How many invertible affine functions $\mathbb{Z_2}\rightarrow \mathbb{Z_2}$

Question

I need to know if there are any non affine invertible functions $\mathbb{Z_2}\rightarrow \mathbb{Z_2}$ and the tip given in the book is to take in consideration how many invertible affine functions are there.

I thought arguing by the fact that there are only two invertible functions $\mathbb{Z_2}\rightarrow \mathbb{Z_2}$, first one being the identity and the second one being the one that maps 0 to 1 and 1 to 0. Both of these being affine, there aren't any non affine invertible functions.

This argument doesn't sound fully complete to me however. How can I know that there aren't others which might fulfill this condition? Would be grateful for some help.

Thanks in advance.

Answer 1

Because it is injective, then $f(0)\neq f(1)$. So for $f(0)$, there is only two possible assigned values $0$ or $1$, once it is assigned to $f(0)$, by the requirement $f(0)\neq f(1)$, $f(1)$ is automatically assigned. So there are only two possible functions.