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Differences Between Codomain and Range

Differences Between Codomain and Range

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Rama Sharma

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Codomain and Range in Mathematics

There are many concepts in Math that can surprise one and also come as a challenge for the students. Like numbers, there is no end to the mysteries of mathematics. The deeper you go in the world of mathematics, the more magnificent it gets. Codomain and Range is one such concept. So today we are here to learn about the differences between Codomain and Range.

But to understand all this we first need to know what a function is. Do you know what a function is? Any relation defined over two different sets is a function, provided that every element that is a part of the first set has a corresponding element in the second set of the relation. However, the relation or the correspondence of the first set's elements must be with exactly one element of the second set.

Function Explained with Example

To understand the definition clearly let’s take an example. Say there are two sets namely, set A and set B. The relation is from set A to set B. So, for this relationship to be a function, all the elements of set A should have a corresponding and unique element in set B. A function is represented in the following manner:

F(x) = Y

Example:-

y=x² is a function where x and y belong to real numbers. Here, set A and set B have all the numbers that are present in real numbers.

Domain

A domain is a group of possible values that the independent variable can take. This means the set of all the possible values that ‘x’ can take in the function f is the domain of the given function. A domain is a set of pre-images. According to the example taken above, set A is the domain of the function.

Example:-

y=x+4

The domain of the above function: (-∞,∞)

Codomain

A codomain is the group of possible values that the dependent variable can take. This means that the set of all the possible values that ‘y’ can take in the function f is the codomain of the given function. A codomain is a set of images. According to the example taken above, set B is the codomain of the function

Range

The range is all the elements from set B that have the corresponding pre-image in set A. Hence, a range can also be defined as the set of all the possible values of the function that we receive upon taking the different values of x in the function f.

Example:-

Taking the example taken above, y = x + 4

Here y0 is the range of the given function. So we can write this as the range of the function is (-∞,∞).

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Domain and Range

Here are the domain and range of some common functions that we see very frequently while solving mathematics-

S.NO.	Types of function	Domain	Codomain
1.	f(x) = x	(-∞ , + ∞)	(-∞ , + ∞)
2.	f(x) = x²	(-∞ , + ∞)	[0 , + ∞)
3.	f(x) = sin ( x )	(-∞ , + ∞)	[−1,1]
4.	f(x) = cos ( x )	(-∞ , + ∞)	[−1,1]
5.	f(x) = sin⁻¹( x )	[−1,1]	[−π/2,π/2]
6.	f(x) = cos⁻¹( x )	[−1,1]	[0,π]
7.	f(x) = a x	(-∞ , + ∞)	(0 , +∞)
8.	f(x) = Logₐ ( x )	(0 , + ∞)	(-∞ , + ∞)

Types of Functions

There are five types of functions on the basis of how the domain and codomain is related.

One-One functions
Many-one functions
Onto functions
Into functions
One-One and Onto functions

They are discussed in detail below.

One-One Functions

When each element of the domain has a distinct image in the codomain then the function is the One-One function. It is also called the injective function.

Many-One Functions

When two or more elements of the domain do not have a distinct image in the codomain then the function is Many -One function.

Onto Functions

When each element of the codomain has a distinct image in the domain then the function is Onto function. It is also called the surjective function.

Into Functions

When two or more elements of the codomain do not have a distinct image in the domain then the function is Into function.

One-One and Onto Functions

When each element of the domain has a distinct image in the codomain and when each element of the codomain has a distinct image in the domain then the function is One-One and Onto function. It is also called bijective function.

Now let’s summarize the difference between Codomain vs Range

Although it might be easy to understand the concepts pertaining to range and domain, it is not really the case with their distinctions. To summarize, we have the following bases that can clear all your doubts between the two:

Difference between Codomain Vs Range

Basics of distinction	Codomain	Range
MEANING	When it comes to a codomain, it is basically the group of values that are possible in the case of what can be taken by the dependent variable. The meaning of this comes as a simple one: The set of possible values which are all something that a ‘y’ variable can take in the f function is what a codomain is. This pertains only to a given function.	A range is basically what all the elements are from a second set, let’s say B, which has the pre-image that is present in the first set, A. In other words, a range is a subset of the codomain, which pertains to a set of possible values of f as a function. It is one that is received when we take into account the different values of a variable x.
PURPOSE	A codomain has the nature of restricting the output of a particular function.	A range is not something that can restrict the output of a specific function.
RELATION	A codomain is in relation to the meaning of a function.	A range is related to a function’s image.
EXAMPLE	If L = {1, 2, 3, 4} and M = {1, 2, 3, 4, 5, 6, 7, 8, 9} and the relation f: A B is defined by f (x) = x2 Codomain = Set M = {1, 2, 3, 4, 5, 6, 7, 8, 9}	If L = {1, 2, 3, 4} and M = {1, 2, 3, 4, 5, 6, 7, 8, 9} and the relation f: A B is defined by f (x) = x2. Range = {1, 4, 9}

Fun Facts

The domain, codomain, and range are not always equal. In some cases, it can be equal.
The range is a subset of the codomain.
The denominator of the given function can never be zero.

This is all about the differences between codomain and range with proper examples. Focus on the core concepts so that you can easily understand the differences easily.

FAQs on Differences Between Codomain and Range

1. Will Codomain and Range ever be equal?

Yes, the codomain and range can be equal when the number of elements in the range is equal to the number of elements in the codomain. This means that every element of set A has an image in set B for a function.

f: A B.

As we already know, that range is the subset of the codomain. Therefore, a set is a subset of itself indicating that for the range to be equal to codomain it must have the same number of elements. For codomain and range to be equal every element of the domain should have an image in the codomain. This will automatically make the codomain and range equal.

2. Can the range be greater than the codomain? Can domain, range, and codomain ever be equal?

No, it is not possible as the range is a subset of the codomain. A subset can never be greater than the universal set. Thus, implying that the number of elements present in a range can never be more than the number of elements present in the codomain. Codomain will always be greater than or equal to the range.

Yes, it is possible for domain, codomain, and range to be equal. The domain and codomain to be equal to each element of the domain should have an image in the codomain. This will lead to an equal number of the range. Thus, making—domain, codomain, and range to be equal. Things that appear the same may not always be. Similarly, range and codomain are also very different.

3. Is it true that a codomain is bigger than the range?

Now, when it comes to which one is bigger, we can say that it is the codomain that is greater than the range. For this, the reason is that the range is known to be a part or a subset of the codomain. This naturally makes up for the fact that a range is either equal to or less than a codomain, but could never be greater than that.