The て form as auxiliary and giving verbs
On top of just connecting two statements into a single sentence, the て form can also act as an auxiliary. When <verb1>て<verb2> is used this way, the second verb modifies the core meaning of the first. Usually the second verb has an abstract and often purely grammatical meaning.
殺してあげる I’ll do you the favor of killing you
In this example, the verb あげる attaches to the て form of 殺す. あげる has a few meanings, two of them being "to raise (something)" and “to give (something to someone)”. Because it is used in this pattern, it assumes the meaning of "giving" the action 殺す to someone else. It can have the nuance of doing it as some kind of favor, sometimes ironically or sarcastically (as in the example above).
助けてもらう To receive help (directed towards yourself)
Here, the "receive" word, もらう, attaches to 助ける. This addition of てもらう can have a nuance of gratefulness as the subject receives some kind of favor. However, this is not only what is happening here. The phrase 私が助ける means that subject 私 is the one that performs the action of “saving/helping” (someone else). However, the phrase 私が助けてもらう marks the subject 私 as the one that receives the action of being helped. The XてY pattern isn't just attaching the verb Y to an earlier statement, it can make a whole new verb phrase, with different grammar and meaning.
This usage of て auxiliary can be very powerful, and it is at the core of what makes some of these grammatical structures so typically Japanese that are often hard to translate into English.
On top of てあげる and てもらう there is also a third verb, てくれる which completes the giving/receiving “circle”:
<verb>てあげる You/Someone does the favor of <verb> towards someone else <verb>てもらう You receive the favor of <verb> from someone else <verb>てくれる Someone else does the favor of <verb> towards you
It is important to keep in mind the difference between the “outside” group and the “inside” group, but it is too complicated to go over it right now. Just remember that てもらう is almost always used with you as the subject, and you receive a favor from someone else. Likewise, in the opposite direction, てくれる has someone else as subject and they do a favor towards you the recipient. Note the particles in the following examples:
父に(私が)助けてもらった (I) was saved by my father 父が(私に)助けてくれた My father saved (me)
The parts in parenthesis are often dropped or removed because just by using てくれる and てもらう, Japanese can imply a “direction” of the verb which makes it obvious whether the action is towards you or not. In English we say “He saved me” but in Japanese we say “彼が助けてくれた”. The “me” part is implied in てくれる.
We will later see other usages of て auxiliary to add a sense of directionality to phrases like ていく and てくる.
For verbs and adjectives in ない form, there is another version of the て form that is often used with an “auxiliary” nuance. In this form, rather than なくて, it becomes ないで instead.
食べないで待つ To wait while not eating
Here, 食べないで is the て auxiliary of 食べない and can be used to describe the following verb (待つ). It acts somewhat adverbially to add additional meaning and context to the last verb in the chain. We will see more examples of ないで in later lessons, just be aware it exists for now.