what is the Gerund phrase and Noun function in these sentences?

1)Writing a research paper sounds difficult.
2)He must like studying calculus.
3)Her favorite pastime is entertaining friends.
4)Water skiing, sailing, and swimming are my favorite summer sports.
5)I enjoy playing the piano.
6)She is successful in mimicking others' voices.
7)Writing a research paper sounds difficult.

In English, verbs with -ing endings can serve as gerunds, participles, or main verbs.

Look for the -ing verb in each sentence and then decide on the words that go with it. Then look at what that whole phrase does in the sentence.

I'll get you started:

1. gerund = Writing
gerund phrase = Writing a research paper
function in the sentence = subject (Ask yourself, "What sounds difficult?")

Let us know what you decide about the others.

I'll get you started.

1) "Writing a research paper" is the gerund phrase and functions as the subject.

I'll be glad to check your answers for the rest.

so in the second one 2)Brett earns his income by repairing cars.

the gerund phrase is =by repairing cars
and the function is= direct object?

No.

Repairing cars is the gerund phrase but it is not the direct object.

would it be the subject?

cause i don't know this stuff that much

No, it's not the subject.

Please study your text materials closely.

This site should also help you.

http://www.chompchomp.com/terms/gerundphrase.htm