- Define the domain as the set of all unique first components (x-coordinates) in the relation.
- Extract the first components from the given pairs: \(7, 9, 5, -10, 5\).
- Identify the unique values and arrange them as a set: \(\{-10, 5, 7, 9\}\).
- The correct option is \(\boxed{\text{a}}\) which represents the set \(\boxed{\{-10, 5, 7, 9\}}\).
Explanation
- Understanding the Definition of a Domain
A relation is defined as a set of ordered pairs \((x, y)\). The domain of a relation consists of the set of all unique first components (the x-coordinates) from every ordered pair in the set. For the given relation:
\[\{(7, 10), (9, -3), (5, 8), (-10, 6), (5, 0)\}\]
we identify the first components as the values appearing in the first position of each pair.
- Identifying the Unique x-coordinates
By extracting the x-coordinates from the set of ordered pairs, we get the list of values:
\[7, 9, 5, -10, 5\]
The domain is the set containing each of these values exactly once. To find this, we list the values and remove any duplicates. We see that the value \(5\) appears twice in our list. Thus, the unique first components are \(-10, 5, 7,\) and \(9\). Listing these in numerical order gives the set \(\{-10, 5, 7, 9\}\).
- Selecting the Correct Option
Comparing our set \(\{-10, 5, 7, 9\}\) to the provided options:
A\(\{-10, 5, 7, 9\}\)
B\(\{-3, 0, 6, 8, 10\}\)
C\(\{-10, 0, 5, 7, 9\}\)
D\(\{-3, 7, 9, 10\}\)
Our derived set matches option 'a'. Thus, the domain of the relation is \(\boxed{\{-10, 5, 7, 9\}}\).
Examples
In real-world data analysis, identifying the domain of a relation is essential for understanding the range of input values that can be processed by a function. For example, if you are tracking the temperature of a city at different times, the set of times (the domain) tells you exactly when data was recorded, ensuring that you don't mistakenly look for a temperature at a time that wasn't monitored. Understanding how to extract this information from a dataset is a fundamental skill in statistics and computer programming.