To determine which function has an inverse that is also a function, we need to identify the set where every [tex]\( y \)[/tex]-value (output) is unique. If each [tex]\( y \)[/tex]-value is unique, then the function will be one-to-one, allowing its inverse to also be a function.
Consider the given sets of points:
1. [tex]\(\{(-1,-2),(0,4),(1,3),(5,14),(7,4)\}\)[/tex]
2. [tex]\(\{(-1,2),(0,4),(1,5),(5,4),(7,2)\}\)[/tex]
3. [tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
4. [tex]\(\{(-1,4),(0,4),(1,2),(5,3),(7,1)\}\)[/tex]
Let's evaluate each set to see if the [tex]\( y \)[/tex]-values are unique.
1. Set 1: [tex]\(\{(-1,-2),(0,4),(1,3),(5,14),(7,4)\}\)[/tex]
- The [tex]\( y \)[/tex]-values are: [tex]\(-2, 4, 3, 14, 4\)[/tex]
- The [tex]\( y \)[/tex]-value 4 repeats, so this set is not one-to-one.
2. Set 2: [tex]\(\{(-1,2),(0,4),(1,5),(5,4),(7,2)\}\)[/tex]
- The [tex]\( y \)[/tex]-values are: 2, 4, 5, 4, 2
- The [tex]\( y \)[/tex]-values 2 and 4 repeat, so this set is not one-to-one.
3. Set 3: [tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
- The [tex]\( y \)[/tex]-values are: 3, 4, 14, 6, 2
- All [tex]\( y \)[/tex]-values are unique, so this set is one-to-one.
4. Set 4: [tex]\(\{(-1,4),(0,4),(1,2),(5,3),(7,1)\}\)[/tex]
- The [tex]\( y \)[/tex]-values are: 4, 4, 2, 3, 1
- The [tex]\( y \)[/tex]-value 4 repeats, so this set is not one-to-one.
After analyzing each set, we find that Set 3 is the only set where all [tex]\( y \)[/tex]-values are unique, making the function one-to-one and ensuring that its inverse is also a function.
Therefore, the function with an inverse that is also a function is:
[tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
The answer is:
Set 3
