#### Answer

$fog=\sqrt {x+1}$
The domain of $fog$: {x: x$\geq$-1}
$gof=\sqrt x +1$
The domain of $gof$: {x: x$\geq$0}

#### Work Step by Step

$fog=f(g(x))=f(x+1)=\sqrt {x+1}$
Domain of g is the set of all reals and that of f is all non-negative reals. This implies that x+1 should be non-negative.
$\implies\,\,$ x should be greater than or equal to -1.
That is, the domain of $fog$ is the set {x: x$\geq$-1}
$gof=g(f(x))=g(\sqrt x)=\sqrt x +1$.
As the domain of g is the set of all real numbers, the domain of $gof$ is simply the domain of f and that is {x: x$\geq$0}