Answer :
To determine which function has an inverse that is also a function, we need to understand the criterion for a function to have such an inverse. For the inverse of a function to be a function, each value in the range (or output values) must be unique among the given points. Here are the detailed steps to identify the correct function from the given sets of points:
### Step-by-Step Solution
1. Identify the y-values for each set of points.
2. Check if all the y-values within a set are unique.
Let's evaluate each set of points in detail:
#### Set 1: [tex]\(\{(-1,-2),(0,4),(1,3),(5,14),(7,4)\}\)[/tex]
- The y-values are: \{-2, 4, 3, 14, 4\}
- Observe that the value `4` appears twice.
- Therefore, y-values are not unique.
#### Set 2: [tex]\(\{(-1,2),(0,4),(1,5),(5,4),(7,2)\}\)[/tex]
- The y-values are: \{2, 4, 5, 4, 2\}
- The value `2` appears twice, and the value `4` appears twice.
- Therefore, y-values are not unique.
#### Set 3: [tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
- The y-values are: \{3, 4, 14, 6, 2\}
- All values are unique: `3`, `4`, `14`, `6`, `2` appear only once.
- Therefore, y-values are all unique.
#### Set 4: [tex]\(\{(-1,4),(0,4),(1,2),(5,3),(7,1)\}\)[/tex]
- The y-values are: \{4, 4, 2, 3, 1\}
- The value `4` appears twice.
- Therefore, y-values are not unique.
### Conclusion
The only set with unique y-values is Set 3:
[tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
Therefore, the function that has an inverse which is also a function is:
[tex]\[ \boxed{3} \][/tex]
This identification process confirms that Set 3 is the correct choice, as it is the only set where all y-values are distinct.
### Step-by-Step Solution
1. Identify the y-values for each set of points.
2. Check if all the y-values within a set are unique.
Let's evaluate each set of points in detail:
#### Set 1: [tex]\(\{(-1,-2),(0,4),(1,3),(5,14),(7,4)\}\)[/tex]
- The y-values are: \{-2, 4, 3, 14, 4\}
- Observe that the value `4` appears twice.
- Therefore, y-values are not unique.
#### Set 2: [tex]\(\{(-1,2),(0,4),(1,5),(5,4),(7,2)\}\)[/tex]
- The y-values are: \{2, 4, 5, 4, 2\}
- The value `2` appears twice, and the value `4` appears twice.
- Therefore, y-values are not unique.
#### Set 3: [tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
- The y-values are: \{3, 4, 14, 6, 2\}
- All values are unique: `3`, `4`, `14`, `6`, `2` appear only once.
- Therefore, y-values are all unique.
#### Set 4: [tex]\(\{(-1,4),(0,4),(1,2),(5,3),(7,1)\}\)[/tex]
- The y-values are: \{4, 4, 2, 3, 1\}
- The value `4` appears twice.
- Therefore, y-values are not unique.
### Conclusion
The only set with unique y-values is Set 3:
[tex]\(\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}\)[/tex]
Therefore, the function that has an inverse which is also a function is:
[tex]\[ \boxed{3} \][/tex]
This identification process confirms that Set 3 is the correct choice, as it is the only set where all y-values are distinct.
