Answer
$g^{-1}~o~f^{-1}=\frac{1}{2}x+\frac{1}{2}$
Work Step by Step
$f(x)=x+4$
$y=x+4~~$ (Interchange $x$ and $y$):
$x=y+4$
$y=x-4$
$f^{-1}(x)=x-4$
$g(x)=2x-5$
$y=2x-5~~$ (Interchange $x$ and $y$):
$x=2y-5$
$x+5=2y$
$y=\frac{1}{2}x+\frac{5}{2}$
$g^{-1}(x)=\frac{1}{2}x+\frac{5}{2}$
$g^{-1}~o~f^{-1}=g^{-1}(f^{-1})=g^{-1}(x-4)=\frac{1}{2}(x-4)+\frac{5}{2}=\frac{1}{2}x-2+\frac{5}{2}=\frac{1}{2}x+\frac{1}{2}$
