To determine the point of intersection between a function [tex]\( f(x) \)[/tex] and its inverse function [tex]\( f^{-1}(x) \)[/tex], we need to identify where the graphs of [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex] cross each other. By definition, the point of intersection occurs where [tex]\( f(x) = f^{-1}(x) \)[/tex]. This implies that at the intersection point, the x-coordinate and y-coordinate are equal, resulting in a point [tex]\((x, x)\)[/tex].
Given the two possible points of intersection:
1. [tex]\((0, -2)\)[/tex]
2. [tex]\((1, -1)\)[/tex]
We will check each point to see if it fits the form [tex]\((x, x)\)[/tex], meaning both coordinates are the same.
Checking the first point [tex]\((0, -2)\)[/tex]:
- The x-coordinate is [tex]\(0\)[/tex] and the y-coordinate is [tex]\(-2\)[/tex].
- Since [tex]\(0 \neq -2\)[/tex], this point does not satisfy the condition [tex]\( x = y \)[/tex].
Checking the second point [tex]\((1, -1)\)[/tex]:
- The x-coordinate is [tex]\(1\)[/tex] and the y-coordinate is [tex]\(-1\)[/tex].
- Since [tex]\(1 \neq -1\)[/tex], this point also does not satisfy the condition [tex]\( x = y \)[/tex].
Hence, neither of the given points [tex]\((0, -2)\)[/tex] nor [tex]\((1, -1)\)[/tex] satisfy the condition required for the point of intersection between [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex].
Therefore, based on the provided choices,
there is no valid intersection point in the provided choices.
