To determine which function has an inverse that is also a function, we need to understand what it means for a function to have an inverse that is itself a function. Specifically, this requires that every [tex]\( y \)[/tex]-value in each set is unique because the inverse function would map each unique [tex]\( y \)[/tex]-value back to a unique [tex]\( x \)[/tex]-value.
Let's evaluate each set of pairs:
1. Set 1: [tex]\(\{(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)\}\)[/tex]
- [tex]\( y \)[/tex]-values: [tex]\[3, 7, 0, -3, -7\][/tex]
- All [tex]\( y \)[/tex]-values are unique.
2. Set 2: [tex]\(\{(-4,6),(-2,2),(-1,6),(4,2),(11,2)\}\)[/tex]
- [tex]\( y \)[/tex]-values: [tex]\[6, 2, 6, 2, 2\][/tex]
- There are repeated [tex]\( y \)[/tex]-values (6 and 2 appear more than once).
3. Set 3: [tex]\(\{(-4,5),(-2,9),(-1,8),(4,8),(11,4)\}\)[/tex]
- [tex]\( y \)[/tex]-values: [tex]\[5, 9, 8, 8, 4\][/tex]
- There are repeated [tex]\( y \)[/tex]-values (8 appears more than once).
4. Set 4: [tex]\(\{(-4,4),(-2,-1),(-1,0),(4,1),(11,1)\}\)[/tex]
- [tex]\( y \)[/tex]-values: [tex]\[4, -1, 0, 1, 1\][/tex]
- There are repeated [tex]\( y \)[/tex]-values (1 appears more than once).
Upon checking each set, we find that only the first set [tex]\(\{(-4, 3), (-2, 7), (-1, 0), (4, -3), (11, -7)\}\)[/tex] has all unique [tex]\( y \)[/tex]-values. This means it is the only set for which the inverse would also qualify as a function.
Therefore, the function whose inverse is also a function is the first set.
