The domain of a composite function [tex]f(g(x))[/tex] is, technically speaking, the set of those inputs [tex]x[/tex] in the domain of [tex]g[/tex] for which [tex] g(x)[/tex] is in the domain of [tex]f[/tex]. What this is really saying, though, is that the domain is for those [tex]x[/tex] that overlap in both functions, or that are defined for both [tex]f(x)[/tex] and [tex]g(x)[/tex]. You can write {[tex]x \ | \ D_f[/tex] ∩ [tex]D_g[/tex]}.
For this specific example, the domain of [tex]f(x)[/tex] is {all real numbers [tex]x[/tex]}. The domain of [tex]g(x)[/tex] is {[tex]x \ | \ x \neq 13[/tex]}. These overlap for all [tex]x[/tex] such that [tex]x \neq 13[/tex], so the domain of [tex]f(g(x))[/tex] is {[tex]x \ | \ x \neq 13[/tex]}.
