- A relation is any set of ordered pairs, and this dataset consists of eight such pairs.
- A function is a specific type of relation where each input \(x\) must map to exactly one output \(y\).
- By examining the data, we see that the input \(x=1\) maps to \(y=3\) and \(y=2\), and the input \(x=3\) maps to \(y=4\) and \(y=3\).
- Since these inputs map to multiple outputs, the set is not a function. The final answer is \(\boxed{\text{a relation only}}\).
Explanation
- Analysis of Data and Definitions
We are given a survey dataset representing the number of siblings and the number of pets for eight classmates. We can represent this data as a set of ordered pairs \((x, y)\), where \(x\) is the number of siblings and \(y\) is the number of pets. The data points are as follows:
\((3, 4), (1, 3), (0, 7), (2, 4), (4, 6), (1, 2), (5, 8), (3, 3)\).
A relation is defined as any set of ordered pairs. Since we have a collection of eight ordered pairs, this set clearly qualifies as a relation.
- Checking the Function Condition
To determine if this relation is also a function, we must check if each input value \(x\) maps to exactly one output value \(y\). A relation is classified as a function if no two distinct ordered pairs have the same first coordinate but different second coordinates.
Let us examine the \(x\)-values in our set of pairs:
- For \(x = 0\), we have the pair \((0, 7)\).
- For \(x = 1\), we have the pairs \((1, 3)\) and \((1, 2)\).
- For \(x = 2\), we have the pair \((2, 4)\).
- For \(x = 3\), we have the pairs \((3, 4)\) and \((3, 3)\).
- For \(x = 4\), we have the pair \((4, 6)\).
- For \(x = 5\), we have the pair \((5, 8)\).
- Conclusion
A function requires that every input corresponds to exactly one output. However, our analysis shows:
- The input \(x = 1\) is associated with two different outputs: \(y = 3\) and \(y = 2\).
- The input \(x = 3\) is also associated with two different outputs: \(y = 4\) and \(y = 3\).
Since there are inputs that map to more than one output, the set of data fails the vertical line test (or the definition of a function). Therefore, the data represents a relation but not a function. The correct classification is that it is a relation only.
Examples
In real life, a function can be thought of as a vending machine: if you press the button for a specific soda (the input), you should always get the same type of soda (the output). If you pressed the button for a cola and sometimes received a cola but other times received a root beer, that machine would not be functioning correctly. Similarly, in mathematics, a rule is only a function if every input consistently produces exactly one output. Sorting datasets helps us identify when a process behaves predictably, which is foundational for data science, economics, and engineering.