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"It is important to grasp the overall meaning of the clause, perhaps putting an answer to it."

I read this sentence (on this forum) by a good writer; I don't understand the use of the participle phrase in bold.

If it were written as below, I think the participle phrase works:

"You should grasp the overall meaning of the clause, perhaps putting an answer to it."

Now, the participle phrase functions adjectivally and modifies the subject 'you.'

But how does the participle phrase in the original work? Does it work in the original sentence?

Also, how is the infinitive phrase 'to grasp...' functioning?

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"It is important to grasp the overall meaning of the clause, perhaps putting an answer to it."
...

"You should grasp the overall meaning of the clause, perhaps putting an answer to it."

But how does the participle phrase in the original work? Does it work in the original sentence?

Also, how is the infinitive phrase 'to grasp...' functioning?

Both are acceptable. In both cases the implied subject of the participial construction is you.

Consider that ... important to grasp the overall meaning ... is really ... important for you to grasp the overall meaning ....

Of course the impersonal one may be used in the paraphrase instead:

"It is important for one to grasp the overall meaning of the clause, perhaps putting an answer to it."

...

"One should grasp the overall meaning of the clause, perhaps putting an answer to it."

CJ

Comments  
I would like to ask a further/related question on this.

Pariciple phrases function adjectivally or adverbially. They are set off from the main clause by comma(s).

Since they function as such, they can modify virtually anything sentence--any noun, pronoun, infinitive phrases, the whole clause, or the verb phrase.

I thought that he was walking home, thinking about what he had done.

This could technically modify either I or he, correct?
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 CalifJim's reply was promoted to an answer.
English 1b3I thought that he was walking home, thinking about what he had done.

This could technically modify either I or he, correct?
As written, it's hard to see it modifying I. You have two clauses there, and your participle phrase is positioned closer to the second clause. But if the participle phrase were first, it would be seen as a modifier of I. There may be cases where the path of modification is more ambiguous, but I don't think this is one of them.

CJ
1) So is it correct to say that a participle phrase can modify any noun/pronoun in a sentence and usually modifies the word to which it is closest?

Example:

I saw the man who was fighiting my friend earlier this week, trying to beat him to death with nothing but a stick and two fists.

2) Does this modify the (relative) pronoun who, (the man)?

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I stood still, deciding whether I needed another wine, since I had already drank three bottles.

3) Does the adverbial phrase in bold modify the verbish aspect of the participle phrase? In other words, do verbals, such as participle phrases, still retain some of their verb like qualities?

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English 1b3 usually modifies the word to which it is closest?
Yes. That's it. Usually. But it's not required. The meaning may point obviously to another interpretation.
English 1b3I saw the man who was fighiting my friend earlier this week, trying to beat him to death with nothing but a stick and two fists.

2) Does this modify the (relative) pronoun who, (the man)?
Yes. The implied subject of the participial is the man. The near parallelism points to this interpretation. man fighting / man trying

man who was fighting / (man who was) trying to beat him

CJ