To determine which function has an inverse that is also a function, we need to check if the function passes the Horizontal Line Test. This means that each output (y-value) is associated with only one input (x-value). In other words, for a function to have an inverse that is also a function, all y-values must be unique.
Let's analyze each given set of ordered pairs in the functions step by step:
[tex]\[
\begin{array}{l}
\{(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)\} \\
\{(-4,5),(-2,9),(-1,8),(4,8),(11,4)\} \\
\{(-4,6),(-2,2),(-1,6),(4,2),(11,2)\} \\
\{(-4,4),(-2,-1),(-1,0),(4,1),(11,1)\}
\end{array}
\][/tex]
1. First function: [tex]\((-4,3), (-2,7), (-1,0), (4,-3), (11,-7)\)[/tex]
The y-values are [tex]\(3, 7, 0, -3, -7\)[/tex]. All of these y-values are unique. Therefore, this function passes the Horizontal Line Test and has an inverse that is also a function.
2. Second function: [tex]\((-4,5), (-2,9), (-1,8), (4,8), (11,4)\)[/tex]
The y-values are [tex]\(5, 9, 8, 8, 4\)[/tex]. Notice that the y-value [tex]\(8\)[/tex] appears twice. This means that the function does not pass the Horizontal Line Test, so it does not have an inverse that is also a function.
3. Third function: [tex]\((-4,6), (-2,2), (-1,6), (4,2), (11,2)\)[/tex]
The y-values are [tex]\(6, 2, 6, 2, 2\)[/tex]. Notice that both [tex]\(6\)[/tex] and [tex]\(2\)[/tex] appear multiple times. This means the function does not pass the Horizontal Line Test, so it does not have an inverse that is also a function.
4. Fourth function: [tex]\((-4,4), (-2,-1), (-1,0), (4,1), (11,1)\)[/tex]
The y-values are [tex]\(4, -1, 0, 1, 1\)[/tex]. Notice that the y-value [tex]\(1\)[/tex] appears twice. This means that the function does not pass the Horizontal Line Test, so it does not have an inverse that is also a function.
After evaluating each function, we conclude that only the first function \{(-4,3),(-2,7),(-1,0),(4,-3),(11,-7)\} has all unique y-values. Therefore, it is the function with an inverse that is also a function.
