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An ordered pair is a fundamental concept in mathematics, especially in the study of relations and functions. An ordered pair is a set of two elements arranged in a specific sequence, typically written as \((x, y)\). Each element has a particular position:

• The first element is called the "x-coordinate" or "x-value."
• The second element is the "y-coordinate" or "y-value."

The order in which these elements appear is crucial since \((x, y)\) is distinct from \((y, x)\) unless \(x = y\). Ordered pairs are commonly used to represent points in a two-dimensional coordinate system, where "x" represents the horizontal position and "y" represents the vertical position.

Let's visualize ordered pairs using the given example: \((15, -3), (0, 0), (4, 6), (-3, -8)\). Each of these pairs corresponds to a point on a graph. For instance, the pair \((15, -3)\) indicates a point where the "x" value is 15 and the "y" value is -3. Understanding ordered pairs is key to figuring out concepts like domain and range, as they directly provide these values.

When we talk about the domain in the context of a relation, we're referring to the set of all possible first elements from each ordered pair, often known as the "x-values."

This means any value that could potentially be an "x" in the pairs you are working with. For the relation \(\{(15,-3), (0,0), (4,6), (-3,-8)\}\), we need to extract the "x" value from each pair. This gives us:

• 15 from \((15,-3)\)
• 0 from \((0,0)\)
• 4 from \((4,6)\)
• -3 from \((-3,-8)\)

Thus, the domain of this particular relation is \(\{15, 0, 4, -3\}\).

Identifying the domain is crucial as it represents all possible inputs or x-coordinates that the relation can have. It's like listing all starting points from which any potential output could be derived.

The range of a relation consists of all possible second elements from each ordered pair, also known as the "y-values."

This set represents all potential outputs that the relation can produce based on the elements of the domain. For our relation \(\{(15,-3), (0,0), (4,6), (-3,-8)\}\), you can determine the range by identifying the "y" value from each ordered pair:

• -3 from \((15,-3)\)
• 0 from \((0,0)\)
• 6 from \((4,6)\)
• -8 from \((-3,-8)\)

So, the range for this particular set is \(\{-3, 0, 6, -8\}\).

The range encompasses all possible outputs or y-coordinates, which tells us all the possible values that the resultant outputs can achieve. It's essential in understanding the behavior and the extent of outcomes a particular relation can provide.