CBSE Class 12 Mathematics Relations and Functions Important Questions Set 02

Fill in the blanks.

Question. A relation \( R \) in a set \( A \) is called _____________ relation, if \( (a_1, a_2) \in R \) implies that \( (a_2, a_1) \in R, a_1, a_2 \in A \).
Answer: symmetric

Question. A relation \( R \) on set \( N \) defined by \( R = \{(x, y) : x + 2y = 8\} \). The domain of \( R \) is _____________ .
Answer: \( \{2, 4, 6\} \)

Question. Let \( A = \{1, 2, 3\} \). Then the number of relations containing \( (1, 2) \) and \( (1, 3) \) which are reflexive and symmetric but not transitive, is _____________ .
Answer: 1

Question. Let \( f : R \to R \) be defined by \( f(x) = \frac{x}{\sqrt{1 + x^2}} \). Then \( (fofof)(x) = \) _____________ .
Answer: \( \frac{x}{\sqrt{3x^2 + 1}} \)

Very Short Answer Questions

Question. If \( A = \{3, 5, 7\} \) and \( B = \{2, 4, 9\} \) and \( R \) is a relation from \( A \) to \( B \) given by “is less than”, then write \( R \) as a set of ordered pairs.
Answer: \( R = \{(3, 4), (3, 9), (5, 9), (7, 9)\} \)

Question. Check whether the relation \( R \) in the set \( \{1, 2, 3\} \) given by \( R = \{(1, 2), (2, 1)\} \) is transitive.
Answer: No, it is not transitive.

Question. If \( f(x) = x + 7 \) and \( g(x) = x - 7, x \in R \), then find \( fog (7) \).
Answer: 7

Question. If \( f(x) \) is an invertible function, then find the inverse of \( f(x) = \frac{3x - 2}{5} \).
Answer: \( f^{-1}(x) = \frac{5x + 2}{3} \)

Question. If \( f(x) = 27x^3 \) and \( g(x) = x^{1/3} \), find \( gof(x) \).
Answer: \( 3x \)

Question. For the set \( A = \{1, 2, 3\} \), define a relation \( R \) in the set \( A \) as follows \( R = \{(1, 1), (2, 2), (3, 3), (1, 3)\} \). Write the ordered pair to be added to \( R \) to make it the smallest equivalence relation.
Answer: \( (3, 1) \)

Question. If \( f : R \to R \) is defined by \( f(x) = 3x + 2 \), define \( f[f(x)] \).
Answer: \( 9x + 8 \)

Question. If \( f(x) = x^2 + 4 \), then find \( f^{-1}(x) \).
Answer: \( f^{-1}(x) = \sqrt{x - 4} \)

Short Answer Questions–I

Question. Let the relation \( R \) be defined on the set \( A = \{1, 2, 3, 4, 5\} \) by \( R = \{(a, b) : |a^2 - b^2| < 8\} \). Then write the set \( R \).
Answer: \( \{(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3, 3), (3, 4), (4, 3), (4, 4), (5, 5)\} \)

Question. Let \( A = \{0, 1, 2, 3\} \) and define a relation \( R \) on \( A \) as follows: \( R = \{(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)\} \). Is \( R \) reflexive? symmetric? transitive?
Answer: Reflexive, Symmetric but not transitive.

Question. For real numbers \( x \) and \( y \), a relation \( R \) is defined as \( xRy \) if \( x - y + \sqrt{2} \) is an irrational number. Write whether \( R \) is reflexive, symmetric or transitive.
Answer: Reflexive but neither symmetric nor transitive.

Question. Let the function \( f : R \to R \) be defined by \( f(x) = 4x - 1, \forall x \in R \). Then show that \( f \) is one – one.
Answer: Let \( f(x_1) = f(x_2) \Rightarrow 4x_1 - 1 = 4x_2 - 1 \Rightarrow 4x_1 = 4x_2 \Rightarrow x_1 = x_2 \). Hence, \( f \) is one-one.

Question. If the function \( f : R \to R \), defined by \( f(x) = \frac{2x - 1}{3} \), \( x \in R \) is one-one and onto function then find the inverse of \( f \).
Answer: \( f^{-1}(y) = \frac{3y + 1}{2} \)

Short Answer Questions–II

Question. Show that the relation \( R \) in the set \( N \times N \) defined by \( (a, b)R(c, d) \) iff \( a^2 + d^2 = b^2 + c^2 \forall a, b, c, d \in N \), is an equivalence relation.
Answer: Reflexivity: \( a^2 + b^2 = b^2 + a^2 \Rightarrow (a, b)R(a, b) \). Symmetry: \( a^2 + d^2 = b^2 + c^2 \Rightarrow c^2 + b^2 = d^2 + a^2 \Rightarrow (c, d)R(a, b) \). Transitivity: If \( (a, b)R(c, d) \) and \( (c, d)R(e, f) \), then \( a^2 + d^2 = b^2 + c^2 \) and \( c^2 + f^2 = d^2 + e^2 \). Adding these gives \( a^2 + d^2 + c^2 + f^2 = b^2 + c^2 + d^2 + e^2 \Rightarrow a^2 + f^2 = b^2 + e^2 \Rightarrow (a, b)R(e, f) \). Thus, it is an equivalence relation.

