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.
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A function
Hence, Range = co-domain for an onto function
Example: Consider,
As every element in
Method to show onto or surjective
Find the range of
Number of Onto functions-
If there is a function
Then, the number of onto functions
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.
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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
3) Range
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
wherein
The range of
for
For remaining numbers
Total ways
Example 3: Let
Solution:
As we learned in
Number of Onto functions -
Such that
and
Number of onto functions
No. of onto functions.
Since if there exist exactly three elements
Hence, the answer is
Example 4: If
Solution:
Hence, the answer is 150.
Example 5: If
Solution:
As we have learned
Number of Onto functions-
Number of onto functions
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.
Functions are one of the basic concepts in mathematics that have numerous applications in the real world.
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.
For onto function, range =codomain
All possible values of x for f(x) is defined (f(x) is a real number) is known as a domain.
If a function is defined from A to B i.e. f: A⇾B, then set B is called the Co-domain of the function.
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