First off, let me just say that this isn't a rigorous proof. It's more like me trying to establish why the domain of the inverse function = range of the original function.
Second, let me clarify what I want to establish. What I'm worried about, however, is the circularity of it. Namely, I'm worried if I'm being circular in the first step, assuming that any inverse function does the exact opposite as the original function. I just get the nagging feeling that there is something circular about this proof.
Can someone please verify it, and explain to me why it is or isn't circular? Thank you!
The link to the proof is here: https://docs.google.com/document/d/1TBA0ZktRyfltN-4fFBEVI60pxCe33uqRnaUHiNATuGM/edit