To correctly identify the domain and range of the given function set [tex]\(\{(6, -8), (9, 3), (-3, 5), (1, -6), (5, 7)\}\)[/tex], let's break it down:
1. Domain:
- The domain of a function is the set of all the first elements (x-coordinates) from each ordered pair.
- From the given pairs, the x-coordinates are [tex]\(6\)[/tex], [tex]\(9\)[/tex], [tex]\(-3\)[/tex], [tex]\(1\)[/tex], and [tex]\(5\)[/tex].
So, the domain is: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex].
2. Range:
- The range of a function is the set of all the second elements (y-coordinates) from each ordered pair.
- From the given pairs, the y-coordinates are [tex]\(-8\)[/tex], [tex]\(3\)[/tex], [tex]\(5\)[/tex], [tex]\(-6\)[/tex], and [tex]\(7\)[/tex].
So, the range is: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex].
Putting it all together, the domain and range of the given function are:
- Domain: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex]
- Range: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex]
Now compare these with the provided answer options:
A. Domain: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex]; Range: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex]
- Incorrect. This option has the domain and range swapped.
B. Domain: [tex]\(\{-3, 1, 5, 6, 3\}\)[/tex]; Range: [tex]\(\{-8, -6, 5, 7, 9\}\)[/tex]
- Incorrect. This option includes an incorrect x-value (duplicated [tex]\(\{3\}\)[/tex]) in the domain and an incorrect range value ([tex]\(\{9\}\)[/tex]).
C. Domain: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex]; Range: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex]
- Correct. The domain and range match our determined values.
D. Domain: [tex]\(\{-3, 1, 5, 6, 3\}\)[/tex]; Range: [tex]\(\{-8, -6, 6, 7, 9\}\)[/tex]
- Incorrect. This option includes an incorrect x-value (duplicated [tex]\(\{3\}\)[/tex]) in the domain and incorrect range values ([tex]\(\{6\}\)[/tex] and [tex]\(\{9\}\)[/tex]).
Thus, the correct answer is:
C. Domain: [tex]\(\{-3, 1, 5, 6, 9\}\)[/tex]; Range: [tex]\(\{-8, -6, 3, 5, 7\}\)[/tex]
