Section 1.3: Functions

Consider the functions \(f_1, f_2, f_3\) and \(f_4\) given by the following diagrams.

In Fig 1.2, we observe that the images of distinct elements of \(X_1\) under the function \(f_1\) are distinct, but the image of two distinct elements 1 and 2 of \(X_1\) under \(f_1\) is same, namely \(b\). Further, there are some elements like \(e\) and \(f\) in \(X_2\) which are not images of any element of \(X_1\) under \(f_1\), while all elements of \(X_3\) are images of some elements of \(X_1\) under \(f_3\). The above observations lead to the following definitions:

Definition: Injdective Functions or One-One Functions
A function \(f : X \to Y\) is defined to be one-one (or injective), if the images of distinct elements of \(X\) under \(f\) are distinct, i.e. for every \(x_1, x_2 \in X\), \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\). Otherwise, \(f\) is called many-one.

The functions \(f_1\) and \(f_4\) in Fig 1.2 (i) and (iv) are one-one and the function \(f_2\) and \(f_3\) in Fig 1.2 (ii) and (iii) are many-one.

Definition: Onto Functions or Surjective Functions
A function \(f : X \to Y\) is said to be onto (or surjective), if every element of \(Y\) is the image of some element of \(X\) under \(f\), i.e., for every \(y \in Y\), there exists an element \(x\) in \(X\) such that \(f(x) = y\).

The function \(f_3\) and \(f_4\) in Fig 1.2 (iii), (iv) are onto and the function \(f_1\) in Fig 1.2 (i) is not onto as elements \(e, f\) in \(X_2\) are not the image of any element in \(X_1\) under \(f_1\).

Remark
\(f : X \to Y\) is onto if and only if \(\text{Range of } f = Y\).

Definition: Bijective functions
A function \(f : X \to Y\) is said to be one-one and onto (or bijective), if \(f\) is both one-one and onto.

The function \(f_4\) in Fig 1.2 (iv) is one-one and onto.

Example 7
Let \(A\) be the set of all 50 students of Class X in a school. Let \(f : A \to \mathbb{N}\) be function defined by \(f(x) = \text{roll number of the student } x\). Show that \(f\) is one-one but not onto.

Example 8
Show that the function \(f : \mathbb{N} \to \mathbb{N}\), given by \(f(x) = 2x\), is one-one but not onto.

Example 9
Prove that the function \(f : \mathbb{R} \to \mathbb{R}\), given by \(f(x) = 2x\), is one-one and onto.

Example 10
Show that the function \(f : \mathbb{N} \to \mathbb{N}\) given by \(f(1) = f(2) = 1\) and \(f(x) = x - 1\), for every \(x > 2\), is onto but not one-one.

Example 11
Show that the function \(f : \mathbb{R} \to \mathbb{R}\), defined as \(f(x) = x^2\), is neither one-one nor onto.

Solution

Since \(f(-1) = 1 = f(1)\), \(f\) is not one-one.
Also, the element \(-2\) in the co-domain \(\mathbb{R}\) is not image of any element \(x\) in the domain \(\mathbb{R}\) (Why?).
Therefore, \(f\) is not onto.

Example 12
Show that \(f : \mathbb{N} \to \mathbb{N}\), given by
\[ f(x) = \begin{cases} x + 1, & \text{if } x \text{ is odd} \\ x - 1, & \text{if } x \text{ is even} \end{cases} \]
is both one-one and onto.

Example 13
Show that an onto function \(f : \{1, 2, 3\} \to \{1, 2, 3\}\) is always one-one.

Example 14
Show that a one-one function \(f : \{1, 2, 3\} \to \{1, 2, 3\}\) must be onto.

Show that the function \(f : \mathbb{R} \to \mathbb{R}\), defined by \(f(x) = \frac{1}{x}\) 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 \(|x| = -x\), 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.
(i) \(f : \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = 3 - 4x\)
(ii) \(f : \mathbb{R} \to \mathbb{R}\) 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
\[ f(n) = \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] 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
\[ f(x) = \frac{x - 2}{x - 3} \] 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