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Mathematics · Class 11 Science

Ch 1Relations and Functions — Class 11 Mathematics, concept-first.

Mathematics, at its heart, is the study of patterns. When you look at two quantities that change — say, the number of hours you study and the marks you score — you are looking for a connection, a predictable link between them. That link is what we call a relation.

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Ordered Pair Equality

Think about a simple list of two things — say, your name and your age. If I write (Ravi, 15), that's an ordered pair. The word "ordered" is the key: the first position and the second position mean different things.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

2.1

Introduction

Mathematics, at its heart, is the study of patterns. When you look at two quantities that change — say, the number of hours you study and the marks you score — you are looking for a connection, a pred…

2.2

Cartesian Products of Sets

16 Q

Before we can talk about the Cartesian product, we need to be absolutely clear about what an ordered pair is. You have seen pairs of numbers before, like coordinates on a graph.

+Worked Examplesi6 questions
  1. Example 1If $(x + 1,\ y - 2) = (3, 1)$, find the values of $x$ and $y$.Free
  2. Example 2If $P = \{a, b, c\}$ and $Q = \{r\}$, form the sets $P \times Q$ and $Q \times P$. Are these two products equal?Free
  3. Example 3Let $A = \{1, 2, 3\}$, $B = \{3, 4\}$ and $C = \{4, 5, 6\}$. Find (i) $A \times (B \cap C)$ (ii) $(A \times B) \cap (A \times C)$ (iii) $A \…Preview
  4. Example 4If $P = \{1, 2\}$, form the set $P \times P \times P$.Preview
  5. Example 5If $\mathbb{R}$ is the set of all real numbers, what do the cartesian products $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb…Preview
  6. Example 6If $A \times B = \{(p, q), (p, r), (m, q), (m, r)\}$, find $A$ and $B$.Preview
+Exercise 2.1i10 questions
  1. Q1If $\left(\dfrac{x}{3} + 1,\ y - \dfrac{2}{3}\right) = \left(\dfrac{5}{3},\ \dfrac{1}{3}\right)$, find the values of $x$ and $y$.Free
  2. Q2If the set $A$ has 3 elements and the set $B = \{3, 4, 5\}$, then find the number of elements in $(A \times B)$.Free
  3. Q3If $G = \{7, 8\}$ and $H = \{5, 4, 2\}$, find $G \times H$ and $H \times G$.Free
  4. Q4State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If $…Preview
  5. Q5If $A = \{-1, 1\}$, find $A \times A \times A$.Preview
  6. Q6If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$. Find $A$ and $B$.Preview
  7. Q7Let $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$, $C = \{5, 6\}$ and $D = \{5, 6, 7, 8\}$. Verify that (i) $A \times (B \cap C) = (A \times B) \cap…Preview
  8. Q8Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Write $A \times B$. How many subsets will $A \times B$ have? List them.Preview
  9. Q9Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1)$, $(y, 2)$, $(z, 1)$ are in $A \times B$, find $A$ and $B$, wher…Preview
  10. Q10The Cartesian product $A \times A$ has 9 elements among which are found $(-1, 0)$ and $(0, 1)$. Find the set $A$ and the remaining elements…Preview
2.3

Relations

12 Q

When we have two sets, the Cartesian product gives us every possible pairing of elements. But in mathematics, we rarely need all possible pairs — we usually care about pairs that satisfy some specific…

