Intransitive or transitive

As the common definition runs, an intransitive verb doesn't need an object as in "I run in the street." But my question is why some verbs are labeled intransitive, and at the same time,they take a preposition followed by an object: For example, agonize plus over/about and then an object:She agonized about what she should do. Why shouldn't we call it a transitive verb? Other examples: apologize to, object to

Solution 1:

An object is a special type of complement of a verb. We can think of a complement as some phrase that fills a special slot set up by another word or phrase. In the phrase:

• apologise to someone

We have a verb apologise, which selects a preposition phrase as a complement. The preposition phrase here is to someone. This preposition phrase has an internal structure, it has a preposition as its head, to and this preposition has its own complement, the noun phrase someone. The word someone here doesn't have a special relationship with the verb, it has a relationship with the preposition. It is the preposition heading the preposition phrase which has a special relationship with the verb apologise. So because someone isn't a complement of the verb here, it cannot be considered a direct object.

Solution 2:

In grammar, “transitive” means that the verb takes a direct object, not just an object. An object introduced by a preposition is not direct.