The gerund phrase functions as the direct object of the verb appreciate.

my (possessive pronoun adjective form, modifying the gerund)

offering (gerund)

you (indirect object of action expressed in gerund)

this opportunity (direct object of action expressed in gerund) The gerund phrase functions as the subject complement.

lying to (gerund)

his constituents (direct object of action expressed in gerund) The gerund phrase functions as the object of the preposition for.

faking (gerund)

an illness (direct object of action expressed in gerund)

to avoid work (infinitive phrase as adverb) The gerund phrase functions as the subject of the sentence.

Being (gerund)

the boss (subject complement for Jeff, via state of being expressed in gerund)