Difference between Codomain and Range
Codomain is the set that contains all possible values that the function can output, Range of a function, on the other hand, is the set of all output values that are actually attained by the function. Although they might seem similar initially, but they have different meanings and uses.
This article will explain the meaning of co-domain and range of a function along with the difference between codomain and range.
What is Codomain?
Codomain of a function refers to the set of all possible values that the function can theoretically map to, irrespective of whether all these values are achieved by the function. In other words, it represents the set of all possible values that the function could map elements from the domain.
The codomain of a function f, denoted as codomain(f) or cod(f), is the set that contains all possible output values of the function.
For example, consider a function f: X→Y, where X is the set of inputs (also known as the domain) and Y is the set of outputs (the codomain). If f(x) = x2 and X is the set of real numbers (R), the codomain Y can be defined as the set of all real numbers (R).
What is Range?
The range of a function, also known as the image, is the actual set of values that the function maps to when applied to the elements of the domain. Essentially, the range consists of all the output values that the function produces.
We can also say, range of a function is a set of all the images of elements in the domain.
Example:
f(x) = x2 with the domain X being the set of real numbers (R), the range of this function would be the set of non-negative real numbers (R+) because squaring any real number results in a non-negative value.
Difference between Range and Codomain
The differences between range and codomain is described in a tabular form below:
|
Aspect |
Codomain |
Range |
|---|---|---|
|
Definition |
Codomain of a function is the set that contains all possible values that the function can output. |
Range of a function is the set of all actual outputs produced by the function. |
|
Relation to Function |
Defined when the function is specified. |
Derived from the function’s actual mapping. |
|
Inclusiveness |
Includes all potential outputs, whether or not they are produced by the function. |
Includes only those outputs that are actually produced. |
|
Example |
For f(x)=x2 with R as domain: Codomain is R. |
For f(x) = x2 with R as domain: Range is R+. |
|
Specification |
Determined when the function is defined, not dependent on the actual values taken by the function. |
Dependent on the actual values taken by the function when applied to its domain. |
Also Check,
Conclusion
In conclusion, we can say that codomain is the set of all possible output values that a function could produce, while the range is the subset of those values that the function actually does produce. The range is a subset of the codomain, and it represents the actual output values of the function.
FAQs on Difference between Codomain and Range
Can the range and the codomain be the same?
Yes, the range and the codomain can be the same if the function maps to every element of the codomain. This happens in cases where the function is surjective (onto).
Is it possible for a function to have an empty range?
No, a function must have at least one output for every input in its domain, as the range cannot be empty.
Can the codomain be smaller than the range?
No, the codomain cannot be smaller than the range. The codomain is always at least as large as the range since the range is a subset of the codomain.
How to differentiate between range and codomain?
Codomain defines all possible output values, whereas the range consists only of the actual output values attained by the function.
How do you determine the range of a function?
To determine the range, you need to apply the function to all possible inputs in the domain and collect all unique outputs.
Why is the distinction between codomain and range important?
The distinction is important for understanding the properties of functions, such as injectivity and surjectivity, and for correctly applying mathematical concepts in various fields.
Contact Us