To find the point on the graph of the inverse function [tex]\( F^{-1}(x) \)[/tex] when [tex]\((-9, 2)\)[/tex] is on the graph of [tex]\( F(x) \)[/tex], we need to understand the relationship between a function and its inverse.
For any function [tex]\( F(x) \)[/tex], if a point [tex]\((a, b)\)[/tex] lies on its graph, then the point [tex]\((b, a)\)[/tex] lies on the graph of its inverse function [tex]\( F^{-1}(x) \)[/tex]. Essentially, the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates are swapped for the inverse function.
Given the point [tex]\((-9, 2)\)[/tex] on the graph of [tex]\( F(x) \)[/tex], we swap the coordinates:
[tex]\[ (x, y) = (-9, 2) \][/tex]
Swapping the coordinates gives us:
[tex]\[ (y, x) = (2, -9) \][/tex]
This means that the point [tex]\((2, -9)\)[/tex] must be on the graph of the inverse function [tex]\( F^{-1}(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(2, -9)} \][/tex]
So, from the given multiple choices:
- A. [tex]\((2, -9)\)[/tex]
- B. [tex]\((-2, 9)\)[/tex]
- C. [tex]\((9, -2)\)[/tex]
- D. [tex]\((-9, 2)\)[/tex]
The point that must be on the graph of [tex]\( F^{-1}(x) \)[/tex] is choice A, [tex]\((2, -9)\)[/tex].
