Answer
The domain of both $(f \circ g) (x)$ and $(g \circ f)(x)$ is $(-\infty,\infty)$
Work Step by Step
We are given:
$f(x)=2^x$
$g(x)=x+1$
$(f \circ g) (x)=f(g(x))=2^{x+1}$
$(g \circ f)(x)=g(f(x))=2^x+1$
The domain of both $(f \circ g) (x)$ and $(g \circ f)(x)$ is $(-\infty,\infty)$