Should we worry about what precedes a preposition?

Solution 1:

You have understood one thing correctly: a preposition is combined with what follows it to form a prepositional phrase, and they form a single constituent in the sentence. Generally speaking, words in English govern (ie. control or specify) the words that come after them. In linguistics, we say that English is right-branching, meaning that new syntactic elements come after (to the right of, in writing) the elements that govern them.

Note that there are exceptions, such as adjectives, which precede the nouns that govern them. English is not exclusively right-branching, but it is predominantly right-branching.

But what does this have to do with prepositions? Well, just as a preposition governs the noun phrase that comes to its right, the preposition itself is governed by something to its left. And in many cases, that thing is a verb. English is full of idiomatic combinations of verb + preposition, where the verb requires a specific preposition to follow it, and anything else is an error. To take some obvious examples cribbed from other answers:

I converse with you. [Not to/at/of you]

They rely on the bus. [Not with/to/at the bus]

These combinations are highly idiomatic, meaning that the correct choice of preposition cannot be predicted simply by knowing the general meaning of the words involved. So the people who ask about what preposition follows a certain word are asking a reasonable and intelligent question. The choice of preposition very, very often depends on what came before it.

Solution 2:

I don't really have an "answer" so much as some observations. It's useful in this case, I think, to consider a couple things: First, in English nearly all prepositions can also be used adverbially (as a prepositional adverb). Second, the combination of a verb + prepositional adverb can often form a phrasal verb. The phrasal verbs have a chunk of meaning that's different from the bare verb.

Some examples: you might analyze a sentence like We look after our children as either

1 We look (after (our children)) S - V - PP

or

2 We (look after) (our children) S - Phrasal V - Obj

That is, "look after" is a phrasal verb with a given meaning ("to supervise"). But the sentence can alternatively be viewed as having a prepositional phrase "after the children". It's almost ambiguous. We certainly can't say *We look our children after.

Now compare the following cases which look parallel on the surface:

3 I gave up the keys / I gave the keys up

vs.

4 I walked up the stairs / *I walked the stairs up

In 3, we can use either form with equivalent meaning. So the "up" must not be a preposition, since it can be moved after "the keys". Give up is indeed a phrasal verb meaning "relinquish". But in 4 we don't have that freedom; the "up" must be part of a PP with "the stairs". Walked up is not a phrasal verb, and is just "walked" + "up".

So in short you've got at least 3 cases:

a I gave up the keys / I gave the keys up Phrasal verb, separable parts

b I looked after the children / *I looked the children after Phrasal verb, inseparable parts

c I walked up the stairs / *I walked the stairs up Not a phrasal verb, just V+PP

Solution 3:

No, we should not worry.

You ask "what should precede a preposition?" as if there were some rule that says for example, that prepositions that indicate direction should follow a verb involving motion. Any such rules would be arbitrary at best. At worst, they would be so convoluted (to avoid the inevitable exceptions because, as JSB said, it's a question of idiom) as to be useless.

Solution 4:

The answer is yes. In some cases the correct preposition is determined by the word that follows it: on Monday, in the morning, at Easter, etc. In other cases the correct preposition is determined by the word that precedes it and with which it forms a phrase: interested in (adjectival), to rely on (verbal), remedy for (noun), etc.