A one-to-one function, also known as an injective function, is a function where each domain element is mapped to a unique element in the codomain. In other words, no two different elements in the domain map to the same element in the codomain. Understanding one-to-one functions is fundamental in various branches of mathematics, particularly in algebra and calculus.
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In this article, we will explore the concept of one-to-one functions, a key topic within the broader category of relations and functions. This concept is crucial for board exams and competitive exams like the Joint Entrance Examination (JEE Main), as well as other entrance tests such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the past decade (2013-2023) in the JEE Main exam, a total of six questions have addressed this concept, with one question in 2017, two in 2019, one in 2021, one in 2022, and one in 2023.
Function
A relation from a set
OR
Function Function Not a function
Not a function
Third one is not a function because d is not related(mapped) to any element in
Fourth is not a function as element a in
An injective function, sometimes referred to as a one-to-one function, is one in which distinct elements of A have distinct relationships with B or distinct images with B. If a function has distinct images, it can only be one-to-one if the pre-images are different. Similarly, if the elements in B set differ, it can only be one-to-one if the elements in A set had different pre-images.
A function
Consider,
Graphically it can be shown that for every x, there is a unique y (or no y has more than one x corresponding to it) as below and hence it is one-one.
Now, consider,
Method to check One-One Function
If
A function is one-one if no line parallel to the X-axis meets the graph of the function at more than one point.
Even degree polynomials are NOT one-one functions
Every distinct input x corresponds to a distinct output f(x), making it a one-to-one function.
This is a one-to-one function because no matter what value of
Number of One-One Function
If A and B are finite sets having elements m and n respectively, then the number of one-one functions from A to B is
1. Uniqueness: Each element of the domain maps to a unique element in the codomain.
2. Inverse Function: If f is a one-to-one function, then it has an inverse function
3. Horizontal Line Test: A function
A one-to-one function is a critical concept in mathematics, ensuring that each input maps to a unique output. Recognizing and proving that a function is one-to-one allows for deeper analysis and application, including the determination of inverse functions and solving unique solutions to equations. Understanding and identifying one-to-one functions is foundational for advanced mathematical studies and various practical applications.
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Example1: The function
1) injective but not surjective
2) surjective but not injective
3) neither injective nor surjective
4) invertible
Solution:
Solution:
So that it is not strictly increasing or decreasing function.
So that it is not one-one.
So, the given function is surjective but not injective.
Hence, the answer is the option 2.
Example 2: Which of the following functions are one - one functions?
1)
2)
3)
4)
Solution:
A line parallel to the
Clearly, this function is one - one function.
Hence, the answer is the option (4).
Example 3: Which of the following functions are one-one functions?
1)
2)
3)
4) None of these
Solution:
As we learned
In the case of composite functions,
If both
Hence, the answer is the option 4 .
Example 4: Let
1) injective but not surjective
2) neither injective nor surjective
3) not injective
4) surjective but not injective
Solution:
One - One or Injective function -
A function in which every element of the range of function corresponds to exactly one element.
- wherein
A line parallel to the
This can be written as
Hence, the answer is option 1 .
Example 5: Which of the functions
1)
2)
3)
4)
Solution:
When
Then
Hence, the answer is option 2.
Frequently Asked Questions(FAQ)-
1. What is a function?
Ans:
2. What is one-one function?
Ans: A function
3. What is the difference between one-one and many functions?
Ans: One-one function has a single value for the single domain but many-one function has multiple values for a single input.
4. Is degree polynomial a one-one function?
Ans: Degree polynomials may not be one-one function.
5. Give some examples of one-one function.
Ans: Logarithmic function, constant function, odd degree polynomial, etc are some one-one functions.
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