doubt 2

if they are inverse of each other they will intersect only at line y=x

f(x) = f^-1(x)
As the graphs of f(x) and its inverse are symmetric about y=x. So there will be solution to f(x)=g(x) when the graph of f(x) or g(x) cuts the line y=x.
So putting y=x in f(x), we have x²-5x+3=0
Solving for roots we have x = 5±√132
But if we put y=x in g(x) we have 1+x ≥0 So, x ≥-1

Hence one solution of f(x) =g(x) is (5+√13)/2