To determine the domain and range of the given function, we need to identify the set of all possible input values (the domain) and the set of all possible output values (the range) from the given set of points:
[tex]\[
\{(6, -8), (9, 3), (-3, 5), (1, -6), (5, 7)\}
\][/tex]
Step-by-Step Solution:
1. Identify the domain: The domain of the function is the set of all [tex]\(x\)[/tex]-coordinates from the given points.
- From the point [tex]\((6, -8)\)[/tex], the [tex]\(x\)[/tex]-coordinate is [tex]\(6\)[/tex].
- From the point [tex]\((9, 3)\)[/tex], the [tex]\(x\)[/tex]-coordinate is [tex]\(9\)[/tex].
- From the point [tex]\((-3, 5)\)[/tex], the [tex]\(x\)[/tex]-coordinate is [tex]\(-3\)[/tex].
- From the point [tex]\((1, -6)\)[/tex], the [tex]\(x\)[/tex]-coordinate is [tex]\(1\)[/tex].
- From the point [tex]\((5, 7)\)[/tex], the [tex]\(x\)[/tex]-coordinate is [tex]\(5\)[/tex].
Collecting these, the domain is:
[tex]\[
\{6, 9, -3, 1, 5\}
\][/tex]
Rewriting the domain in ascending order for clarity, we get:
[tex]\[
\{-3, 1, 5, 6, 9\}
\][/tex]
2. Identify the range: The range of the function is the set of all [tex]\(y\)[/tex]-coordinates from the given points.
- From the point [tex]\((6, -8)\)[/tex], the [tex]\(y\)[/tex]-coordinate is [tex]\(-8\)[/tex].
- From the point [tex]\((9, 3)\)[/tex], the [tex]\(y\)[/tex]-coordinate is [tex]\(3\)[/tex].
- From the point [tex]\((-3, 5)\)[/tex], the [tex]\(y\)[/tex]-coordinate is [tex]\(5\)[/tex].
- From the point [tex]\((1, -6)\)[/tex], the [tex]\(y\)[/tex]-coordinate is [tex]\(-6\)[/tex].
- From the point [tex]\((5, 7)\)[/tex], the [tex]\(y\)[/tex]-coordinate is [tex]\(7\)[/tex].
Collecting these, the range is:
[tex]\[
\{-8, 3, 5, -6, 7\}
\][/tex]
Rewriting the range in ascending order for clarity, we get:
[tex]\[
\{-8, -6, 3, 5, 7\}
\][/tex]
3. Select the correct option: Comparing our findings with the provided options, we match the domain and range with the following option:
- Domain: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex]
- Range: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex]
The correct option is:
[tex]\[
\boxed{D: \text{ Domain: } \{-3, 1, 5, 6, 9\} \text{; Range: } \{-8, -6, 3, 5, 7\}}
\][/tex]
