91Ó°ÊÓ

In mathematics, when we speak about the **domain** of a relation, we refer to the set of all possible inputs. For a set of ordered pairs, these inputs are the first numbers in each pair, commonly known as the x-values. Let’s take a look at the relation given in the exercise: \( \{(8,0),(5,4),(9,3),(3,8)\} \).To identify the domain, simply extract these x-values. Thus, the domain \(D\) for this set is \( \{3, 5, 8, 9\} \). It’s important to list these values in ascending order as a standard, although not necessary.The **range**, on the other hand, is all about the outputs. These are represented by the second numbers in the ordered pairs, or the y-values. For the same set of ordered pairs, the y-values are \( \{0, 4, 3, 8\} \). Hence, the range \(R\) for this relation is \( \{0, 3, 4, 8\} \). It’s always good practice to ensure no values are repeated and that they’re listed in order. In summary:

• Domain \(D\): collection of x-values \( \{3, 5, 8, 9\} \).
• Range \(R\): collection of y-values \( \{0, 3, 4, 8\} \).

Ordered pairs are a fundamental concept in understanding relations. An **ordered pair** is simply a pair of numbers that defines a point on a coordinate plane. The first number represents the x-coordinate (or input), and the second number represents the y-coordinate (or output). Consider the ordered pairs from the exercise: \( (8,0), (5,4), (9,3), (3,8) \)The x-coordinate comes first in each pair and the y-coordinate follows. This order is crucial, as it dictates the exact location on the graph, just like coordinates in geography tell you where a place is located.By organizing and analyzing these ordered pairs, we can determine the domain and range of a relation. Each ordered pair provides a specific input-output connection, which is essential for identifying if a relation can be termed a function, as we’ll discuss next. But firstly, it's essential to recognize that without understanding ordered pairs, the concepts of domain and range would not be accessible.

Understanding if a relation is also a function is key in mathematics. A **relation** is simply a set of ordered pairs. But to decide if this relation is a **function**, we look at how the values correspond.A relation is deemed a function if each x-value (input) is paired with exactly one y-value (output). In simple terms, no x-value should be paired with more than one y-value.Let’s examine the relation from the exercise: \( \{(8,0),(5,4),(9,3),(3,8)\} \)Looking at each x-value:

• 8 corresponds to 0


• 5 corresponds to 4


• 9 corresponds to 3


• 3 corresponds to 8

This correspondence shows that each x-value pairs with only one y-value, verifying that this relation is indeed a function.If, for example, x-value 8 paired with both 0 and another y-value like 2, it would not be a function. But in this case, the relation passes the function test. This forms a basic yet essential part of relation analysis, influencing how we graph, interpret, and utilize mathematical models in higher problem-solving.