The image set (yellow) is a subset of the codomain (violet).
The idea of the range of a function is used in the field of mathematics . The expression usually refers to the image of the function , although it can also refer to the codomain .
The variable values
Typically, older academic papers and books appeal to the term rank to focus on the co-domain of function . On the other hand, the most recent publications resort to the rank of a function with the intention of mentioning the image . That is why it is essential to pay attention to the context to avoid making an error due to a misunderstanding.
Currently, in short, the range of a function is related to the set of real values that the variable f(x) acquires. Calculating the range of a function requires determining what the domain of its inverse function is.
While the domain of the function f(x) is equivalent to the set of values for which said function is defined, the range of the function f(x) is the set of values that f takes.
The image
Thus, when the range of a function is linked to the image , it is the set that includes all the results of said function . In this sense, it is important to keep in mind that the image constitutes a subset of the codomain .
It is important to note that the set we refer to in this way does not always have the same number of elements as the domain, since in some cases more than one element of the first has the same image. Suppose we have a set with numbers from 1 to 10 and we want to pass them through a function that returns whether they are even or odd ; In this case, the image set will be very small, since it will simply have these two values, for each of which there will be several relationships with those of the domain.
There may also be the case in which the second set is not directly the image of the first, but rather we find it as a subset of it. This occurs if one or more of its elements are not an image of any of the values of the first. Such would be the case of a set with the names of certain people and another with a series of nationalities, if the latter had some that did not belong to any of the people.
And this example helps us to establish a main difference between image and co-domain, beyond the fact that both are related to the concept of range of a function : the first is always a subset of the second, even if it is one that includes all its elements. Speaking in purely mathematical terms, if a given function has the real numbers as its codomain but its equation simply consists of squaring a variable, the image will not include the negative numbers, so it will be only part of the codomain.
In this example, the negative numbers in the codomain are not part of the picture
The Codomain
The range of a function applied to the codomain, meanwhile, is associated with the set of complex numbers (in the case of the branch known as complex analysis ) or the set of real numbers (in real analysis ).
Other names by which the codomain is known are tour, arrival set , and counterdomain . As we mentioned above, this is the set in which we can find the output values of the function. It is as if the definition of the function itself tells us the area in which we can find the results to prevent us from wasting time looking in others, but it does not tell us exactly which of them will be useful to us. The image represents the end of this search.