What is the function of the adverb clause in the sentence: since you're getting up early, can you take the dog for a walk in the morning?

It answers the question why and modifies the verb take.

Explanation:

An adverb clause is a type of dependent clauses that represents a group of words that functions as an adverb. Unlike independent clauses, dependent clauses do not express a complete thought, so they can't stand alone, as without the rest of the sentence they don't mean anything.

The adverb clause in the given sentence is Since you're getting up early. Adverb clauses modify verbs, providing additional information about when, where, why, how, how much or under what condition the action expressed by the verb takes place. The clause from our example modifies the verb take and is an adverb clause of reason, informing us about why the action described by the verb is taken. Adverb clauses of reason usually begin with conjunctions such as because, since, as, and that.