Verified Answer :

  • The domain consists of all unique first coordinates (x-values) from the given set of ordered pairs.
  • By listing the x-coordinates \(8, -8, 3, -2, 4, 8\), we identify the set of unique x-values.
  • Sorting these unique values gives the domain \(\{-8, -2, 3, 4, 8\}\).
  • The final answer is \(\boxed{\{-8, -2, 3, 4, 8\}}\).

Explanation

  1. Defining the Domain of a Relation

A relation is a set of ordered pairs of the form \((x, y)\). The domain of a relation is defined as the set of all unique first coordinates (the \(x\)-values) present in the ordered pairs. We are given the set of ordered pairs: \((8, -8), (-8, 10), (3, 7), (-2, 1), (4, 6),\) and \((8, 2)\). To find the domain, we extract the first coordinate from each pair.

  1. Extracting the x-coordinates

We list the \(x\)-coordinates from each of the given pairs:

  • For \((8, -8)\), the \(x\)-coordinate is \(8\).
  • For \((-8, 10)\), the \(x\)-coordinate is \(-8\).
  • For \((3, 7)\), the \(x\)-coordinate is \(3\).
  • For \((-2, 1)\), the \(x\)-coordinate is \(-2\).
  • For \((4, 6)\), the \(x\)-coordinate is \(4\).
  • For \((8, 2)\), the \(x\)-coordinate is \(8\).

Collecting these values, we have the set \(\{8, -8, 3, -2, 4, 8\}\).

  1. Finalizing the Domain

To determine the final domain, we identify the unique values in the collected set and write them in ascending order. The unique values are \(-8, -2, 3, 4,\) and \(8\). Comparing this to the provided options, we see that it matches the set \(\{-8, -2, 3, 4, 8\}\). The other provided set, \(\{-8, 1, 2, 6, 7, 10\}\), consists of the unique \(y\)-coordinates, which forms the range of the relation, not the domain. Therefore, the correct domain is \(\{-8, -2, 3, 4, 8\}\).

Examples

In data science or database management, defining the domain is essential when you have a list of entries mapping users (IDs) to their activity scores. For example, if you have user records \((101, 85), (102, 90), (101, 88)\), the "domain" would be the set of unique user IDs: \(\{101, 102\}\). This helps in quickly identifying all active participants without duplicates, ensuring efficient data processing and analysis.

Related Questions