The domain of
\[f(g(x))\]
is the domain of \[g(x)\]
and the range of \[f(g(x))\]
is restricted to \[f(Range \: of \: g(x))\]
.Example:
\[h(x)=e^{\sqrt{x-1}}\]
is the composition \[h(x)=f(g(x))\]
of the two functions \[g(x)=\sqrt{x-1}\]
and \[f(x)=e^x\]
.The domain of
\[g(x)=\sqrt{x-1}\]
(and of \[f(g(x))=e^{\sqrt{x-1}}\]
) is \[x \ge 1\]
and the range is the set of values taken by \[e^x\]
with \[x \ge 0\]
i.e. the range is \[y \ge 1\]
.