FUNCTION

 When we talk about function in mathematics, the first thing that comes to mind is that it is like a machine that gives an output for an input. Initially, the functions were the idealization of how a variable term relates to another quantity. The function is a relationship between a set of values(inputs) and a set of possible outputs in mathematics. The characteristics of functions is that every input corresponds to exactly one output. Functions are the foundation of calculus in mathematics. The functions are the various forms of relationships. In mathematics, a function is represented as a rule that produces a unique result for each input x, also the mapping or transformation is used to denote a function. These functions are commonly represented by letters like f, g, and h. The domain is defined as the collection of all the values that the function can accept when it is defined. The range includes all the values returned by the function in question. A co-domain is a set of values that have the potential to be outputs of a function. 

A function in mathematics is a relation from a set X to a set Y that assigns exactly one element of Y to each element of X. The set X is domain and set Y is the codomain of the given function.

The domain can be defined as the collection of all possible input values for the given function when it is defined earlier. The range is the collection of all the values that the function’s output produces. The collection of all the possible values that could be outputs of the given function is known as the co-domain of that function.

A mathematical function is a rule that determines the value of a dependent variable based on the values of one or more independent variables. A function can be represented in a variety of ways, including a table, a formula, or a graph. Apart from isolated points, the mathematical functions encountered in physical chemistry are single-valued. The mathematical functions that arise in physical chemistry, apart from isolated points, are continuous. The number of independent variables for the functions that arise in these areas is specified by thermodynamic theory and quantum mechanical theory.

TYPES OF FUNCTION: –

There are various types of functions based on the element of the set are: –

1. One One function,
2. Many to One Function,
3. Onto Function,
4. One One and Onto Function (Bijection),
5. Into Function,
6. Constant Function.

• One One function

      A one-to-one function is defined as f: A to B, which connects each element in set A to a unique element in set B. An injective function is sometimes known as a one-to-one function. For the provided function, each domain element has a unique picture or co-domain element.

• Many to One Function

The function f: A to B defines a many to one function, in which more than one element of the set A is associated to the same element in the set B. More than one element has the same co-domain or image in a many to one function. A constant function is one in which a many to one function in the codomain has a single value or the domain elements are all related to a single element.

• Onto Function

In an onto function, every codomain element is associated to the domain element in an onto function. Every element in set B has a pre-image in set A for a function defined by f: A B. The subjective function is also known as the onto function.

• One one and Onto Function (Bijection)

A bijective function is a function that is both a one and an onto function. Every element in the domain is linked to a different element in the codomain, and each codomain element has a pre-image. In other words, every element in set A is linked to a distinct element in set B, and there isn’t a single element in set B that is missing.

• Into Function

 In terms of properties, an into function is the opposite of an onto function. In the co-domain, there are some elements that do not have a pre-image. Set B contains extra items that are not connected to any of the elements in set A.

• Constant Function

A constant function is one of the most common types of many to one function. A constant function has a single picture for all domain items. The constant function has the form f(x) = K, where K is a positive integer. For a constant function, the same range value of K is produced for different values of the domain (x value).

CONCLUSION

A function is a sort of relation in which every element in the domain is included and every input only creates one output.

A function relates elements in a set (the domain) to elements in another set (the codomain). The range refers to all the outputs (the actual values associated with them).