To find the point of intersection of a function [tex]\( f(x) \)[/tex] and its inverse function [tex]\( f^{-1}(x) \)[/tex] on the same coordinate plane, we need to find a point where [tex]\( f(x) = x \)[/tex]. This is because, at the point of intersection, the function and its inverse will have the same x and y coordinates, i.e., [tex]\( f(a) = a \)[/tex].
Let’s analyze the given points to see if they satisfy [tex]\( f(x) = x \)[/tex]:
1. Point (0, -2):
- Here, [tex]\( x = 0 \)[/tex] and [tex]\( y = -2 \)[/tex].
- We need to check if [tex]\( 0 = -2 \)[/tex]. Clearly, [tex]\( 0 \neq -2 \)[/tex]. Therefore, this point is not where [tex]\( f(x) \)[/tex] intersects [tex]\( f^{-1}(x) \)[/tex].
2. Point (1, -1):
- Here, [tex]\( x = 1 \)[/tex] and [tex]\( y = -1 \)[/tex].
- We need to check if [tex]\( 1 = -1 \)[/tex]. Clearly, [tex]\( 1 \neq -1 \)[/tex]. Therefore, this point is also not where [tex]\( f(x) \)[/tex] intersects [tex]\( f^{-1}(x) \)[/tex].
Given the points to check:
- [tex]\((0, -2)\)[/tex]
- [tex]\((1, -1)\)[/tex]
We can conclude that neither of these points satisfy the condition [tex]\( f(x) = x \)[/tex]. Therefore, there isn't an intersection point among the given options.
Thus, the answer is:
[tex]\[
(0, -2), (1, -1)
\][/tex]
