set $B$. The range is only the subset of elements in $B$ that actually get hit by an arrow. Range $\subseteq$ Codomain. They are equal only if every element in $B$ has a preimage (onto function).
Trap 2: Square Root Output is Never Negative
In real functions, $\sqrt{x}$ denotes the principal (positive) square root. So $\sqrt{4} = 2$, NOT $\pm 2$. If someone writes $\text{Range of } \sqrt{9 - x^2} = [-3, 3]$, they lose all marks! The range is $[0, 3]$.
Trap 3: Greatest Integer of Negative Numbers
Students often write $[-2.3] = -2$. That is WRONG! $[-2.3]$ rounds DOWN (to the left) on the number line, so $[-2.3] = -3$. Remember: $[x] \le x$. Since $-2 > -2.3$, it cannot be $-2$.
Trap 4: Forgetting the Overlap Check in Piecewise Functions
When asked if a split relation is a function, always evaluate both formulas at the boundary point! If they disagree, the single input has two different images and it fails to be a function.
10. Quick Self-Check Quiz
Q1: If set $A$ has 3 elements and set $B$ has 2 elements, how many non-empty relations can be defined from $A$ to $B$?
Q4: What is the domain of $f(x) = \frac{1}{\sqrt{x - 5}}$?
Answer
Since the square root is in the denominator, $x - 5$ must be strictly positive: $x - 5 > 0 \implies x > 5$. Domain is $(5, \infty)$. (Note: open bracket at 5 because division by zero is undefined).