Question. Show that the relation \( S \) in the set \( R \) of real numbers, defined as \( S = \{(a, b): a, b \in R \text{ and } a \le b^3\} \) is neither reflexive, nor symmetric nor transitive.
Answer: Counterexamples: (i) \( 1/2 \not\le (1/2)^3 \) (not reflexive). (ii) \( 1 \le 2^3 \) but \( 2 \not\le 1^3 \) (not symmetric). (iii) \( 3 \le 2^3 \) and \( 2 \le 1.5^3 \) but \( 3 \not\le 1.5^3 \) (not transitive).

Question. Prove that the relation \( R \) in the set \( A = \{1, 2, 3, \dots, 12\} \) given by \( R = \{(a, b) : |a - b| \text{ is divisible by } 3\} \), is an equivalence relation. Find all elements related to the element 1.
Answer: Relation is reflexive (\( |a-a|=0 \)), symmetric (\( |a-b|=|b-a| \)), and transitive (\( a-b=3k, b-c=3m \Rightarrow a-c=3(k+m) \)). Elements related to 1 are \( \{1, 4, 7, 10\} \).

Question. Prove that the relation \( R \) on the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) given by \( R = \{(a, b) : |a - b| \text{ is even } \} \), is an equivalence relation.
Answer: \( |a-a|=0 \) is even (Reflexive). \( |a-b|=|b-a| \) (Symmetric). If \( a-b \) and \( b-c \) are even, their sum \( a-c \) is even (Transitive).

Question. If \( f(x) = \frac{4x + 3}{6x - 4}, x \ne \frac{2}{3} \), then show that \( fof(x) = x, \forall x \ne \frac{2}{3} \). What is the inverse of \( f \)?
Answer: \( f(f(x)) = \frac{4(\frac{4x+3}{6x-4}) + 3}{6(\frac{4x+3}{6x-4}) - 4} = x \). Inverse of \( f \) is \( f^{-1}(x) = \frac{4x + 3}{6x - 4} \).

Question. If the function \( f : R \to R \) be given by \( f(x) = x^2 + 2 \) and \( g : R \to R \) be given by \( g(x) = \frac{x}{x - 1}, x \ne 1 \), find \( fog \) and \( gof \) and hence find \( fog(2) \) and \( gof(-3) \).
Answer: \( fog(x) = \frac{x^2}{(x-1)^2} + 2 = \frac{3x^2 - 4x + 2}{(x-1)^2} \); \( gof(x) = \frac{x^2 + 2}{x^2 + 1} \). Values: \( fog(2) = 6 \); \( gof(-3) = 11/10 \).

Question. Show that the relation \( R \) on \( R \) defined as \( R = \{(a, b) : a \le b\} \), is reflexive and transitive but not symmetric.
Answer: \( a \le a \) (Reflexive). \( a \le b \) and \( b \le c \Rightarrow a \le c \) (Transitive). \( 1 \le 2 \) but \( 2 \not\le 1 \) (Not symmetric).

Question. Let \( Z \) be the set of all integers and \( R \) be relation on \( Z \) defined as \( R = \{(a, b) : a, b \in Z \text{ and } (a - b) \text{ is divisible by } 5\} \). Prove that \( R \) is an equivalence relation.
Answer: Standard proof for reflexivity, symmetry, and transitivity in modular arithmetic.

Question. Show that the function \( f \) in \( A = R - \{2/3\} \) defined as \( f(x) = \frac{4x + 3}{6x - 4} \) is one-one and onto. Hence, find \( f^{-1} \).
Answer: \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \) (one-one). \( y = f(x) \Rightarrow x = \frac{4y+3}{6y-4} \) (onto). \( f^{-1}(x) = \frac{4x + 3}{6x - 4} \).

Question. Prove that the function \( f : N \to N \), defined by \( f(x) = x^2 + x + 1 \) is one-one but not onto. Find inverse of \( f : N \to S \), where \( S \) is range of \( f \).
Answer: \( x_1^2 + x_1 + 1 = x_2^2 + x_2 + 1 \Rightarrow (x_1 - x_2)(x_1 + x_2 + 1) = 0 \Rightarrow x_1 = x_2 \) (one-one). Not onto because \( f(x) = 2 \) has no solution in \( N \). \( f^{-1}(y) = \frac{-1 + \sqrt{4y - 3}}{2} \).

