a)Having had a good season, they hoped to win the trophy.
having had a good season: adjectival participial phrase right
to win the trophy: infinitival noun phrase, direct object right
b)In the last few weeks, they have often been seen going to the theatre together.
In the last few weeks : adverbial prepositional phrase right
going to the theatre together: adjectival participial phrase right
c)The car parked in the garage is not for rent
parked in the garage: adjectival participial phrase right
for rent: adverbial prepositional phrase It's a prepositional phrase, all right, but I'm not sure about "adverbial."
d)Rejecting their offer, she said she was no longer interested in doing business with them
Rejecting their offer: adverbial participial phrase right
doing business: gerund, subject complement gerund, yes; but you'll need to explain the rest to me
with them: adverbial prepositional phrase right
e)To be honest, I donít want to take extra work at the moment
to be honest: infinitival adverb phrase right
to take extra work: infinitival noun phrase, direct object right
at the moment: adjectival prepositional phrase prepositional phrase, yes; please explain how it's adverbial
f)Eating and Drinking was forbidden during the performance.
Eating and Drinking: gerund, subject right
during the performance: adjectival prepositional phrase right
g)During the good weather, it is hard to stay indoors and study
During the good weather: adjectival prepositional phrase right
to stay indoors and study: infinitival extraposed noun phrase subject right
h)Bored with the game, he walked to the exit whistling loudly
Bored with the game: adverbial participial phrase right
whistling loudly: adjectival participial phrase right
i)If it continues to rain like this, we will have to leave
If it continues: adverbial prepositional phrase adverbial, yes; but not a prepositional phrase
to rain like this: infinitival noun phrase, direct object right
to leave: infinitival noun phrase, direct object right, I think
j)The way to lose weight is to exercise a lot.
to lose weight: infinitival adjective phrase, direct object no adjective (word or phrase) will be a direct object
to exercise a lot: infinitival noun phrase, subject complement right
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