Functions are a fundamental concept in mathematics and computer science. They are mathematical entities that take inputs

The significance of functions lies in their inherent connection to a multitude of phenomena within the realm of reality. Therefore, when we calculate the square of a given real number, we are essentially performing an operation on the number x to obtain the number x^2. Therefore, a function can be conceptualized as a set of instructions that generates novel elements based on a given set of elements. The term “function” is also commonly referred to as “mapping” or “map” in academic discourse.

The independent variable is a factor or condition that is manipulated or controlled by the researcher in an experiment. It is the variable that is hypothesized to have an effect on the outcome. An independent variable refers to a symbol that has the ability to assume any value from a specified set.

The dependent variable is a variable that is being measured or observed in an experiment or study, and is expected to be influenced by the independent variable(s).The variable that is contingent upon independent variables is referred to as the dependent variable.

The term “function” can be defined as a mathematical concept that establishes a relationship between two sets, known as the domain and the codomain, in which each element in the domain is associated with exactly one element

Definition 1: A function f can be defined as a relation between a non-empty set A and a non-empty set B, where the domain of f is A and there are no two distinct ordered pairs in f that share the same first element.

Definition 2 refers to the second specified meaning or interpretation of a concept or term. Let A and B denote two non-empty sets. A rule that establishes a correspondence between each element of A and a distinct element of B is referred to as a mapping or a function from A to B, denoted as f: A → B. If a function f maps element x to element y, it is said that y is the image of x under f. This can also be expressed as the f image of x, denoted as f(x), where y = f(x). The term “pre-image” or “inverse-image” is used to refer to the element x. 

Therefore, considering a function mapping elements from set A to set B:

(i) Both A and B must be non-empty.

(ii) Every element of set A must have a corresponding image in set B.

(iii) Each element in set A should have at most one corresponding image in set B.

Question: Analyze the given relations and determine, with justification, whether each one is a function or not. The set R can be represented as {(1, 2), (2, 2), (3, 1), (4, 2)}. The relation R can be represented as a set of ordered pairs: R = {(2, 2), (1, 2), (1, 4), (4, 4)}. The relation R can be represented as a set of ordered pairs: R = {(1, 2), (2, 3), (4, 5), (5, 6), (6, 7)}.

The proposed resolution:

(i) As the first element of each ordered pair is distinct, it can be concluded that this relation satisfies the criteria of a function.

(ii) The relation is not a function because the same first element (1) is associated with two different images (2 and 4).

(iii) Given that the first element of each ordered pair is distinct, it can be concluded that this relation satisfies the criteria of a function.