- Review the definition of a function: every input (x) must map to exactly one output (y).
- Check each set for duplicate x-coordinates that correspond to different y-coordinates.
- Observe that in the set \(\{(-1,8), (8,-4), (8,3), (-5,-6)\}\), the input \(8\) maps to both \(-4\) and \(3\).
- Conclude that this set is not a function: \(\boxed{\{(-1,8), (8,-4), (8,3), (-5,-6)\}}\).
Explanation
- Analyzing the Definition of a Function
A relation is defined as a set of ordered pairs \((x, y)\). A relation is a function if and only if each input value \(x\) in the domain is associated with exactly one output value \(y\) in the range. If any input value \(x\) is associated with more than one output value \(y\), then that relation fails to be a function. To determine which set does not represent a function, we must check each set to see if any x-coordinate (the first value in the pair) is repeated with a different corresponding y-coordinate.
- Testing Each Set for the Function Property
We examine the four sets provided:
- Set 1: \(\{(-6,-1), (-4,6), (-8,2), (3,-1)\}\). The domain values are \(\{-6, -4, -8, 3\}\). All these values are distinct, so each x maps to exactly one y. This is a function.
- Set 2: \(\{(7,4), (9,-8), (-5,-7), (-2,4)\}\). The domain values are \(\{7, 9, -5, -2\}\). All these values are distinct, so each x maps to exactly one y. This is a function.
- Set 3: \(\{(-3,-3), (4,5), (8,9), (0,5)\}\). The domain values are \(\{-3, 4, 8, 0\}\). All these values are distinct, so each x maps to exactly one y. This is a function.
- Set 4: \(\{(-1,8), (8,-4), (8,3), (-5,-6)\}\). The domain values include \(-1, 8, 8, -5\). The value \(8\) appears twice, mapping to \(-4\) and \(3\), which are different values. Since one input maps to two different outputs, this set does not represent a function.
- Conclusion
Based on the analysis, we have identified that the fourth set, \(\{(-1,8), (8,-4), (8,3), (-5,-6)\}\), violates the fundamental rule of functions because the domain element \(8\) maps to two distinct values (\(-4\) and \(3\)). Therefore, this set is not a function. The final answer is \(\boxed{\{(-1,8), (8,-4), (8,3), (-5,-6)\}}\).
Examples
Imagine you have a simple machine, like a vending machine. If you press a button labeled "A", you expect to get a soda. If sometimes pressing button "A" gives you a soda and other times it gives you a bag of chips, the machine is not functioning correctly—it does not map one input (button A) to a single specific output. This is exactly how we evaluate functions in math: a valid "machine" or function must always yield the same unique result for every single input you provide.