what is the gerund phrase and the noun function of the gerund in the sentence

The gerund phrase is "entertaining friends." It functions as the predicate nominative.

thank you ms.sue

i have one more question

what is the function of the italicized infinitive phrase ins the sentence.

1.I want to go with you to the store.
2.May I have permission to use the car tonight?

1. "To go" is the direct object of the verb.

I'll be glad to check your answer for the second sentence.

okay and the answer can only be a noun.adverb or adjective.

Direct objects are nouns.

okay and the second one is i have permission and its a adverb?

No. "I have permission" is not an infinitive phrase.

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

so it is to use and it would be a adverb

2. Yes, the infinitive is "to use." It's an adjective modifying permission.

thank you and for this one: Meredith is to be restored as captain.the infinitive is to be and is a noun?