Hello & Goodbye

Being words that are used a lot in daily conversation, the words for hello and goodbye are shortened a lot.

Originally, exclamations like

Ἁeιc καᴧεн ευтιεн - May you (sg) have a good day

or

Cου καᴧυ мεнeι ευтιε - May your day be good

could be used either to begin or end a conversation.
The latter would be shortened to

Καᴧυ ευтιε - good day

or with less emphasis on good:

Єυтιε καᴧυ - good day

Which is interchangeable with adding the prefix ευ- to καᴧυ:

Єυκαᴧυ - goodday

And finally, children and young adults often just say:

Καᴧυ - day; hi, hey, hello, bye

The same happens with

Ιeιc ἁοнтιι тυxεн ευтιεн - May you go having good fate/fortune/luck

Ιeιc (cυн) тυxει ευтιει - May you go (with) good luck

Both meaning "bye".
This would lead to

Ιeιc ευтυxει - May you go goodfortuned

Deformalised to

Ιтι ευтυxει - Go goodfortuned

Finally

Єυтυxει - goodfortune; goodbye

And ofcourse

Тυxει - with fate; bye

Obviously, there are many more ways of expressing greetings or good luck, but these are all formed in a similar way.