The concepts of domain, co-domain, and range are fundamental in the study of functions. The domain of a function refers to the set of all possible input values for which the function is defined. It represents the independent variable in the function. The co-domain, on the other hand, is the set of all possible output values that

The set A is referred to as the domain of the function f, while the set B is referred to as the co-domain. The range of the function f, denoted by f(A), refers to the set of all images of elements in A under the mapping f.

The range of function f, denoted as R(f), is defined as the set of all possible output values that the function can produce.

It is evident that 

Consequently,

It is imperative that every image of f is contained within B. However, there may exist elements in B that are not the image of any element in A, meaning their pre-image under f is not found in A.

It is possible for two or more elements within set A to possess identical images within set B.

The notation “f: A → B” indicates that for a given function f, an element x in set A is mapped to an element y in set B.

In instances where it is unnecessary to explicitly state the domain and range of a function, it is common practice to represent the function f as y = f(x) and interpret this notation as indicating that y is a function dependent on x.

A function that has the set of real numbers, R, or one of its subsets as its range is referred to as a “real-valued function.” Moreover, if the domain of the function is R or a subset of R, it is referred to as a real function, where R represents the set of real numbers. 

The term “domain” refers to a specific area or field of study.

The domain of a function refers to the collection of all possible input values for which the function is defined. An illustration of this can be seen in the domain of the function f(x) = x, which consists of all non-negative real numbers due to the undefined nature of taking the square root of a negative number.

The codomain refers to the set of all possible values that a function can output.

The codomain of a function refers to the collection of all potential output values that the function can produce. As an illustration, the codomain of the function f(x)=x^2 encompasses the entire set of real numbers, as any real number can be squared to yield another real number.

The concept of “range” refers to the extent or scope of something. 

The range of a function refers to the collection of all possible output values that the function can produce. The range of the function f(x)=x^2 can be defined as the collection of non-negative real numbers, as the square of any real number is inherently non-negative.

The interconnection among the domain, codomain, and range of a function can be succinctly described in the following manner:

The domain of a function is a subset of its codomain.

The range is a subset that is contained within the codomain.

The range of a function is always a subset of its domain.

This limitation arises from the fact that the function is constrained to generating outputs exclusively within the codomain, and it is further restricted to producing outputs solely for input values that fall within the domain.