Answer
$f^{-1}(g(1)) = -1$
Work Step by Step
1. $f^{-1}(g(1))$
$g(1) = 1$
$f^{-1}(1) = -1$
For the inverse, remember that the $x$ and $y$ (In this case, $f(x)$) swaps positions. So $f(x) = 1$ and $x = -1$ is swapped when you have the inverse function.
$f(-1) = 1$ (Given Function)
$f^{-1}(1) = -1$ (Inverse Function)