as you already know, to get the inverse of any expression, we start off by doing a quick switch on the variables, and then solve for y.
[tex] \bf \stackrel{f(x)}{y}=-4\sqrt{x-1}\implies \stackrel{\textit{quick switcheroo}}{\underline{x}=-4\sqrt{\underline{y}-1}}\implies \cfrac{x}{-4}=\sqrt{y-1}\\\\\\\stackrel{\textit{squaring both sides}}{\left( -\cfrac{x}{4} \right)^2=(\sqrt{y-1})^2}\implies \cfrac{x^2}{16}=y-1\implies \cfrac{x^2}{16}+1=\stackrel{f^{-1}(x)}{y} [/tex]
for that inverse to be a function, it has to be a one-to-one function, and that can only happen if it can pass the horizontal and vertical lines tests, and that can only be if x ⩽ 0, check the picture below.
