On $1$-way functions and formalizing my intuitions about them
What would a rigorous proof look like?  It would start with the assumptions that are given, and derive the conclusion you want to derive, with each step being logically and rigorously justified.

How should you approach this?  Start with the definitions.  If you know what the definition of a 1-way function is, then you know what you have to prove, to prove that $a \circ b$ is a 1-way function.  Start by trying to prove that, using the assumptions you are given.  You'll probably use a reduction at some point: if there exists an adversary against $a \circ b$, then....

This is a basic question, which suggests to me you should either review the material in your textbook on reduction-based proofs of security, or check whether you have the mathematical maturity for this topic.