To determine the point of intersection between a function [tex]\( f(x) \)[/tex] and its inverse [tex]\( f^{-1}(x) \)[/tex] plotted on the same coordinate plane, we need to understand the relationship between a function and its inverse.
The function [tex]\( f(x) \)[/tex] and its inverse [tex]\( f^{-1}(x) \)[/tex] intersect at points where [tex]\( f(x) = x \)[/tex] and [tex]\( f^{-1}(x) = x \)[/tex]. This means that the function and its inverse intersect where the value of the function is equal to the input value.
Given the coordinates [tex]\((0, -2)\)[/tex] and [tex]\((1, -1)\)[/tex], we can check if they satisfy the condition [tex]\( f(x) = x \)[/tex] and [tex]\( f^{-1}(x) = x \)[/tex].
1. Check the point [tex]\((0, -2)\)[/tex]:
- For this point, [tex]\(x = 0\)[/tex] and [tex]\(y = -2\)[/tex].
- Plugging [tex]\(x = 0\)[/tex] into the condition [tex]\( f(x) = x \)[/tex], we have [tex]\( f(0) = 0 \)[/tex]. However, [tex]\(y = -2\)[/tex], so the point [tex]\((0, -2)\)[/tex] does not satisfy [tex]\( f(x) = x \)[/tex].
2. Check the point [tex]\((1, -1)\)[/tex]:
- For this point, [tex]\(x = 1\)[/tex] and [tex]\(y = -1\)[/tex].
- Plugging [tex]\(x = 1\)[/tex] into the condition [tex]\( f(x) = x \)[/tex], we have [tex]\( f(1) = 1 \)[/tex]. However, [tex]\(y = -1\)[/tex], so the point [tex]\((1, -1)\)[/tex] does not satisfy [tex]\( f(x) = x \)[/tex].
Neither of these points meet the criteria where the function [tex]\( f(x) \)[/tex] equals the inverse function [tex]\( f^{-1}(x) \)[/tex] at a given [tex]\(x\)[/tex]-value.
Hence, there is no point of intersection between [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex] among the points provided [tex]\((0, -2)\)[/tex] and [tex]\((1, -1)\)[/tex]. Therefore, the answer is:
[tex]\[
\boxed{\text{None}}
\][/tex]
