## 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
$y \ge 1$
.