Is it possible to find two functions $$f$$ and $$g$$ such that the domain of $$f$$ is not finite, the domain of $$g$$ is finite, $$gof$$ is defined? Justify your answer.

PLACEHOLDER ANSWER - the answer is yes with an example. $$gof$$ only needs the range of $$f$$ to lie inside the domain of $$g$$. For example, $$f \colon \mathbb{R} \rightarrow \{1\}$$ with $$f(x) = 1$$ has an infinite domain, and $$g \colon \{1\} \rightarrow \mathbb{R}$$ with $$g(1) = 5$$ has a finite domain. Then $$gof(x) = 5$$ is defined for all real $$x$$, so it is possible.