To solve this problem, we need to find the inverse function [tex]\( h^{-1}(x) \)[/tex] from the given function [tex]\( h(x) \)[/tex].
The function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[
h(x) = \{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}
\][/tex]
The inverse function [tex]\( h^{-1}(x) \)[/tex] is obtained by swapping each pair of the function [tex]\( h(x) \)[/tex]. For each pair [tex]\((a, b)\)[/tex] in [tex]\( h(x) \)[/tex], the corresponding pair in [tex]\( h^{-1}(x) \)[/tex] will be [tex]\((b, a)\)[/tex].
Let’s perform this swapping step-by-step for each pair in [tex]\( h(x) \)[/tex]:
1. For the pair [tex]\( (3, -5) \)[/tex], the swapped pair is [tex]\( (-5, 3) \)[/tex].
2. For the pair [tex]\( (5, -7) \)[/tex], the swapped pair is [tex]\( (-7, 5) \)[/tex].
3. For the pair [tex]\( (6, -9) \)[/tex], the swapped pair is [tex]\( (-9, 6) \)[/tex].
4. For the pair [tex]\( (10, -12) \)[/tex], the swapped pair is [tex]\( (-12, 10) \)[/tex].
5. For the pair [tex]\( (12, -16) \)[/tex], the swapped pair is [tex]\( (-16, 12) \)[/tex].
Thus, the inverse function [tex]\( h^{-1}(x) \)[/tex] is given by:
[tex]\[
h^{-1}(x) = \{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}
\][/tex]
Now let's evaluate the given options:
1. [tex]\(\{(3, 5), (5, 7), (6, 9), (10, 12), (12, 16)\}\)[/tex]
2. [tex]\(\{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}\)[/tex]
3. [tex]\(\{(3, -5), (5, -7), (6, -9), (10, -12), (12, -16)\}\)[/tex]
4. [tex]\(\{(5, 3), (7, 5), (9, 6), (12, 10), (16, 12)\}\)[/tex]
Based on our calculations, the correct answer is:
[tex]\[
\{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}
\][/tex]
Therefore, the option that gives [tex]\( h^{-1}(x) \)[/tex] is:
[tex]\[
\boxed{\{(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)\}}
\][/tex]
