No details of g(x) given and no relationship between F(x) and g(x) given; but domain of F(x) is all values of x where x-3≥0 since square root is only valid for positive numbers (that is, real roots). So x≥3 is domain.
Assuming g(x) is the inverse of F(x) (or F(x) is the inverse of g(x)), we can write g(x)=x^2+3, then g(1)=4, g(7)=52 and g(5)=28. The minimum value of g(x) is 3 for g(0). Also g(x)=g(-x). The domain for g(x) is all x.