How do you design and program a Computer Algebra System?

While the subject practically says it all, keep in mind the language that I'd be using to do this is C++ rather than LISP, PERL, or some high level language. What I want to do is to create an object that represents a mathematical function [such as y = f(x) using real variables] where the formula can be specified by the user. I know that computer algebras like AXIOM already exist, but how are they constructed?