- Analyze the given set of points to determine the domain and range.
- The domain consists of the x-coordinates: \(\{-1, 0, 5, 10\}\).
- The range consists of the y-coordinates: \(\{1, -1, -11, -21\}\).
- Selecting the set of domain values, the final answer is \(\boxed{\{-1, 0, 5, 10\}}\).
Explanation
- Analyzing the Given Data Points
The problem presents a set of four ordered pairs representing points on a coordinate plane: \((-1, 1)\), \((0, -1)\), \((5, -11)\), and \((10, -21)\). To characterize these points, we can examine their domain and range. The domain is the set of all x-coordinates, and the range is the set of all y-coordinates. By extracting these values from the given points, we can identify which of the provided sets accurately describes the data.
- Identifying the Domain and Range Sets
First, let us identify the set of x-coordinates (the domain). The given x-values are \(-1\), \(0\), \(5\), and \(10\). Combining these into a set, we obtain \(\{-1, 0, 5, 10\}\).
Next, let us identify the set of y-coordinates (the range). The given y-values are \(1\), \(-1\), \(-11\), and \(-21\). Combining these into a set, we obtain \(\{1, -1, -11, -21\}\).
Comparing these results with the provided options, we see that both \(\{1, -1, -11, -21\}\) and \(\{-1, 0, 5, 10\}\) are present. In many mathematical contexts, describing the domain (\(\{-1, 0, 5, 10\}\)) is a primary way to characterize a set of data points. We will select this set as our answer.
- Conclusion
We have established that the set of x-coordinates (domain) is \(\{-1, 0, 5, 10\}\). This set represents the inputs of the relation. Each point corresponds to a y-value determined by the linear relationship \(y = -2x - 1\). Thus, the set \(\{-1, 0, 5, 10\}\) correctly characterizes the domain of the given collection of points. The final answer is \(\boxed{\{-1, 0, 5, 10\}}\).
Examples
Understanding how to identify the domain and range of a set of data points is fundamental in fields like data analysis and statistics. For instance, if you are tracking the temperature at different times of the day, the times would be your domain, and the temperatures would be your range. Identifying these sets allows you to establish trends, build models, and predict future outcomes, such as estimating the temperature at a time not explicitly recorded.