The sentence "to begin my first day of school" is a phrase, since it is missing a subject and a verb that agrees with that subject, as seen below.
What is a phrase?
A phrase is a group of words that can be a part of a sentence, but that is not a sentence in itself. Phrases lack some essential components that would form a sentence and convey a complete thought, such as a subject and a verb that agrees with that subject.
That is why "to begin my fist day of school" is a phrase. An independent clause would convey a full thought by itself. A dependent clause would have a subject and a subordinating conjunction. Therefore, we can eliminate those options.
We can conclude, thus, that "to begin my first day of school" is a phrase, as explained above.
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