To find the inverse of a function [tex]\( g(x) \)[/tex], we need to swap the elements in each ordered pair [tex]\((a, b)\)[/tex] to [tex]\((b, a)\)[/tex].
Given the function [tex]\( g(x) = \{ (4, -5), (-3, 2), (-6, 1), (1, 0) \} \)[/tex], we will find its inverse [tex]\( g^{-1}(x) \)[/tex] by swapping the coordinates of each pair.
Let's swap the coordinates for each pair:
1. For the pair [tex]\((4, -5)\)[/tex], swapping gives [tex]\((-5, 4)\)[/tex].
2. For the pair [tex]\((-3, 2)\)[/tex], swapping gives [tex]\((2, -3)\)[/tex].
3. For the pair [tex]\((-6, 1)\)[/tex], swapping gives [tex]\((1, -6)\)[/tex].
4. For the pair [tex]\((1, 0)\)[/tex], swapping gives [tex]\((0, 1)\)[/tex].
Thus, the inverse of [tex]\( g(x) \)[/tex] is [tex]\( g^{-1}(x) = \{ (-5, 4), (2, -3), (1, -6), (0, 1) \} \)[/tex].
Now, we compare this with the given choices:
A. [tex]\(\{ (5, -4), (-2, 3), (-1, 6), (0, -1) \}\)[/tex]
B. [tex]\(\{ (-4, 5), (3, -2), (6, -1), (-1, 0) \}\)[/tex]
C. [tex]\(\{ (-5, 4), (2, -3), (1, -6), (0, 1) \}\)[/tex]
D. [tex]\(\{ (4, -5), (-3, 2), (-6, 1), (1, 0) \}\)[/tex]
We see that the correct set of ordered pairs that represents the inverse of [tex]\( g(x) \)[/tex] is given in choice C.
Therefore, the answer is
[tex]\[ \boxed{C} \][/tex]
