- A relation is a function if every \(x\)-coordinate appears only once in the set of ordered pairs.
- Examining the sets reveals:
- Set A: Unique \(x\)-coordinates; it is a function.
- Set B: Unique \(x\)-coordinates; it is a function.
- Set C: Repeated \(x=12\); not a function.
- Set D: Repeated \(x=-13\); not a function.
- Set E: Unique \(x\)-coordinates; it is a function.
- The sets that represent functions are A, B, and E.
- Final Answer: \(\boxed{\text{A, B, E}}\)
Explanation
- Understanding the Definition of a Function
To determine which of the given sets of ordered pairs represent a function, we rely on the fundamental definition of a function. A relation is a function if every input (the first coordinate, \(x\)) corresponds to exactly one output (the second coordinate, \(y\)).
If a set contains two or more ordered pairs that share the same \(x\)-value but have different \(y\)-values, that relation fails the definition of a function. Therefore, our task is to examine each set and identify those where every \(x\)-coordinate appears exactly once.
- Evaluating the Sets
Let's examine each set:
- Set A: \(\{(15,-3), (-6,9), (-3,0), (-1,16), (5,5)\}\)
The \(x\)-coordinates are \(\{15, -6, -3, -1, 5\}\). Since all these values are unique, each \(x\) maps to only one \(y\). Thus, Set A is a function.
- Set B: \(\{(6, 0), (-12, -16), (-6, 10), (20, -7), (16, 9)\}\)
The \(x\)-coordinates are \(\{6, -12, -6, 20, 16\}\). All these values are unique, so Set B is a function.
- Set C: \(\{(12,-18), (15,1), (12,5), (0,9), (-5,-17)\}\)
The \(x\)-coordinates are \(\{12, 15, 12, 0, -5\}\). The value \(x=12\) appears twice, mapping to \(y=-18\) and \(y=5\). Since one input maps to two different outputs, Set C is not a function.
- Set D: \(\{(-13, 4), (7, -15), (-13, 9), (6, -12), (-18, 0)\}\)
The \(x\)-coordinates are \(\{-13, 7, -13, 6, -18\}\). The value \(x=-13\) appears twice, mapping to \(y=4\) and \(y=9\). Thus, Set D is not a function.
- Set E: \(\{(10, 9), (-2, -16), (-6, 7), (5, 8), (8, -16), (-11, 9)\}\)
The \(x\)-coordinates are \(\{10, -2, -6, 5, 8, -11\}\). All these values are unique, so Set E is a function.
- Conclusion
Based on our analysis:
- Set A is a function.
- Set B is a function.
- Set E is a function.
These are the three sets that satisfy the definition of a function. The final answer is sets A, B, and E.
Examples
Understanding whether a set of data points constitutes a function is essential in fields like data science and engineering. For example, if you represent a sequence of measurements where \(x\) is the time and \(y\) is the temperature of an object, each time point must yield a single, unique temperature reading to be a well-defined physical process. If you had two different temperatures at the exact same time, your data would not represent a valid functional relationship, suggesting a possible error in measurement or a multi-state system. Identifying these relationships helps maintain data integrity and accurate modeling of real-world phenomena.