When dealing with functions and their inverses, it is important to understand that for any given point [tex]\((a, b)\)[/tex] on the function [tex]\(f(x)\)[/tex], the coordinates swap roles when considering the inverse function [tex]\(f^{-1}(x)\)[/tex]. That means for any point [tex]\((a, b)\)[/tex] on [tex]\(f(x)\)[/tex], the corresponding point on [tex]\(f^{-1}(x)\)[/tex] will be [tex]\((b, a)\)[/tex].
Given that the point [tex]\((5, -7)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex], we need to find the corresponding point on the inverse function [tex]\(f^{-1}(x)\)[/tex].
1. Identify the coordinates of the given point on [tex]\(f(x)\)[/tex]:
[tex]\[
(a, b) = (5, -7)
\][/tex]
2. Swap the coordinates to find the corresponding point on [tex]\(f^{-1}(x)\)[/tex]:
[tex]\[
(b, a) = (-7, 5)
\][/tex]
Thus, the point [tex]\((-7, 5)\)[/tex] must be a point on the graph of the inverse function [tex]\(f^{-1}(x)\)[/tex].
