Answer :
[tex]f(x)=x^2-16\\\\y=x^2-16\\\\x^2=y+16\\\\x=\sqrt{y+16}\\\\f^{-1}(x)=\sqrt{x+16}[/tex]
A quadratic function doesn't have inverse function. But you can find inverse function for each of its "arms".
[tex]f(x)=x^2-16 \Rightarrow x_{vertex}=0\\ y=x^2-16\\ x^2=y+16\\ x=-\sqrt{y+16} \vee x=\sqrt{y+16}\\ f^{-1}(x)=-\sqrt{x+16} \hbox{ for } x<0\hbox{ (left arm)}\\ f^{-1}(x)=\sqrt{x+16} \hbox{ for } x>0\hbox{ (right arm)}[/tex]
[tex]f(x)=x^2-16 \Rightarrow x_{vertex}=0\\ y=x^2-16\\ x^2=y+16\\ x=-\sqrt{y+16} \vee x=\sqrt{y+16}\\ f^{-1}(x)=-\sqrt{x+16} \hbox{ for } x<0\hbox{ (left arm)}\\ f^{-1}(x)=\sqrt{x+16} \hbox{ for } x>0\hbox{ (right arm)}[/tex]
