If [tex]f,g[/tex] are inverses of one another, then
[tex]f\circ g(x) = g\circ f(x) = x[/tex]
Given [tex]f(x)=x^2-1[/tex] and [tex]g(x)=\sqrt{x-8}[/tex], then
[tex]f\circ g (x) = f(g(x)) = f(\sqrt{x-8}) = \left(\sqrt{x-8}\right)^2 - 1 = \boxed{x - 9} \neq x[/tex]
so right away we know [tex]f[/tex] and [tex]g[/tex] are not inverses.
In the other direction, we come to the same conclusion.
[tex]g\circ f(x) = g(f(x)) = g(x^2 - 1) = \sqrt{(x^2-1)-9} = \boxed{\sqrt{x^2 - 9}} \neq x[/tex]
