Function
It’s a reason why we communicate with each other. E.g. Apologising; Greeting; clarifying, making a suggestion etc.
Exponent
It’s the language we use to express a function or functions (depending on the context). E. g. I’m so tired or Thank you so much for this.
Besides an exponent has different levels of formality, from informal to neutral and formal. E.g. Lunch?——> We wouuld be very pleased if you could join us for lunch.
Appropriacy
The act of CHOOSING the level of formality which suits a particular situation.
Register
It´s the LANGUAGE that reflects the situation in which it’s used. E.g. Hi and A very good morning to you.