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Function

F1

Function: The relation defined from set A to Set B such that every element of set A associates with unique element of set B is called a function from set A to set B that can be denoted as f :A→B.

Domain: The set of all values for which a function is defined is called domain. In simple words, if a function is defined as f :A→B, the elements of set A is called domain of the function.

Co-domain: Co-domain is the set of all possible outcomes of a function. In simple words, if a function is defined as f :A→B, the elements of set B is called Co-domain of the function.

Range: Range is the set of actual outcomes of a function. It is the subset of co-domain that consists of elements which associates with elements of domain.

**Co-domain includes the elements which may or may not associate with elements of domain but range ** consists of the elements that associate with the elements in domain.

Let's understand with the help of diagram,



 

The above diagram shows a function defined as f :A→B so the function is defined from set A to set B meaning set A is the domain whereas set B is the co-domain.
Therefore,
     
Domain={1,2,3}
    Co-domain={5,6,7,8}
We can see that every elements of co-domain doesn't associate with the element of domain. We know, the elements of co-domain that associate with the element in domain is range so,
    Range={5,6,8}



***There are few conditions for which a given relation cannot*** be 
considered as function.

i) If a domain remains unassociated i.e. if a relation has one or more domain with no range, then the given relation cannot be called has function. 
Lets understand through diagram,

The relation shown above cannot be called as function because we can see two elements of the domain i.e. {3,5} doesn't associate with any range.

NOTE: One or more range may not have domain but domain must have at least one range for the given relation to be a function.

ii) If domains have more than one range i.e. if in a relation one or more domain associates with two or more range, the given relation cannot be defined as function.
Diagrammatically

Here, we can see an element of domain i.e.{-35} have two range i.e. {1,22} so the above relation is not a function.


Types of Function

One to one: A function is said to one to one if every of the domain associates with unique/different co-domains i.e. distinct domain has distinct range.


Both the function shown above are one to one function as every domain associates with the elements of co-domain uniquely meaning no domain has same range.

Many to one: A function is said to many to one if two or more domain associates with same element in co-domain i.e. if a function has at least two domain with the same range then function is many to one.

          

Both the function shown above are many to one function as at least two domain associates with same element of co-domain meaning domains have same range.

iii) Onto function: A function is said to be onto if co-domain is equal to range. If a function has no co-domain left over after mapping i.e. each and every co-domain associates with at least one domain then the function is onto.

       

All the co-domain above is mapped with at least one domain so both the function above is onto.

iv) Into function: A function is said to be into if co-domain is not equal to range. If a function has at least one co-domain left over after mapping i.e., at least one or more co-domain doesn’t associate with domain then the function is into.


Both the function above are into as at least one co-domain is left over.


From the above terms defined we understand what onto, into, one to one and many to one function are but the types of function can be more specified than this.

i.e.,
i. If a function is one to one and onto at the same time then the function is one to one onto. Here, unique domain has unique range and no co-domain is left over.
Example,

ii. If a function is one to one and into at the same time then the function is one to one into. Here, unique domain has unique range and at least one co-domain is left over.
Example,

iii. If a function is many to one and onto at the same time then the function is many to one onto. Here, at least two domain has same range and no co-domain is left over.
Example,

iv. If a function is many to one and into at the same time then the function is many to one into. Here, at least two domain has same range and at least one co-domain is left over.
Example,




Inverse of a function:  The function obtained by interchanging the roles/position of domain and range in one to one onto function is called

inverse of a function. The inverse of function f(x) is denoted as f-1(x).
The inverse for function any other than one to one onto doesn't exist.


Composite of a function: Let f : A → B and g: B → C be the two functions, then function defined as A→ C is called composite function from A to C. It is denoted as gof in this case as the function which was operated secondly is written before the first one for the denotation.


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