+Worked Examplesi3 questions
  1. Example 7Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : y = x + 1\}$. (i) Depict this relation using an arr…Free
  2. Example 8An arrow diagram relates two sets. Set $P = \{9, 4, 25\}$ and set $Q = \{5, 3, 2, 1, -2, -3, -5\}$. Arrows are drawn from $P$ to $Q$ as foll…Preview
  3. Example 9Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Find the number of relations from $A$ to $B$.Preview
+Exercise 2.2i9 questions
  1. Q1Let $A = \{1, 2, 3, \ldots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : 3x - y = 0,\ \text{where}\ x, y \in A\}$. Write…Free
  2. Q2Define a relation $R$ on the set $\mathbb{N}$ of natural numbers by $R = \{(x, y) : y = x + 5,\ x\ \text{is a natural number less than }4;\…Free
  3. Q3$A = \{1, 2, 3, 5\}$ and $B = \{4, 6, 9\}$. Define a relation $R$ from $A$ to $B$ by $R = \{(x, y) : \text{the difference between }x\text{ a…Free
  4. Q4An arrow diagram relates two sets. Set $P = \{5, 6, 7\}$ and set $Q = \{3, 4, 5\}$. An arrow is drawn from each element of $P$ to exactly on…Preview
  5. Q5Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A,\ b\text{ is exactly divisible by }a\}$. (i) W…Preview
  6. Q6Determine the domain and range of the relation $R$ defined by $R = \{(x, x + 5) : x \in \{0, 1, 2, 3, 4, 5\}\}$.Preview
  7. Q7Write the relation $R = \{(x, x^3) : x\text{ is a prime number less than }10\}$ in roster form.Preview
  8. Q8Let $A = \{x, y, z\}$ and $B = \{1, 2\}$. Find the number of relations from $A$ to $B$.Preview
  9. Q9Let $R$ be the relation on $\mathbb{Z}$ defined by $R = \{(a, b) : a, b \in \mathbb{Z},\ a - b\text{ is an integer}\}$. Find the domain and…Preview
2.4

Functions

A function is a special kind of relation — the most important one in mathematics. You can think of it as a rule that takes an input and gives you exactly one output.

+Worked Examplesi3 questions
  1. Example 10Let $\mathbb{N}$ be the set of natural numbers and the relation $R$ be defined on $\mathbb{N}$ such that $R = \{(x, y) : y = 2x,\ x, y \in \…Free
  2. Example 11Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not? (i) $R = \{(2, 1…Preview
  3. Example 12Let $\mathbb{N}$ be the set of natural numbers. Define a real valued function $f : \mathbb{N} \to \mathbb{N}$ by $f(x) = 2x + 1$. Using this…Preview
2.4.1

Some Functions and Their Graphs

The simplest function you will encounter is the identity function. Let be the set of real numbers. Define by

+Worked Examplesi3 questions
  1. Example 13Define the function $f : \mathbb{R} \to \mathbb{R}$ by $y = f(x) = x^2$, $x \in \mathbb{R}$. Complete the table given below by using this de…Free
  2. Example 14Draw the graph of the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$, $x \in \mathbb{R}$.Preview
  3. Example 15Define the real valued function $f : \mathbb{R} - \{0\} \to \mathbb{R}$ defined by $f(x) = \dfrac{1}{x}$, $x \in \mathbb{R} - \{0\}$. Comple…Preview
2.4.2

Algebra of Real Functions

7 Q

When you have two real functions — that is, functions whose domain and codomain are subsets of the real numbers — you can combine them using the same arithmetic operations you use with ordinary number…

+Worked Examplesi2 questions
  1. Example 16Let $f(x) = x^2$ and $g(x) = 2x + 1$ be two real functions. Find $(f + g)(x)$, $(f - g)(x)$, $(fg)(x)$, $\left(\dfrac{f}{g}\right)(x)$.Free
  2. Example 17Let $f(x) = \sqrt{x}$ and $g(x) = x$ be two functions defined over the set of non-negative real numbers. Find $(f + g)(x)$, $(f - g)(x)$, $(…Preview
+Exercise 2.3i5 questions
  1. Q1Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (i) $\{(2, 1), (5, 1), (8…Free
  2. Q2Find the domain and range of the following real functions: (i) $f(x) = -|x|$ (ii) $f(x) = \sqrt{9 - x^2}$.Free
  3. Q3A function $f$ is defined by $f(x) = 2x - 5$. Write down the values of (i) $f(0)$, (ii) $f(7)$, (iii) $f(-3)$.Preview
  4. Q4The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \dfrac{9C}{5} + 32$. F…Preview
  5. Q5Find the range of each of the following functions. (i) $f(x) = 2 - 3x,\ x \in \mathbb{R},\ x > 0$. (ii) $f(x) = x^2 + 2,\ x$ is a real numbe…Preview

