In general, a Function is a kind of relation in which each element in the domain is paired with only one element in the range. To help you understand the concept of Functions or Mapping we have provided a detailed explanation along with solved examples. Mapping Diagram consists of two columns in which the first column denotes the domain and the second column denotes the range.

A mapping diagram usually represents the relationship between input and output values. A mapping diagram is called a function if each input value is paired with only one output value.

- Relations and Mapping
- Ordered Pair
- Cartesian Product of Two Sets
- Relation in Math
- Domain and Range of a Relation
- Practice Test on Math Relation
- Domain Co-domain and Range of Function
- Math Practice Test on Functions

## Introduction to Mapping or Function

Let us assume there are two sets A and B and the relation between Set A to Set B is called the function or Mapping.

Every element of Set A is associated with a unique element of Set B. Function f from A to B is represented by f : A → B. Relation will have a set of ordered pairs. In the Ordered Pairs, the second element is called the image of the first element and the first element is the preimage of the second element.

If f: A → B and x ∈ A, then f(x) ∈ B where f(x) is called the image of x under f & x is called the pre-image of f(x) under ‘f’.

**For f to be Mapping from A to B**

Different Elements of A can have the Same Image in B. Thus, the following figure represents Mapping.

No element of A should have more than one image. The below figure doesn’t represent mapping since the element in Set A is having two images i.e. I, III.

Every element of A must have an Image in Set B. The adjacent figure doesn’t represent mapping since 1, 2 are not associated with the elements in Set B.

Thus, we can infer that every mapping is a relation but not every relation is mapping.

### Function as a Special Kind of Relation

Suppose A and B are two non-empty sets then rule f associates each element of A with a unique element in B is known as function or mapping from A to B.

We can denote f as a mapping from A to B in f: A → B and read as f is a function from A to B.

If f: A → B and x ∈ A and y ∈ B then y is called the image of element x under f and function f is given by f(x).

Thus can we write as y = f(x)

where x is the pre-image of y.

Therefore, For a function from A to B.

● Set A and Set B should be non-empty.

● Every element of Set A should have an image in Set B.

● No Element in Set A should have more than one image in Set B.

● Two or more elements of Set A can have the same image in Set B.

● f: x → y means that under the function of ‘f’ from A to B, an element x in Set A has image y in Set B.

● It is necessary that every f image is in Set B but there may be some elements in Set B that are not f images of any elements of Set A.

### How to Identify Functions from Mapping Diagrams?

A Mapping Diagram represents a function if each input value is paired with only one output value.

**Example:**

The following figure represents a function since each element in Set A is paired with only one element in Set B.