Domain and Range/Codomain of a Composite Function

The domain and range of a composite function can often be deduced from the functions of which which that function is composed.
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  
\[y \ge 1\]
.