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-
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
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. What is the core difference between the codomain and range of a function?
The core difference is that the codomain is the set of all possible output values a function is defined to have, while the range is the set of all actual output values the function produces. The range is always a subset of the codomain. For a function f: A → B, set B is the codomain, and the set of all f(x) values for every x in A is the range.
2. How is the domain of a function different from its codomain?
The domain and codomain refer to two different sets in a function's definition.
- The domain is the set of all possible input values (the 'x' values) for the function.
- The codomain is the set of all possible output values (the 'y' values) that the function is allowed to map to.
3. Can the range of a function be equal to its codomain? Provide an example.
Yes, the range can be equal to the codomain. This occurs when every element in the codomain is the image of at least one element in the domain. Such a function is called an onto or surjective function. For example, consider the function f(x) = x + 1, where both the domain and codomain are the set of all real numbers (ℝ). In this case, for any real number 'y' in the codomain, we can find a real number x = y - 1 in the domain. Thus, the range is also ℝ, making it equal to the codomain.
4. Why can the range of a function never be larger than its codomain?
The range can never be larger than the codomain because the range is, by definition, a subset of the codomain. The codomain is the pre-defined 'universe' of all possible outputs. The range consists only of the actual outputs generated by the function from its domain. Since every actual output must belong to the set of possible outputs, the range is either equal to or contained within the codomain, and can therefore never have more elements.
5. How do the concepts of codomain and range relate to onto (surjective) functions?
The relationship between codomain and range is the defining characteristic of an onto (surjective) function. A function is classified as 'onto' if and only if its range is equal to its codomain. This means that every single element in the codomain set is an actual output (image) for at least one input element from the domain. If even one element in the codomain is not produced as an output, the function is considered an 'into' function, not an 'onto' one.
6. Using an example, explain how to determine the range from a given function and its codomain.
To determine the range, you must find all the actual output values the function produces. Consider the function f(x) = x² with its domain as the set of all integers {..., -2, -1, 0, 1, 2, ...} and its codomain as the set of all integers.
- First, apply the function to the domain values: f(-2)=4, f(-1)=1, f(0)=0, f(1)=1, f(2)=4, etc.
- Next, collect all the unique output values. The outputs are {0, 1, 4, 9, 16, ...}.
- This set of actual outputs, {0, 1, 4, 9, ...}, is the range. Notice that the range is a subset of the codomain (all integers), as it does not include negative integers or non-perfect squares.
7. What is the significance of the codomain in defining a function? Why not just use the range?
The codomain is significant because it establishes the type of function being discussed and the mathematical space in which it operates. It is part of the function's formal definition (f: Domain → Codomain). Specifying the codomain beforehand allows us to ask critical questions, such as whether the function is surjective (onto) or not. If we only defined the range, every function would be surjective by default, making the concept meaningless. The codomain provides the full picture of the target set, while the range shows what part of that target is actually reached.
8. In the context of a function, what is the difference between an 'image' and the 'range'?
The terms 'image' and 'range' are closely related but distinct.
- An image refers to the single output value produced by a specific input. For a function f(x) = y, 'y' is the image of the element 'x'.
- The range is the complete set of all images. It is the collection of every single output value that the function generates when applied to every element in its entire domain.
