Which function has an inverse that is also a function?

A. [tex]\{(-1,-2),(0,4),(1,3),(5,14),(7,4)\}[/tex]

B. [tex]\{(-1,2),(0,4),(1,5),(5,4),(7,2)\}[/tex]

C. [tex]\{(-1,3),(0,4),(1,14),(5,6),(7,2)\}[/tex]

D. [tex]\{(-1,4),(0,4),(1,2),(5,3),(7,1)\}[/tex]



Answer :

To determine which of the given functions has an inverse that is also a function, we need to ensure that the function in question has unique [tex]\( y \)[/tex]-values for each [tex]\( x \)[/tex]-value. In other words, no [tex]\( y \)[/tex]-value should be repeated for different [tex]\( x \)[/tex]-values. This property ensures that the function is one-to-one and therefore has an inverse that is also a function.

Given sets:

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]

We will check each set to see if [tex]\( y \)[/tex]-values are unique.

### 1. [tex]\(\{(-1, -2), (0, 4), (1, 3), (5, 14), (7, 4)\}\)[/tex]

[tex]\[ y\text{-values: } \{-2, 4, 3, 14, 4\} \][/tex]

Here, the [tex]\( y \)[/tex]-value 4 is repeated (for [tex]\( x = 0 \)[/tex] and [tex]\( x = 7 \)[/tex]). Therefore, this function does not have a unique inverse.

### 2. [tex]\(\{(-1, 2), (0, 4), (1, 5), (5, 4), (7, 2)\}\)[/tex]

[tex]\[ y\text{-values: } \{2, 4, 5, 4, 2\} \][/tex]

Here, the [tex]\( y \)[/tex]-values 2 and 4 are repeated. Thus, this function does not have a unique inverse.

### 3. [tex]\(\{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\}\)[/tex]

[tex]\[ y\text{-values: } \{3, 4, 14, 6, 2\} \][/tex]

Here, all the [tex]\( y \)[/tex]-values are unique. Therefore, this function potentially has an inverse that is also a function, making it one-to-one.

### 4. [tex]\(\{(-1, 4), (0, 4), (1, 2), (5, 3), (7, 1)\}\)[/tex]

[tex]\[ y\text{-values: } \{4, 4, 2, 3, 1\} \][/tex]

Here, the [tex]\( y \)[/tex]-value 4 is repeated (for [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]). Thus, this function does not have a unique inverse.

Based on the analysis, we can conclude that the function:

[tex]\[ \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \][/tex]

has an inverse that is also a function. Thus, the correct function is the third one.