91Ó°ÊÓ

In algebra, understanding the difference between a relation and a function is crucial. A **relation** is simply a set of ordered pairs. This means it pairs up x-values (input) with y-values (output) in a specific way. An example of a relation is the set of pairs given: \({(8, 0), (5, 4), (9, 3), (3, 8)}\).
However, not all relations are functions. A **function** is a special type of relation where each x-value is paired with exactly one y-value. This means no x-value can repeat with a different y-value. In our example, we check each x-value (8, 5, 9, 3) and find that they are all unique. Hence, this relation is a function because all x-values only appear once and are paired with one y-value.

The **domain** and **range** of a function are fundamental concepts. The **domain** refers to all possible x-values (inputs) of the function. For the relation \((8,0), (5,4), (9,3), (3,8)\), the domain is the set of all x-values: \(\{8, 5, 9, 3\}\).
The **range** refers to all possible y-values (outputs) of the function. In the given relation, the range is the set of y-values: \(\{0, 4, 3, 8\}\). Understanding the domain and range is essential as it helps in defining the scope and behavior of a function. The domain tells us what inputs are valid, while the range shows the possible outcomes.

A key characteristic of a function is that each x-value must be unique. This means no x-value can pair with more than one y-value. Why is this important? Because it ensures that the function produces a single output for each input, making it predictable and well-defined.
In the given set \((8,0), (5,4), (9,3), (3,8)\), we observe that all x-values are distinct: 8, 5, 9, and 3. Since these x-values do not repeat, they each have a unique y-value associated with them. This confirms that the given relation is indeed a function. Always remember, if you find duplicate x-values paired with different y-values, the relation cannot be classified as a function.