Miscellaneous Examples

+Miscellaneous Examplesi5 questions
  1. Example 18Let $\mathbb{R}$ be the set of real numbers. Define the real function $f : \mathbb{R} \to \mathbb{R}$ by $f(x) = x + 10$ and sketch the grap…Free
  2. Example 19Let $R$ be a relation from $\mathbb{Q}$ to $\mathbb{Q}$ defined by $R = \{(a, b) : a, b \in \mathbb{Q}\ \text{and}\ a - b \in \mathbb{Z}\}$.…Free
  3. Example 20Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a linear function from $\mathbb{Z}$ into $\mathbb{Z}$. Find $f(x)$.Preview
  4. Example 21Find the domain of the function $f(x) = \dfrac{x^2 + 3x + 5}{x^2 - 5x + 4}$.Preview
  5. Example 22The function $f$ is defined by $f(x) = \begin{cases} 1 - x, & x < 0 \\ 1, & x = 0 \\ x + 1, & x > 0 \end{cases}$ Draw the graph of $f(x)$.Preview

Miscellaneous Exercise on Chapter 2

+Miscellaneous Exercisei12 questions
  1. Q1The 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 b…Free
  2. Q2If $f(x) = x^2$, find $\dfrac{f(1.1) - f(1)}{(1.1 - 1)}$.Free
  3. Q3Find the domain of the function $f(x) = \dfrac{x^2 + 2x + 1}{x^2 - 8x + 12}$.Free
  4. Q4Find the domain and the range of the real function $f$ defined by $f(x) = \sqrt{x - 1}$.Preview
  5. Q5Find the domain and the range of the real function $f$ defined by $f(x) = |x - 1|$.Preview
  6. Q6Let $f = \left\{ \left(x,\ \dfrac{x^2}{1 + x^2}\right) : x \in \mathbb{R} \right\}$ be a function from $\mathbb{R}$ into $\mathbb{R}$. Deter…Preview
  7. Q7Let $f, g : \mathbb{R} \to \mathbb{R}$ be defined, respectively by $f(x) = x + 1$, $g(x) = 2x - 3$. Find $f + g$, $f - g$ and $\dfrac{f}{g}$…Preview
  8. Q8Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$, for some integer…Preview
  9. Q9Let $R$ be a relation from $\mathbb{N}$ to $\mathbb{N}$ defined by $R = \{(a, b) : a, b \in \mathbb{N}\ \text{and}\ a = b^2\}$. Are the foll…Preview
  10. Q10Let $A = \{1, 2, 3, 4\}$, $B = \{1, 5, 9, 11, 15, 16\}$ and $f = \{(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)\}$. Are the following true? (i) $…Preview
  11. Q11Let $f$ be the subset of $\mathbb{Z} \times \mathbb{Z}$ defined by $f = \{(ab, a + b) : a, b \in \mathbb{Z}\}$. Is $f$ a function from $\mat…Preview
  12. Q12Let $A = \{9, 10, 11, 12, 13\}$ and let $f : A \to \mathbb{N}$ be defined by $f(n) = $ the highest prime factor of $n$. Find the range of $f…Preview

Summary

- A relation from set to set is a subset of . The domain is the set of all first coordinates, the range is the set of all second coordinates, and the codomain is .

