Relation and its Domain and Range - Algebra-2

1. Fundamental Concepts

Relation: A set of pairs consisting of input and output values (a set of number pairs). It describes the corresponding relationship between elements in two sets.
Domain: The set of all input values in a relation (i.e., the x-coordinates in ordered pairs).
Range: The set of all output values in a relation (i.e., the y-coordinates in ordered pairs).
Four ways to represent a relation:
- Ordered Pairs: In the form of (x, y), such as (0, 0), (-3, 4), etc.
- Table of Values: Presenting input values and their corresponding output values in a tabular form.
- Mapping Diagram: Using arrows to point elements in the input set to corresponding elements in the output set.
- Graph: A figure formed by plotting points corresponding to ordered pairs in a coordinate system.

2. Key Concepts

A relation is composed of a domain and a range, which are two indispensable parts of a relation.
Elements in the domain are all input values in the relation, and elements in the range are all output values in the relation, which correspond one-to-one with the ordered pairs in the relation.
When representing the domain and range, elements are usually arranged in a certain order (such as from smallest to largest) and expressed in the form of a set.

3. Examples

Easy Level

Example: Given that the ordered pairs of a relation are {(1, 2), (3, 4), (5, 6)}, find its domain and range.

Domain: {1, 3, 5} (the set of all x-coordinates)
Range: {2, 4, 6} (the set of all y-coordinates)

Medium Level

Example: Determine the domain and range of the relation based on the following table of values.

x（Input）	y（Output）
-2	3
0	5
2	3
4	7

Domain: {-2, 0, 2, 4}
Range: {3, 5, 7} (Note: The same element in the range is written only once)

Hard Level

Example: Skydiver Mason jumps from a height of 10,000 feet, and the heights corresponding to different times are as follows: 9677 feet at 5 seconds, 8713 feet at 10 seconds, 7462 feet at 15 seconds, and 5049 feet at 20 seconds. What are the domain and range of this relation?

Domain: {5, 10, 15, 20} (time as the input value)
Range: {5049, 7462, 8713, 9677} (height as the output value, arranged from smallest to largest)

4. Problem-Solving Techniques

Determining the domain:
1. Regardless of the form in which the relation is presented (ordered pairs, table of values, mapping diagram, graph), first find all input values (x-values).
2. Organize these input values into a set, remove duplicate elements, and arrange them in a certain order (usually from smallest to largest) to obtain the domain.
Determining the range:
1. Find all output values (y-values) in the relation.
2. Organize these output values into a set, remove duplicate elements, and arrange them in a certain order to obtain the range.
Notes: When dealing with relations in practical problems, it is necessary to clarify which quantity is the input (corresponding to the domain) and which is the output (corresponding to the range) to avoid confusion. For example, in the relation between time and height, time is the input and height is the output.