Onto Function: Definition, Properties, Examples

An onto function, also known as a surjective function, is a type of function where every element in the co-domain is mapped to at least one element in the domain. In other words, an onto function covers the entire codomain, ensuring that every possible output value is achieved by some input value.

In this article, we will explore the concept of onto functions, an important topic within the broader category of relations and functions, which is a crucial chapter in class 12 Mathematics. Understanding onto functions is essential not only for board exams but also for competitive exams such as the Joint Entrance Examination (JEE Main), SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the past ten years (2013 to 2023) in the JEE Main exam, a total of five questions have been asked on this concept: one in 2013, one in 2015, one in 2019, one in 2022, and one in 2023.

Onto function

A function $f: X \rightarrow Y$ 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 $f(x)=y$

Hence, Range = co-domain for an onto function

Example: Consider, $\mathrm{X}=\left\{\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4\right\}$ and $\mathrm{Y}=\left\{\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3\right\}$

$
f: X \rightarrow Y
$

As every element in $Y$ has a pre-image in $X$, so it is an onto function

Method to show onto or surjective

Find the range of $y=f(x)$ and show that range of $f(x)=\operatorname{co}$-domain of $f(x)$

Properties of Onto Functions

1. Completeness: Every element of the codomain Y is the image of at least one element from the domain A.
2. Range and Codomain: For a function to be onto, its range (the set of all outputs) must be equal to its codomain.
3. Inverse Function: An onto function has a right inverse, meaning that there exists a function $g: Y \rightarrow X$ such that $f(g(y))=y$ for all $b \in Y$.

Number of Onto functions-

If there is a function $f: A \rightarrow B_{\text {such that }} n(A)=m$ and $n(B)=n$, where $m \geq n$.
Then, the number of onto functions $=\displaystyle\sum_{r=1}^n(-1)^{n-r} n_{C_r} r^m$

Solved Examples Based On the Onto Functions:

Example 1: In the case of an onto function, which of the following is true?

1) Codomain = Range
2) Codomain $\subseteq$ Range
3) Range $\subseteq$ Codomain
4) None of these

Solution:

By the definition of onto function, we know that:

Co-domain = Range

Hence, the answer is the option (1).

Example 2: The number of functions f from {1,2,3,........,20} onto {1,2,3,........,20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is:

Solution:

Onto function -

If $f: A \rightarrow B$ is such that each \& every element in $B$ is the $f$ image of at least one element in $A$. Then it is the Onto function.
wherein
The range of $f$ is equal to Co - domain of $f$.

$
f(k)=3,6,9,12,15,18
$

for $\mathrm{k}=4,8,12,16,20$

$
\text { ways }=6 \times 5 \times 4 \times 3 \times 2 \times 1=6!
$
For remaining numbers $=(20-5)!=15$ !
Total ways $=15!\times 6$ !

Example 3: Let $A=\left\{x_1, x_2, x_3 \ldots . ., x_7\right\}$ and $B=\left\{y_1, y_2, y_3\right\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f: \mathrm{A} \rightarrow \mathrm{B}$ that are onto, if there exist exactly three elements x in A such that $f(\mathrm{x})=\mathrm{y}_2$, is equal to :

Solution:

As we learned in

Number of Onto functions -

$f: A \rightarrow B$

Such that $n(A)=m$
and $n(B)=n$

$m \geqslant n$

Number of onto functions

$=\displaystyle\sum_{r=1}^n(-1)^{n-r} n_{C_r} r^m$

No. of onto functions.
Since if there exist exactly three elements $x$ in $A$ which give a single image $y_2$. If this means four elements of $A$ will give an image of $y_1$ and $y_3$. So that $\left(2^4-2\right)$ of ${ }^7 C_3$

$=(16-2)^7 C_3=14 .^7 C_3$

Hence, the answer is $14 \cdot{ }^7 C_3$.

Example 4: If $n(A)=5$ and $n(B)=3$. Find the number of onto functions $A$ to $B$.

Solution:

$\begin{aligned}
& \sum_{r=1}^3(-1)^{3-r} C_r^3 \cdot r^5 \\
& =C_1^3-C_2^3\left(2^5\right)+C_3^3\left(3^5\right) \\
& =3-96+243 \\
& =150
\end{aligned}$

Hence, the answer is 150.

Example 5: If $n(A)=3$ and $n(B)=5$. Find the number of onto functions from $A$ to $B$.
Solution:
As we have learned
Number of Onto functions-
$f: A \rightarrow B$ such that $\mathrm{n}(\mathrm{A})=\mathrm{m}$ and $\mathrm{n}(\mathrm{B})=\mathrm{n}$, where:

$m \geqslant n$
Number of onto functions

$=\sum_{r=1}^n(-1)^{n-r} n_{C_r} r^m$
For onto function n(A) n(B). Otherwise, it will always be a into function.

Hence, there are zero onto functions.

Hence, the answer is 0.

Summary

An onto function ensures that every element of the codomain is mapped to by at least one element of the domain, covering the entire codomain. These functions are crucial in mathematical theory, proofs, and real-world applications, ensuring that every possible outcome or output is achievable. Recognizing and proving that a function is onto is essential for deeper mathematical understanding and application.