Depending on their context, certain words can belong to more than one part of speech. It is necessary to determine the function of these words to classify them properly.
The word "at" is classified as a preposition. However, in the phrasal verb "look at", it is best classified as a part of the verb or verb particle. The functions of "at" in the prepositional phrase "at three o'clock" and in the phrasal verb "Look at my new shoes" are clearly different.
In the case of "after", it can be classified as a preposition, conjunction and adverb. Consider the following examples:
1) It started to rain after I had arrived.
2) You can go after me.
3) You can go after.
In the first sentence, "after" is a conjunction. The dependent clause "after I had arrived" is connected to the independent clause "It started to rain". The dependent clause modifies the verb "rain" and provides more information about the time that it started to rain.
In the second example, "after" is a preposition. It expresses the relationship between the two pronouns in the sentence. The pronoun "me" is an object of the preposition.
In the third sentence, "after" is an adverb. The meaning is similar to that of "later". It modifies the verb "go".
The linguist Dr. Geoffrey Pullum has suggested that the word "after" in both the second and third examples be analyzed as a preposition. He explains that the second example can be analyzed as a transitive preposition and the third one as an intransitive.
This parallels with verbs which can be transitive and intransitive. For example, the verb "walk" is transitive in "I walk my dog every day" but intransitive in "I walk every day". If one adopts the idea of transitive and intransitive prepositions, the word "after" can only be a preposition or a conjunction.
However, many grammarians believe that only verbs should be analyzed as transitive and intransitive. They argue that prepositions always take objects and therefore cannot be considered intransitive.
The idea of transitive and intransitive prepositions is unlikely to be adopted by traditional grammarians. However, it illustrates that the analysis of parts of speech and also grammar can vary from one individual to another. In any case, it is clear that words such as "after" which can be classified into three different parts of speech in traditional grammar nevertheless share much in common.