Hi Joshua,

Suppose the domain of $g$ is $D.$ If $g(x)$ is to exist then it must be that $x$ is in $D.$ But if $f(g(x))$ is also to exist $g(x)$ must be in the domain of $f.$

To be more explicit let $E$ be the set of all $t$ so that there is an $s$ in $D$ so that $t = g(s),$ that is

\[E = \left\{t | \exists \; s \in D \mbox{ so that } t = g(s) \right\}.\]

The domain of $f(g(x))$ is all $x$ in both $D$ and $E,$ that is

\[\mbox{ the domain of f(g(x)) is } D \cap E.\]

Penny