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NCERT Solutions · Class 12 · Chapter 1: Relations and Functions

NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2

Exercise 1.2: Types of functions. One-one: \(f(x_1) = f(x_2) \Rightarrow x_1 = x_2\) (to disprove, find two inputs with the same output). Onto: every element of the codomain is an output (solve \(f(x) = y\) for \(x\) in the domain). Bijective: both.

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Exercise 1.2 questions and solutions

Exercise 1.2, Question 1

Show \(f : \mathbb R_* \to \mathbb R_*\), \(f(x) = \tfrac1x\) is one-one and onto (\(\mathbb R_*\): non-zero reals). Is this still true with domain \(\mathbb N\) and the same codomain?
Show solution
  1. One-one: \[\begin{aligned}&\tfrac1{x_1} = \tfrac1{x_2} \\ \Rightarrow\ &x_1 = x_2\end{aligned}\] Onto: any \(y \ne 0\) is \(f\left(\tfrac1y\right)\).
  2. With domain \(\mathbb N\): still one-one, but not onto (e.g. \(2\) is not \(\tfrac1n\) for any natural \(n\)).
Answer: Bijective on \(\mathbb R_*\); on \(\mathbb N\) one-one but not onto

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Exercise 1.2, Question 2

Check whether each function is one-one (injective) and onto (surjective).
(i) \(f : \mathbb N \to \mathbb N,\ f(x) = x^2\)
Show solution
  1. One-one: \(x_1^2 = x_2^2\) with positive \(x\) gives \(x_1 = x_2\).
  2. Not onto: 2 is not a square of a natural number.
Answer: Injective, not surjective
(ii) \(f : \mathbb Z \to \mathbb Z,\ f(x) = x^2\)
Show solution
  1. \(f(-1) = f(1)\): not one-one. Not onto: \(-1\) is not a square.
Answer: Neither
(iii) \(f : \mathbb R \to \mathbb R,\ f(x) = x^2\)
Show solution
  1. \(f(-1) = f(1)\): not one-one. Not onto: \(-1\) has no real square root.
Answer: Neither
(iv) \(f : \mathbb N \to \mathbb N,\ f(x) = x^3\)
Show solution
  1. One-one: cubes of different naturals differ.
  2. Not onto: 2 is not a cube.
Answer: Injective, not surjective
(v) \(f : \mathbb Z \to \mathbb Z,\ f(x) = x^3\)
Show solution
  1. One-one: \(x^3\) is strictly increasing.
  2. Not onto: 2 is not the cube of an integer.
Answer: Injective, not surjective

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Exercise 1.2, Question 3

Show that the greatest integer function \(f : \mathbb R \to \mathbb R\), \(f(x) = [x]\), is neither one-one nor onto.
Show solution
  1. \(f(1.2) = f(1.9) = 1\): not one-one.
  2. Outputs are integers only, so \(0.5\) has no pre-image: not onto.
Answer: Shown

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Exercise 1.2, Question 4

Show that the modulus function \(f : \mathbb R \to \mathbb R\), \(f(x) = |x|\), is neither one-one nor onto.
Show solution
  1. \(|-1| = |1|\): not one-one.
  2. \(|x| \ge 0\), so \(-1\) has no pre-image: not onto.
Answer: Shown

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Exercise 1.2, Question 5

Show that the signum function (\(1\) for \(x > 0\), \(0\) for \(x = 0\), \(-1\) for \(x < 0\)) from \(\mathbb R\) to \(\mathbb R\) is neither one-one nor onto.
Show solution
  1. \(f(1) = f(2) = 1\): not one-one.
  2. The only outputs are \(-1, 0, 1\), so \(2\) has no pre-image: not onto.
Answer: Shown

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Exercise 1.2, Question 6

\(A = \{1, 2, 3\}\), \(B = \{4, 5, 6, 7\}\), \(f = \{(1, 4), (2, 5), (3, 6)\}\). Show f is one-one.
Show solution
  1. The images \(4, 5, 6\) are all different.
Answer: f is one-one (not onto: 7 has no pre-image)