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 42 questions42 questions
  1. Q1Let $A = \{-1, 2, 3\}$ and $B = \{1, 3\}$. Determine (i) $A \times B$ (ii) $B \times A$ (iii) $B \times B$ (iv) $A \times A$Free
  2. Q2If $P = \{x : x < 3,\ x \in \mathbf{N}\}$, $Q = \{x : x \le 2,\ x \in \mathbf{W}\}$. Find $(P \cup Q) \times (P \cap Q)$, where $\mathbf{W}$…Free
  3. Q3If $A = \{x : x \in \mathbf{W},\ x < 2\}$, $B = \{x : x \in \mathbf{N},\ 1 < x < 5\}$, $C = \{3, 5\}$ find (i) $A \times (B \cap C)$ (ii) $A…Free
  4. Q4In each of the following cases, find $a$ and $b$. (i) $(2a + b,\ a - b) = (8, 3)$ (ii) $\left(\dfrac{a}{4},\ a - 2b\right) = (0,\ 6 + b)$Preview
  5. Q5Given $A = \{1, 2, 3, 4, 5\}$, $S = \{(x, y) : x \in A,\ y \in A\}$. Find the ordered pairs which satisfy the conditions given below: (i) $x…Preview
  6. Q6Given $R = \{(x, y) : x, y \in \mathbf{W},\ x^2 + y^2 = 25\}$. Find the domain and Range of $R$.Preview
  7. Q7If $R_1 = \{(x, y) \mid y = 2x + 7,\ \text{where } x \in \mathbf{R} \text{ and } -5 \le x \le 5\}$ is a relation. Then find the domain and R…Preview
  8. Q8If $R_2 = \{(x, y) \mid x \text{ and } y \text{ are integers and } x^2 + y^2 = 64\}$ is a relation. Then find $R_2$.Preview
  9. Q9If $R_3 = \{(x,\ |x|) \mid x \text{ is a real number}\}$ is a relation. Then find domain and range of $R_3$.Preview
  10. Q10Is the given relation a function? Give reasons for your answer. (i) $h = \{(4, 6), (3, 9), (-11, 6), (3, 11)\}$ (ii) $f = \{(x, x) \mid x \t…Preview
  11. Q11If $f$ and $g$ are real functions defined by $f(x) = x^2 + 7$ and $g(x) = 3x + 5$, find each of the following (a) $f(3) + g(-5)$ (b) $f\left…Preview
  12. Q12Let $f$ and $g$ be real functions defined by $f(x) = 2x + 1$ and $g(x) = 4x - 7$. (a) For what real numbers $x$, $f(x) = g(x)$? (b) For what…Preview
  13. Q13If $f$ and $g$ are two real valued functions defined as $f(x) = 2x + 1$, $g(x) = x^2 + 1$, then find. (i) $f + g$ (ii) $f - g$ (iii) $fg$ (i…Preview
  14. Q14Express the following functions as set of ordered pairs and determine their range. $f : X \to \mathbf{R}$, $f(x) = x^3 + 1$, where $X = \{-1…Preview
  15. Q15Find the values of $x$ for which the functions $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$ are equalPreview
  16. Q16Is $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$ a function? Justify. If this is described by the relation, $g(x) = \alpha x + \beta$, then what…Preview
  17. Q17Find the domain of each of the following functions given by (i) $f(x) = \dfrac{1}{\sqrt{1 - \cos x}}$ (ii) $f(x) = \dfrac{1}{\sqrt{x + |x|}}…Preview
  18. Q18Find the range of the following functions given by (i) $f(x) = \dfrac{3}{2 - x^2}$ (ii) $f(x) = 1 - |x - 2|$ (iii) $f(x) = |x - 3|$ (iv) $f(…Preview
  19. Q19Redefine the function $f(x) = |x - 2| + |2 + x|$, $-3 \le x \le 3$Preview
  20. Q20If $f(x) = \dfrac{x - 1}{x + 1}$, then show that (i) $f\left(\dfrac{1}{x}\right) = -f(x)$ (ii) $f\left(-\dfrac{1}{x}\right) = \dfrac{-1}{f(x…Preview
  21. Q21Let $f(x) = \sqrt{x}$ and $g(x) = x$ be two functions defined in the domain $\mathbf{R}^+ \cup \{0\}$. Find (i) $(f + g)(x)$ (ii) $(f - g)(x…Preview
