Syntactic Functions of Clauses

Solution 1:

1. The fact that Lionel writes fast cannot be true.

2. Jonny became happy that they were finally leaving

3. Everybody expected we would win the match.

4. The house, if care is not taken, will be demolished.

The first three are all tensed that-clauses, complements of one sort or another. The last is adverbial.

In particular,

• (1) is a Noun Phrase Complement, a construction only possible with nouns like fact, story, rumor, or claim that refer to propositions instead of things. It modifies The fact, and forms a common idiomatic phrase the fact that which serves billions as a hesitation phrase when they haven't completely figured out the next thing they're going to say. This works often enough because the fact means essentially nothing -- if you are telling the truth, of course it's a fact -- and only comments on its factness. That means you can put it in front of almost anything without changing the meaning.

  Most complement clauses are noun clauses like (2) and (3), but if you want to call (1) an adjective clause because it modifies a noun, go ahead. It doesn't function as the subject, or in apposition to the subject, though -- the noun phrase beginning with the fact and containing the complement clause is the subject.

    (note, incidentally, introductory that is obligatory in noun complement clauses;

  • *The fact Lionel writes fast cannot be true.

• (2) is a that-complement, with optional that, of the predicate adjective happy. Predicates like happy, upset, scared, convinced that refer to mental or emotional states can take complement clauses indicating the cause of those states, just the way nouns like story can take complements indicating their content.

  (2) is grammatical without that, and works with was or got instead of became:

  • Jonny became/got/was happy they were finally leaving.

• (3) is a straightforward that-complement (without the optional that in this case),
  it's a noun clause because it functions as the complete direct object of expected.

  • What did everybody expect? That we would win the match.
    (That is required when such a complement clause begins a sentence.)

• Finally, (4) is an adverbial clause, introduced by if, and set off with commas in writing and intonations in speech, because like most adverbials it can occur in several different positions, all grammatical:

  • The house, if care is not taken, will be demolished.
  • The house will be demolished, if care is not taken.
  • If care is not taken, the house will be demolished.

Solution 2:

You seem to be confusing types of clauses with their syntactic functions.

1. Noun clause in apposition.
2. Adverbial clause of reason complementizing the adjective "happy".
3. Noun clause, direct object.
4. Adverbial clause of condition.