Does the object include the preposition or not?

Question

Don’t be afraid of making mistakes in speaking English.

Is this the structure of
[verb phrase: don’t be afraid of][object: making mistakes] or
[verb phrase: don’t be afraid][object: of making mistakes]?

Answer 1

The idiom be afraid of is ambiguous, and how you parse it depends on what you want to do with it.

1. One approach is to treat the entire phrase as a ‘phrasal verb’ equivalent to the lexical verb fear:

   Don't fear [making mistakes]

   On this reading, the bracketed non-finite clause is the Direct Object of the verb. This reading is supported by recasting the sentence as a question:

   What do you fear?
   What are you afraid of?

2. Another, very traditional, approach is to treat be as the ordinary ‘linking verb’ (copula), with afraid as its subject complement. This is clearer if we recast the sentence from the imperative into the indicative mood:

   You are afraid [of making mistakes].

   On this reading, of making mistakes is a prepositional phrase modifying afraid. This reading is supported by replacing afraid with similar adjectives; note that in each the prepositional phrase can be deleted and a well-formed sentence remains.

   You are frightened [of making mistakes]. You are proud [of making mistakes].
   You are sad [at making mistakes].
   You are embarrassed [at making mistakes].

3. Note, however, that some of those adjectives take a different preposition. This leads me to suggest that a third approach is possible here. Afraid of is a ‘phrasal adjective’ which differs from the base lexical adjective in that it takes an object representing its agent:

   You are afraid of [making mistakes] = Making mistakes makes you afraid.
   You are proud of [making mistakes] = Making mistakes makes you proud.
   You are sad at [making mistakes] = Making mistakes makes you sad.
   You are embarrassed at [making mistakes] = Making mistakes makes you embarassed.

   This is in effect to see such ‘phrasal adjectives’ as equivalent to past participles of active verbs—which is in fact the case with embarassed! Don't be afraid of making mistakes is then the equivalent of:

   Don't be frightened by making mistakes.

Answer 2

Grammatically, object is a, "noun or noun phrase governed by an active transitive verb or by a preposition."
It doesn't include the preposition.