
Define injective function. Give an example of injective function.
Answer
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Hint: We solve this problem by defining the injective function and then we take some examples.
The injective function is also called as one – one function which is defined as “for every element in the codomain there is the image of exactly one element in the domain”. The graphical representation of a function is as shown below.
Here, X is the domain and the set Y is called the codomain.
By using this definition we take some examples of injective functions.
Complete step by step answer:
We are asked to define the injective function.
The injective function is defined as a function in which for every element in the codomain there is an image of exactly one in the domain.
Let us assume that a function mapping as then the graphical representation of this function if it is injective is given as
Here, we can see that for every element in the domain X there is exactly one image in the codomain Y.
So, we can say that this function is an injective function.
There is other definition of injective in mathematical form as follows:
“A function of domain X is said to an injective function if for any two elements in the domain X, whenever then ”
Symbolically we can say that
Now, let us take one example as follows
(1) The function is defined as where is constant.
Let us check whether this function is injective or not.
Let us assume two values such that
Now by using the function in above equation we get
We know that if a function is defined as then we can say that the function is injective.
By using this definition we can conclude that the function is defined as where is constant is a injective function.
Let us take some more examples of injective functions as follows:
(2) The function is defined as where is constant.
(3) The function is defined as where is constant.
Note: We come across some situations where we are asked to check whether a given function is injective or not. Then we use the mathematical definition in the case that is
But students may do from the direct definition that is “for every element in the codomain there is the image of exactly one element in the domain”
Here we need to check for each and every element in the domain.
If the function is mapped as then we need to check each and every element in the domain that is real numbers. But we have infinite real numbers then it will be impossible to check the injection from the general definition.
So, we need to use the mathematical definition for solving the problem easily.
The injective function is also called as one – one function which is defined as “for every element in the codomain there is the image of exactly one element in the domain”. The graphical representation of a function

Here, X is the domain and the set Y is called the codomain.
By using this definition we take some examples of injective functions.
Complete step by step answer:
We are asked to define the injective function.
The injective function is defined as a function in which for every element in the codomain there is an image of exactly one in the domain.
Let us assume that a function mapping as

Here, we can see that for every element in the domain X there is exactly one image in the codomain Y.
So, we can say that this function is an injective function.
There is other definition of injective in mathematical form as follows:
“A function of domain X is said to an injective function if for any two elements
Symbolically we can say that
Now, let us take one example as follows
(1) The function
Let us check whether this function is injective or not.
Let us assume two values
Now by using the function
We know that if a function is defined as
By using this definition we can conclude that the function
Let us take some more examples of injective functions as follows:
(2) The function
(3) The function
Note: We come across some situations where we are asked to check whether a given function is injective or not. Then we use the mathematical definition in the case that is
But students may do from the direct definition that is “for every element in the codomain there is the image of exactly one element in the domain”
Here we need to check for each and every element in the domain.
If the function is mapped as
So, we need to use the mathematical definition for solving the problem easily.
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