About Relations and Functions
What are Relations?
A relation is the depiction of input and output, it is represented by ordered pairs or mapping diagrams. In ordered pairs, the “x-coordinate” is called the input, and the “y-coordinate” is called the output.
For example, in an ordered pair (1, 2) the 1 is called an input or “x-coordinate” and the 2 is called an output or “y-coordinate”.
For example,
Relations can also be represented by using mapping diagrams
A mapping diagram depicts the relationship between the inputs and outputs. In mapping diagrams, the input is mapped or matched to its corresponding output. To represent the relationship, lines or arrows are used.
What is Function?
A function is a special kind of relation in which each input has only one output.
For example, determining whether the relation is a function or not for the ordered pairs, { (1, – 1) (0, 3)(4, – 2)(2, – 3)(3, – 3) }
We first map the ordered pairs into a diagram.
In the given ordered pairs each input is paired with exactly one output. So, the relation represents a function.
For example, determining whether the relation is a function or not for the ordered pairs, { (1, – 1) (0, 3)(2, – 2)(2, – 3)(3, – 5) }
Mapping the ordered pairs into the diagram.
The input ‘2’ has two output values which are “- 2” and “- 3”.
So, the relation doesn’t represent a function.
Solved Relations and Functions Examples
1) In the given diagram, explain the relationship between the inputs and outputs. After that, finish the mapping diagram. Is it possible to have more than one answer? Explain, and list the ordered pairs.
Solution:
The input column indicates the type of sports and the output column indicates the starting alphabets of the sports. As there are more than 1 sport that starts with “s” it is possible to have more than one answer.
So one of the answers can be surfing as it starts with “s”.
The list of ordered pairs are
(Wrestling, W) (Water Polo, W) (Swimming, S) (Surfing, S)(Golf, G) .
2) In the given diagram, explain the relationship between the inputs and outputs. List the ordered pairs.
Solution:
The given mapping diagram represents the names of individuals and their identity numbers. The mapping is done according to the student and the identity number of that student.
The list of ordered pairs are:
(John, 002) (Joseph, 004) (Tom, 001) (Mary, 009).
3) The given ordered pairs represent the roll number of students and the number of hours spent studying for the upcoming test. Determine whether the relation represents a function or not for the given data.
{ (1, 5) (2, 5)(3, 2)(4, 5)(5, 2) }
Solution:
Mapping the ordered pairs into the diagram.
In the given ordered pairs each input is paired with exactly one output. So, the relation represents a function.
4) The given mapping diagram represents the cost of a mobile phone in 4 different cities. Determine whether the relation represents a function or not for the given data.
Solution:
Mapping the diagram into ordered pairs.
{ (1, 1000) (2, 1200) (3, 1400) (4, 1600) }
In the given ordered pairs each city is paired with exactly one dollar value for the cost of the mobile phone. Hence the relation is a function.
Relations relate inputs and outputs. When every input in a relation has exactly one output then the relation is a function. As a result, all functions are relations, but not all relations are functions.
A function is a relation in which each input corresponds to one and only one output. Even if one output corresponds to 2 inputs, but if each input corresponds to only one output, then the relation is still a function.
For example,
In the above mapping diagram, each input corresponds to only one output. But inputs 2 and 3 correspond to output -3, that is two inputs have the same output. Even then, this relation is a function.