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Exercise 1.2, Question 7

Is each function one-one, onto or bijective?
(i) \(f : \mathbb R \to \mathbb R,\ f(x) = 3 - 4x\)
Show solution
  1. One-one: \[\begin{aligned}&3 - 4x_1 = 3 - 4x_2 \\ \Rightarrow\ &x_1 = x_2\end{aligned}\]
  2. Onto: \[\begin{aligned}&y = 3 - 4x \\ \Rightarrow\ &x = \dfrac{3 - y}{4}\end{aligned}\], a real number for every real \(y\).
Answer: Bijective
(ii) \(f : \mathbb R \to \mathbb R,\ f(x) = 1 + x^2\)
Show solution
  1. \(f(-1) = f(1) = 2\): not one-one.
  2. \(1 + x^2 \ge 1\): \(0\) has no pre-image, not onto.
Answer: Neither

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Exercise 1.2, Question 8

For sets A and B, show \(f : A \times B \to B \times A\), \(f(a, b) = (b, a)\), is bijective.
Show solution
  1. One-one: \[\begin{gathered}(b_1, a_1) = (b_2, a_2) \Rightarrow a_1 = a_2, \\ b_1 = b_2\end{gathered}\]
  2. Onto: any \((b, a) \in B \times A\) is \(f(a, b)\).
Answer: Bijective

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Exercise 1.2, Question 9

\(f : \mathbb N \to \mathbb N\), \(f(n) = \dfrac{n + 1}{2}\) if n is odd, \(\dfrac n2\) if n is even. Is f bijective?
Show solution
  1. \(f(1) = 1 = f(2)\): not one-one.
  2. Onto: any \(m \in \mathbb N\) equals \(f(2m)\).
Answer: Onto but not one-one, so not bijective

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Exercise 1.2, Question 10

\(A = \mathbb R - \{3\}\), \(B = \mathbb R - \{1\}\), \(f : A \to B\), \(f(x) = \dfrac{x - 2}{x - 3}\). Is f one-one and onto?
Show solution
  1. One-one: \[\begin{aligned}&\dfrac{x_1 - 2}{x_1 - 3} = \dfrac{x_2 - 2}{x_2 - 3} \\ \Rightarrow\ &-3x_1 - 2x_2 = -2x_1 - 3x_2 \\ \Rightarrow\ &x_1 = x_2\end{aligned}\]
  2. Onto: \[\begin{aligned}&y = \dfrac{x - 2}{x - 3} \\ \Rightarrow\ &x = \dfrac{3y - 2}{y - 1}\end{aligned}\], defined for \(y \ne 1\), and \(x \ne 3\) (else \(3y - 2 = 3y - 3\)).
Answer: Bijective

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Exercise 1.2, Question 11

\(f : \mathbb R \to \mathbb R\), \(f(x) = x^4\). Choose: (A) one-one onto (B) many-one onto (C) one-one not onto (D) neither one-one nor onto
Show solution
  1. \(f(-1) = f(1)\): not one-one. \(x^4 \ge 0\): not onto.
Answer: (D)

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Exercise 1.2, Question 12

\(f : \mathbb R \to \mathbb R\), \(f(x) = 3x\). Choose: (A) one-one onto (B) many-one onto (C) one-one not onto (D) neither
Show solution
  1. \(3x_1 = 3x_2 \Rightarrow x_1 = x_2\); any \(y\) is \(f\left(\tfrac y3\right)\).
Answer: (A)

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Relations and Functions in our sample papers: Sample paper 1 (questions 1, 38) · Sample paper 2 (questions 1, 21, 36) · Sample paper 3 (questions 1, 21, 38) · Sample paper 4 (questions 1, 21, 36) · Sample paper 5 (questions 1, 21, 38).

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Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.