NCERT Solutions
Class 12 - Mathematics Part-1 - Chapter 1: RELATIONS AND FUNCTIONS

EXERCISE 1.2

Show that the function \( f : \mathbb{R}_* \to \mathbb{R}_* \) defined by \( f(x) = \dfrac{1}{x} \) is one-one and onto, where \( \mathbb{R}_* \) is the set of all non-zero real numbers. Is the result true, if the domain \( \mathbb{R}_* \) is replaced by \( \mathbb{N} \) with co-domain being same as \( \mathbb{R}_* \)?

Check the injectivity and surjectivity of the following functions:

1. \( f : \mathbb{N} \to \mathbb{N} \) given by \( f(x) = x^2 \).
2. \( f : \mathbb{Z} \to \mathbb{Z} \) given by \( f(x) = x^2 \).
3. \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^2 \).
4. \( f : \mathbb{N} \to \mathbb{N} \) given by \( f(x) = x^3 \).
5. \( f : \mathbb{Z} \to \mathbb{Z} \) given by \( f(x) = x^3 \).

Prove that the Greatest Integer Function \( f : \mathbb{R} \to \mathbb{R} \), given by \( f(x) = [x] \), is neither one-one nor onto, where \([x]\) denotes the greatest integer less than or equal to \( x \).

Show that the Modulus Function \( f : \mathbb{R} \to \mathbb{R} \), given by \( f(x) = |x| \), is neither one-one nor onto, where \(|x|\) is \(x\) if \(x\) is positive or 0 and \(|x|\) is \(-x\) if \(x\) is negative.

Show that the Signum Function \( f : \mathbb{R} \to \mathbb{R} \), given by

\[ f(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if } x < 0 \end{cases} \]

is neither one-one nor onto.

Let \( A = \{1,2,3\} \), \( B = \{4,5,6,7\} \) and let \( f = \{(1,4), (2,5), (3,6)\} \) be a function from \( A \) to \( B \). Show that \( f \) is one-one.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

1. \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3 - 4x \).
2. \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 1 + x^2 \).

Let \( A \) and \( B \) be sets. Show that \( f : A \times B \to B \times A \) such that \( f(a,b) = (b,a) \) is a bijective function.

Let \( f : \mathbb{N} \to \mathbb{N} \) be defined by

\[ f(n) = \begin{cases} \dfrac{n+1}{2}, & \text{if } n \text{ is odd} \\ \dfrac{n}{2}, & \text{if } n \text{ is even} \end{cases} \]

for all \( n \in \mathbb{N} \). State whether the function \( f \) is bijective. Justify your answer.

Let \( A = \mathbb{R} - \{3\} \) and \( B = \mathbb{R} - \{1\} \). Consider the function \( f : A \to B \) defined by \( f(x) = \dfrac{x - 2}{x - 3} \). Is \( f \) one-one and onto? Justify your answer.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = x^4 \). Choose the correct answer.

(A) \( f \) is one-one onto

(B) \( f \) is many-one onto

(C) \( f \) is one-one but not onto

(D) \( f \) is neither one-one nor onto.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 3x \). Choose the correct answer.

(A) \( f \) is one-one onto

(B) \( f \) is many-one onto

(C) \( f \) is one-one but not onto

(D) \( f \) is neither one-one nor onto.