Function and One to One function

Question

Do one-to-one functions have to comply to the rules of a function? Does this explain why y^2=x is not a part of the one-to-one function? And why do only one-to-one functions are inverse functions?

Answer 1

Read the definitions carefully. Yes, a one-to-one function has to comply with the rules of a function. One-to-one is and adjective modifying function, so you start with a function and impose the requirement of one-to-one. Only one-to-one functions have inverses because if there are $a\neq b$ with $f(a)=f(b)$ (so the function is not one-to-one) you don't have a single answer for $f^{-1}(f(a))=a$ because it could be $f^{-1}(f(b))=b$