About Relation Vs Function
Relation
In mathematics, a relation is defined as a collection of ordered pairs containing inputs and outputs. The pairing of an input with an output is expressed as a relation. If we have an ordered pair (a, b) in a relation, a will represent the input and b will represent the respective output.
Functions
A relation that connects each input to exactly one output is termed as a function.
In the above mapping diagram, each input has exactly one output. Hence, the relation is a function.
What is the Distinction between Functions and Relations?
Differentiating Parameter | Relations | Functions |
|---|---|---|
Definition | A relation pairs inputs with outputs. | A function is a relationship in which each input has only one output. |
Example | Ordered pairs: {(3, x), (7, y), (3,z)} "3" has an output of x and z, so it is not a function, though it is a relation. | Ordered pairs: {(3, x), (9, y), (5, z)} Here, each input has exactly one output. Hence, it is a function. |
Note: | Every relation need not be a function. | Every function is a relation. |
Solved Examples
Example 1: Examine the following to see if they are functions or not.
a) Ordered pairs: {(1,2), (2,3), (3,4), (4,5)}
b) Ordered pairs: {(1,6), (2,5), (1,9), (4,3)}
Solution:
a)
Each input has exactly one output. Hence, the relation is a function.
b)
The input 1 has two outputs, 6 and 9. Hence, the relation is not a function.
Example 2: Although all relations are functions, not all functions are relations. Justify.
Solution:
Assume we have two relations as shown in the table.
Example 3:
The ordered pairs stated below represent the roll number of students as the input and the number of hours spent studying for the upcoming test as the output. Determine whether the relation represents a function or not for the data provided.
{(1, 5), (2, 5), (3, 2), (4, 5), (5, 2)}
Solution:
Map the ordered pairs:
In the given ordered pairs each input is paired with exactly one output. So, the relation represents a function.
A function rule is an equation between inputs (independent variables) and outputs (dependent variables).
Relations contain inputs and outputs. The relationship between inputs and outputs can be depicted using ordered pairs and mapping diagrams.
When each input in a relation is mapped to exactly one output, the relation is said to be a function.