This word is annoyingly and confusingly used with different meanings in English language teaching.
1. Language functions (also called pragmatic functions) are the things we do with language, such as promising, inviting, giving advice,
asking for permission, etc. Exponents are the actual pieces of language used to express that
function.
2. When looking at a particular grammatical form which can be used in different ways with different meanings, these different meanings or uses are also called functions. It is very common for tenses to have multiple functions, for example:
a) He will get up early when I am trying to have a lie in.
b) He will get up early tomorrow because he has to catch a train at 7.30.
c) I expect he will get up early tomorrow because he's so excited.
In these examples, the future simple: He will get up serves the functions of expressing a) annoying habit/insistence b) future as fact c) prediction.
It is function in this second sense that is referred to when we talk about teaching the function of new language in a grammar or functional language lesson.
3. Grammatical function can refer to either the grammatical category a word in a sentence belongs to (See Unit 4, Part 1 Parts of speech):
Smoking is bad for you. - Smoking is a noun They were smoking. Smoking is a verb.
We had a party on the beach. On is a preposition.
or the role a word or phrase serves in the structure of a clause (see Unit 4, Part 2 Clause structure):
Smoking is bad for you. - Smoking is functioning as a subject.
We had a partyon the beach. - On the beach is a prepositional phrase functioning as an adverbial.