  22. Q22Find the domain and Range of the function $f(x) = \dfrac{1}{\sqrt{x - 5}}$.Preview
  23. Q23If $f(x) = y = \dfrac{ax - b}{cx - a}$, then prove that $f(y) = x$.Preview
  24. Q24Let $n(A) = m$, and $n(B) = n$. Then the total number of non-empty relations that can be defined from $A$ to $B$ is (A) $m^n$ (B) $n^m - 1$…Preview
  25. Q25If $[x]^2 - 5[x] + 6 = 0$, where $[\,.\,]$ denote the greatest integer function, then (A) $x \in [3, 4]$ (B) $x \in (2, 3]$ (C) $x \in [2, 3…Preview
  26. Q26Range of $f(x) = \dfrac{1}{1 - 2\cos x}$ is (A) $\left[\dfrac{1}{3},\ 1\right]$ (B) $\left[-1,\ \dfrac{1}{3}\right]$ (C) $(-\infty,\ -1] \cu…Preview
  27. Q27Let $f(x) = \sqrt{1 + x^2}$, then (A) $f(xy) = f(x) \cdot f(y)$ (B) $f(xy) \ge f(x) \cdot f(y)$ (C) $f(xy) \le f(x) \cdot f(y)$ (D) None of…Preview
  28. Q28Domain of $\sqrt{a^2 - x^2}$ $(a > 0)$ is (A) $(-a,\ a)$ (B) $[-a,\ a]$ (C) $[0,\ a]$ (D) $(-a,\ 0]$Preview
  29. Q29If $f(x) = ax + b$, where $a$ and $b$ are integers, $f(-1) = -5$ and $f(3) = 3$, then $a$ and $b$ are equal to (A) $a = -3,\ b = -1$ (B) $a…Preview
  30. Q30The domain of the function $f$ defined by $f(x) = \sqrt{4 - x} + \dfrac{1}{\sqrt{x^2 - 1}}$ is equal to (A) $(-\infty,\ -1) \cup (1,\ 4]$ (B…Preview
  31. Q31The domain and range of the real function $f$ defined by $f(x) = \dfrac{4 - x}{x - 4}$ is given by (A) Domain $= \mathbf{R}$, Range $= \{-1,…Preview
  32. Q32The domain and range of real function $f$ defined by $f(x) = \sqrt{x - 1}$ is given by (A) Domain $= (1,\ \infty)$, Range $= (0,\ \infty)$ (…Preview
  33. Q33The domain of the function $f$ given by $f(x) = \dfrac{x^2 + 2x + 1}{x^2 - x - 6}$ (A) $\mathbf{R} - \{3,\ -2\}$ (B) $\mathbf{R} - \{-3,\ 2\…Preview
  34. Q34The domain and range of the function $f$ given by $f(x) = 2 - |x - 5|$ is (A) Domain $= \mathbf{R}^+$, Range $= (-\infty,\ 1]$ (B) Domain $=…Preview
  35. Q35The domain for which the functions defined by $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$ are equal is (A) $\left\{-1,\ \dfrac{4}{3}\right\}$ (B) $…Preview
  36. Q36Let $f$ and $g$ be two real functions given by $f = \{(0, 1), (2, 0), (3, -4), (4, 2), (5, 1)\}$, $g = \{(1, 0), (2, 2), (3, -1), (4, 4), (5…Preview
  37. Q37The ordered pair $(5, 2)$ belongs to the relation $R = \{(x, y) : y = x - 5,\ x, y \in \mathbf{Z}\}$Preview
  38. Q38If $P = \{1, 2\}$, then $P \times P \times P = \{(1, 1, 1), (2, 2, 2), (1, 2, 2), (2, 1, 1)\}$Preview
  39. Q39If $A = \{1, 2, 3\}$, $B = \{3, 4\}$ and $C = \{4, 5, 6\}$, then $(A \times B) \cup (A \times C) = \{(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),…Preview
  40. Q40If $(x - 2,\ y + 5) = \left(-2,\ \dfrac{1}{3}\right)$ are two equal ordered pairs, then $x = 4$, $y = \dfrac{-14}{3}$Preview
  41. Q41If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$, then $A = \{a, b\}$, $B = \{x, y\}$Preview
  42. Q42Let $f = \{(2, 4), (5, 6), (8, -1), (10, -3)\}$, $g = \{(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)\}$ be two real functions. Then, match the…Preview