Why do we say “he let her down” but not “he looked her after”? It has nothing to do with the fact that the first sentence expresses something negative and the second one is positive.
There are in fact two types of phrasal verbs, the so called prepositional phrasal verbs, e.g. “look after someone”, and particle phrasal verbs, e.g. “let someone down”. The terminology here is somewhat unfortunate, and in fact, linguists usually do not call the first type “phrasal verbs”. The construction
exists in all European languages. For example:
Er glaubt an Gott. (German)
Il croit en Dieu. (French)
As you can see, there’s nothing special about this kind of verbs. Prepositions always precede their objects; nobody is surprised we cannot say “he believes God in”.
True phrasal verbs
Then there are true phrasal verbs, such as “let down”, where “down” is not a preposition but a particle which belongs to the verb, not to the object. To see the difference, consider the following two examples:
Down whom does he let? Down her.
The latter sentence does not make sense. We cannot say “down whom” and “down her”, because the inseparable pair is “let down” not “down somebody”.
If a particle phrasal verb has an object, it can, in theory, always be put between the verb and the particle:
She has to hand it over.
He took his hat off.
In practice, however, when the object phrase is longer, it sounds usually awkward when the verb and its particle are split:
He let down his mother who has been ill for a long time. (better)
When the object consists of a single pronoun (e.g. “me”, “him”, “her”, “them”), it always goes between the verb and the particle:
How to recognize a true phrasal verb?
So far, so good, but the real problem is how to recognize when a word after a verb is a particle and when it is a preposition. This is an easy task for a native speaker: If you can put an object in between, as in “take off → take it off”, or there cannot be any object at all, as in “fall him apart / fall apart him”, it is a particle. In all other cases, it is a preposition.
Non-native speakers often don’t have the luxury of being able to feel whether you can or cannot do that. Sometimes the distinction is easy; for example “out” in “check out” can only be a particle because “out” is never a preposition, in any context (we would say “out of something”, not “out something”). However, in most cases, non-native speakers just have to learn by heart whether or not a particle/preposition can be separated from a particular verb.
Sometimes the distinction even changes the meaning of the whole phrase. For example, you can “look something up” in the sense of searching and finding it, but you can also “look up something” in the sense of directing your eyes upwards in its direction, for example:
He looked up the stairs. (he looked upwards at the top of the stairs)