The phrase to play basketball is an Infinitive Phrase. An infinitive Phrase has verbs as base forms.
The word To is a verb and verbs are action words, and so, the phrase to play basketball is an action word and the phrase in the sentence is an infinitive phrase.
Infinitive phrase is a noun phrase that begins with an infinitive verb.
There are several types of phrases and some are
Therefore, the correct answer is Option C:
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