The answer that you have is correct.
You would complete the square using +9
g(x)= (x-3)2 +7-9 This is done to keep it balanced
g(x)= (x-3)2 +2
Switch the G(x) to y and reserve that x and y
x= (y-3)2+2 Now solve for y
y= Squrt (x+2) +3
The domain is limited by the square rt. It can't be <0 So the domain is x< and = to -2.
You did a great job