Long Answer Questions

Question. Let \( A = \{x \in Z : 0 \le x \le 12\} \). Show that \( R = \{(a, b) : a, b \in A, |a - b| \text{ is divisible by } 4\} \) is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].
Answer: Equivalence relation proof follows standard steps. Elements related to 1: \( \{1, 5, 9\} \). Equivalence class [2]: \( \{2, 6, 10\} \).

Question. Let \( f : N \to N \) be a function defined as \( f(x) = 9x^2 + 6x - 5 \). Show that \( f : N \to S \), where \( S \) is the range of \( f \), is invertible. Find the inverse of \( f \) and hence find \( f^{-1} (43) \) and \( f^{-1} (163) \).
Answer: \( f^{-1}(x) = \frac{-1 + \sqrt{x+6}}{3} \). Values: \( f^{-1}(43) = 2 \); \( f^{-1}(163) = 4 \).

Question. Show that the function \( f : R \to R \) defined by \( f(x) = \frac{x}{x^2 + 1}, \forall x \in R \) is neither one-one nor onto. Also, if \( g : R \to R \) is defined as \( g(x) = 2x - 1 \), find \( fog(x) \).
Answer: \( f(2) = f(1/2) = 2/5 \) (not one-one). Range is \( [-1/2, 1/2] \) (not onto). \( fog(x) = \frac{2x - 1}{4x^2 - 4x + 2} \).

Question. Let \( f : N \to R \), be a function defined as \( f(x) = 4x^2 + 12x + 15 \). Show that \( f : N \to S \), where \( S \) is the range of \( f \), is invertible. Also find the inverse of \( f \).
Answer: \( f^{-1}(x) = \frac{\sqrt{x - 6} - 3}{2} \).

Question. Show that the relation \( R \) defined by \( (a, b) R (c, d) \Leftrightarrow a + d = b + c \) on the \( A \times A \), where \( A = \{1, 2, 3, \dots, 10\} \) is an equivalence relation. Hence write the equivalence class of [(3, 4)]; \( a, b, c, d \in A \).
Answer: Equivalence class of [(3, 4)] is \( \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10)\} \).

Question. Let \( f : N \to N \) be a function defined as \( f(x) = 4x^2 + 12x +15 \). Show that \( f : N \to S \) is invertible (where \( S \) is range of \( f \)). Find the inverse of \( f \) and hence find \( f^{-1}(31) \) and \( f^{-1}(87) \).
Answer: \( f^{-1}(y) = \frac{\sqrt{y - 6} - 3}{2} \). Values: \( f^{-1}(31) = 1 \); \( f^{-1}(87) = 3 \).

Question. Let \( f : W \to W \) be defined as \( f(n) = \begin{cases} n - 1, & \text{if } n \text{ is odd} \\ n + 1, & \text{if } n \text{ is even} \end{cases} \). Show that \( f \) is invertible and find the inverse of \( f \). Here, \( W \) is the set of all whole numbers.
Answer: The function is its own inverse: \( f^{-1}(x) = \begin{cases} x + 1, & \text{if } x \text{ is odd} \\ x - 1, & \text{if } x \text{ is even} \end{cases} \).

Question. If the function \( f : R \to R \) be defined by \( f(x) = 2x - 3 \) and \( g : R \to R \) by \( g(x) = x^3 + 5 \), then find \( fog \) and show that \( fog \) is invertible. Also, find \( (fog)^{-1} \), hence find \( (fog)^{-1} (9) \).
Answer: \( fog(x) = 2x^3 + 7 \). \( (fog)^{-1}(x) = (\frac{x - 7}{2})^{1/3} \). Value: \( (fog)^{-1}(9) = 1 \).

Self-Assessment Test

Question. In the set \( Z \) of all integers, which of the following relation \( R \) is not an equivalence relation?
(a) \( x R y : \text{if } x \le y \)
(b) \( x R y : \text{if } x = y \)
(c) \( x R y : \text{if } x - y \text{ is an integer} \)
(d) \( x R y : \text{if } x \cong y \text{ (Mod 3)} \)
Answer: (a)

Question. Let \( A = \{1, 2, 3\} \) and \( R = \{(1, 2), (2, 3), (1, 3)\} \) be a relation on set \( A \). Then \( R \) is
(a) neither reflexive nor transitive
(b) neither symmetric nor transitive
(c) transitive
(d) None of these
Answer: (c)