To determine which function has an inverse that is also a function, we need to check each set of pairs to see if it satisfies the criteria. Specifically, the function's range (y-values) must be unique for each x-value. This ensures that the inverse relation will assign one and only one value of [tex]\( x \)[/tex] to each [tex]\( y \)[/tex].
Let's analyze each set one by one:
1. Set 1: [tex]\(\{(-1, -2), (0, 4), (1, 3), (5, 14), (7, 4)\}\)[/tex]
- For this set, we check the y-values: [tex]\(\{-2, 4, 3, 14, 4\}\)[/tex].
- The y-value [tex]\( 4 \)[/tex] appears twice.
- Therefore, this function does not have an inverse that is also a function because the y-values are not unique.
2. Set 2: [tex]\(\{(-1, 2), (0, 4), (1, 5), (5, 4), (7, 2)\}\)[/tex]
- For this set, we check the y-values: [tex]\(\{2, 4, 5, 4, 2\}\)[/tex].
- The y-values [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex] each appear twice.
- Therefore, this function does not have an inverse that is also a function because the y-values are not unique.
3. Set 3: [tex]\(\{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\}\)[/tex]
- For this set, we check the y-values: [tex]\(\{3, 4, 14, 6, 2\}\)[/tex].
- All y-values are unique: [tex]\( 3, 4, 14, 6, 2 \)[/tex].
- Therefore, this function has an inverse that is also a function because the y-values are unique.
4. Set 4: [tex]\(\{(-1, 4), (0, 4), (1, 2), (5, 3), (7, 1)\}\)[/tex]
- For this set, we check the y-values: [tex]\(\{4, 4, 2, 3, 1\}\)[/tex].
- The y-value [tex]\( 4 \)[/tex] appears twice.
- Therefore, this function does not have an inverse that is also a function because the y-values are not unique.
Based on the analysis, the third set [tex]\(\{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\}\)[/tex] has unique y-values, meaning that this function has an inverse that is also a function.
The correct answer is Set 3.
