Question

Evaluate fog and go f and use the results to determine if g is the inverse function of f.

Answer 1

If
`f,g` are inverses of one another, then

`f\circ g(x) = g\circ f(x) = x`

Given
`f(x)=x^2-1` and
`g(x)=√(x-8)`, then

`f\circ g (x) = f(g(x)) = f(√(x-8)) = \left(√(x-8)\right)^2 - 1 = \boxed{x - 9} \\eq x`

so right away we know
`f` and
`g` are not inverses.

In the other direction, we come to the same conclusion.

`g\circ f(x) = g(f(x)) = g(x^2 - 1) = √((x^2-1)-9) = \boxed{√(x^2 - 9)} \\eq x`