Domain, Codomain, Range Functions

Dhristi JEE 2022-24

Domain, Codomain, Range

A function in mathematics is defined within a specified range, and we define domain terms for that. However, this topic is not only limited to this aspect. It would help if you dive in to understand it in a better way. First, you need to understand the proper definition for a function, its domain, range and codomain. Considering the simplest form of a function, it is defined as the values that can satisfy a function's conditions. The range is defined as the output that we get after solving a function.

 

The domain can be defined as the set which is the input of the function. Or in simple terms, it is the input values that are used for a function. A function’s values can be defined as the values that are defined on a set.

  

The range can be defined as the actual output which we are supposed to get after we enter the function’s domain. The range is dependent on the variables of the functions. The codomain can be defined as the total number of values present in a set. They are thus the values which are expected to come out when the domain values are entered. The set of all the possible values which qualifies the inputs of a function is called the domain or it can be defined as the entire set of values which is possible for variables that are independent. The domain can be found in the fraction’s denominator which is not equal to zero and the digit present under the square root bracket. 


How will you Define a Function for a Domain Range?

A function is a way to relate input to get its output. In real-time, functions are the necessary part of understanding and implementing. Also, functions are required for methodical applications. Thus, you can solve different real-world problems with it.

 

If you want to understand a function and relation between two functions, this is possible with a cartesian product. The basic points to define a function includes:

  • A function may not satisfy all mathematical values. 

  • You can define a function with the help of sets.

A function will relate each value of one set to the values of another set. It can be the same set or a different one. A set is the collection of values, numbers or things. 

Consider the below diagram:

 

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In the above diagram, X and Y are two sets and function is defined from values of X to that of Y.

 

Domain and Range of a Function

Not all the values are specified for a function. Some specifications define it as what can be put into a function to get the desired results. There are three terms that are to be defined for a function:

 

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According to the diagram, Domain is the entire set A and codomain is set as the whole B, and Range is the outcome after entering domain values. Or simply saying, the range is the pointed values of set B. 

 

How to Relate Codomain and Range?

In short terms, we can say that range is the subset of the codomain. It is not important that a function might satisfy all the values of the codomain. However, the values that we get after entering domain values in a function are the range. Thus, it is part of the codomain set. 

 

Difference Between Codomain and Range

Without a doubt, both codomain and range are present on the output side. However, there is a difference between the two of them. Codomain is defined as the possibility of the values as an outcome. Thus we can say that codomain is one part while defining a function. However, on the other hand, the range is the actual output that we are supposed to get.

FAQs on Domain, Codomain, Range Functions

1.What is a range?

Range of a function can be defined as the set of all the outputs of a function or the set of all outputs that is found after substituting the domain. The entire set of all the values that are possible as the dependent variables’ outcomes is considered in the domain. Let us understand this with an example, the range of the function F is {1,3,6,8}. The whole set B is known as the function’s codomain. It is this set which has all the outputs of the function. The set of real numbers can be defined as the codomain for the real-valued fraction that exists. 

2. What is a natural domain?

When a real function f is given by a formula then it may or may not be defined for some values of the variables. In such situations, they are called the partial function and the set of real numbers on which the formula can be considered for evaluation with a real number is called the domain of f or natural domain. In other terms, the partial function is simply defined as the function and the natural domain is expressed as the domain.  

3. What is injective and surjective?

Injective is a function which is also called one-to-one when each element of the codomain is mapped by one element at the highest from the domain. Or in simple terms, if the distinct elements of the domain map to the distinct elements in the codomain then it is an injective function. It is also named as injection. On the other hand, surjective function or on-to function can be defined when each element of the codomain is mapped by at least one element of the domain. This means that the codomain and image of the function are the same. This function is also known as surjection. 

4. What is a codomain?

The codomain of a function can be defined as the set in which all the output of the function exists. If a set Y exist then it is expressed as X🡪Y. The term range is sometimes used interchangeably with it which refers to codomain or image of a function. A codomain is part of a function f if f is called a triple written as (X, Y, G).Here, X is the domain of the function f, G is its graph and Y is its codomain. The image of a function is defined as the subset of its codomain so it doesn’t coincide with it. A function is not surjective when its element y is in the codomain which can be written as the equation f(x) = y and this doesn’t have a solution. 

5. Can I get mathematics solutions on Vedantu?

Yes, you can get mathematics solutions for all the classes. These have been solved with an explanation which will help you understand the concepts better and help in solving questions in your examinations. All you have to do is log in or sign up to Vedantu platforms i.e. the app or website where you can access the solutions and they are present as a PDF file. This will help you for quick revisions before exams as well and help you get good scores.  

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