Functions or Mapping



Now, in functions or mapping we will study about special type of relations called functions or mapping. To understand them, let us take few real life examples.

  All these questions have 
  unique answers. Let us
  see how we can relate 
  this in learning mapping.  

● From where does the sun rise?
    East

● Which is the capital of India?
    Delhi

● What is the successor of 4?
    5

● What is the sum o f 5 and 3?
    8

Mapping or Functions:



If A and B are two non-empty sets, then a relation ‘f‘ from set A to set B is said to be a function or mapping,

● If every element of set A is associated with unique element of set B.
● The function ‘f’ from A to B is denoted by f : A → B.
● If f is a function from A to B and x ∈ A, then f(x) ∈ B where f(x) is called the image of x under f and x is called the pre image of f(x) under ‘f’.

Note:
For f to be a mapping from A to B:

● Every element of A must have image in B. Adjoining figure does not represent a mapping since the element d in set A is not associated with any element of set B.

Mapping or Functions




● No element of A must have more than one image. Adjoining figure does not represent a mapping since element b in set A is associated with two elements d, f of set B.

Represent a Mapping




● Different elements of A can have the same image in B. Adjoining figure represents a mapping.

Figure Represents a Mapping




Note:
Every mapping is a relation but every relation may not be a mapping.

Function as a special kind of relation:



Let us recall and review the function as a special kind of relation suppose, A and B are two non-empty sets, then a rule 'f' that associates each element of A with a unique element of B is called a function or a mapping from A to B.

If 'f' is a mapping from A to B,
we express it as f: A → B
we read it as 'f' is a function from A to B.

If ‘f ' is a function from A to B and x∈A and y∈B, then we say that y is the image of element x under the function ' f ' and denoted it by f(x).

Therefore, we write it as y = f(x)
Here, element x is called the pre-image of y.
Thus, for a function from A to B.

● A and B should be non-empty.
● Each element of A should have image in B.
● No element of 'A' should have more than one image in 'B’.

Note:
● Two or more elements of A may have the same image in B.
● f : x → y means that under the function of 'f' from A to B, an element x of A has image y in B.
● It is necessary that every f image is in B but there may be some elements in B which are not f images of any element of A.



Related Concepts

Ordered Pair
Cartesian Product of Two Sets
Relation
Domain and Range of a Relation
Domain Co-domain and Range of Function

7th Grade Math Problems

8th Grade Math Practice

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