Inverse of Root

It's more intuitive (at least for most people) if we swap the letters $x$ and $y$ on our relation before starting the calculation, we get $x=\sqrt{y-3}+6$ . We do this because on the inverse function the the roles are interchanged, hence now the task is to clear the variable $x$

$x=\sqrt{y-3}+6 \\ (x-6)^2=y-3 \\ x^2-12x+39=y$

Looking at the first equation (or the graph) it's clear that the smallest value $x$ can take is $6$ when the square root is $0$ , hence $a+b+c=6-12+39=33$