Answer
$\frac{x+1}{2} $
Work Step by Step
To find the inverses, we replace $f(x)$ by $x$ and $ x$ by $ f^{-1}(x)$ and solve for $ f^{-1}(x)$.
Thus $ x = f^{-1}(x)+4 ⇒ f^{-1}(x) = x-4$
And $x= 2g^{-1}(x)-5⇒ g^{-1}(x) = \frac{x+5}{2} $
To find $(f\circ g)^{-1}$ we can either calculate $fog$ and find its inverse or we can use the relation:
$(f\circ g)^{-1} = g^{-1}\circ f^{-1}$
Let's do it by the second method
So, $ (f\circ g)^{-1} =(g^{-1} \circ f^{-1})(x) = g^{-1}(f^{-1}(x)) = g^{-1}(x-4) = \frac{(x-4)+5}{2} = \frac{x+1}{2} $
