To find which ordered pairs are found on the graph of the inverse function, we need to understand what an inverse function does. If [tex]\( f(x) \)[/tex] is a function that maps values of [tex]\( x \)[/tex] to [tex]\( f(x) \)[/tex], the inverse function [tex]\( f^{-1}(x) \)[/tex] swaps the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. In other words, given a pair [tex]\((x, y)\)[/tex] in the original function, the inverse will have the pair [tex]\((y, x)\)[/tex].
Given the table of values for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline x & f(x) \\
\hline
-2 & 12 \\
\hline
-1 & 10 \\
\hline
0 & 8 \\
\hline
1 & 6 \\
\hline
2 & 4 \\
\hline
\end{array}
\][/tex]
To determine the pairs for the inverse function, we swap each [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex]:
1. The pair [tex]\((-2, 12)\)[/tex] becomes [tex]\((12, -2)\)[/tex]
2. The pair [tex]\((-1, 10)\)[/tex] becomes [tex]\((10, -1)\)[/tex]
3. The pair [tex]\((0, 8)\)[/tex] becomes [tex]\((8, 0)\)[/tex]
4. The pair [tex]\((1, 6)\)[/tex] becomes [tex]\((6, 1)\)[/tex]
5. The pair [tex]\((2, 4)\)[/tex] becomes [tex]\((4, 2)\)[/tex]
Now let's look at the given options and see which match the pairs we found:
A. [tex]\((12, -2), (10, -1), (8, 0), (6, 1), (4, 2)\)[/tex]
This option matches all the pairs derived from inversing the given function.
B. [tex]\((-2, -12), (-1, -10), (0, 8), (1, -6), (2, -4)\)[/tex]
These pairs do not match the pairs for the inverse function.
C. [tex]\((2, -12), (1, -10), (0, -8), (-1, -6), (-2, -4)\)[/tex]
These pairs also do not match the pairs for the inverse function.
D. [tex]\((-12 / 2), (-10, 1), (-8, 0), (-6, -1), (-4, -2)\)[/tex]
This option lists pairs that are not even logically constructed properly; it is incorrect.
Thus, the correct choice for the ordered pairs found on the graph of the inverse function is:
A. [tex]\((12, -2), (10, -1), (8, 0), (6, 1), (4, 2)\)[/tex]
