***** NOT A TEACHER *****
I have a few ideas (not "answers") to share with you.
1. Your sentence contains the prepositional phrase "on purpose." So do you think that "to bother" in your sentence is maybe an infinitive of purpose?
2. Let's delete (remove) the prepositional phrase. We get: "She made a noise only to bother me at work." Maybe we could also express that idea like:
a. At work, she made a noise only to bother me.
b. At work, she only made a noise to bother me.
c. At work, she made a noise only in order to bother me.
d. At work, she only made a noise in order to bother me.
I did some googling, and found this sentence: "Her sinuses would feel better, only to bother her again a few weeks later." (Source: desertspringshospital.com/health)
In my opinion, the purpose of the sinuses feeling better was NOT to bother her again later. So can we say that "to bother" in that sentence is NOT an infinitive of purpose?
Here is a good example of an infinitiive of result: "Thema looked up suddenly to find a cat on the chair." (Source: Bruce Liles, A Basic Grammar of Modern English, 1979).
Only my ideas: (1) I do not think that Thema looked up with the purpose of finding a cat. In fact, it was a surprise. (2) Some people, I think, would add "only" in front of the word "to." They might also add a comma after "suddenly." This pause would further indicate that a clause of result was coming.
One more example of an infinitive of result: "He raised our expectations only to disappoint them." (Source: Otto Jespersen, Essentials of English Grammar (1933).
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