## Domain and Range/Codomain of a Composite Function

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 \[y \ge 1\]

.