What is an infinitive verb? In a simple sentence like “Teachers taught…” How would I expand it by beginning with an infinitive verb?

Example: To teach students is the teachers’ main responsibility.
Infinitive is an example of a verbal or fake verb. You can easily identify it in a sentences for it uses the formula to + base form of the verb.* Examples are to see, to talk, to walk, to look, to post. *
For further explanation, let’s have its functions.
##1.## Noun – an infinitive functioning as a noun is the subject of the sentence or what is being talked about.
Example: To walk a mile is one of the challenges I want to do.
Question: What is one of the challenges that I want to do?
Answer: To walk a mile.
*It is usually found at the beginning of the sentence.
##2.## Adjective – when an infinitive describes a subject or something in a sentence; it is mostly found after a noun.
Example: You only have five minutes to think.
Does the sentence have an action word in which you see movement? NONE.*
What word comes before the infinitive to think? FIVE MINUTES
Five minutes is what part of speech? A NOUN*
What word is being described by the word phrase to think? *FIVE MINUTES
Therefore, “to think” in the sentence is an adjective.
##3.## Adverb- modifies or comes AFTER THE VERB.
Example: We live in this world to serve God.
Does the sentence have an action word in which you see movement? YES, IT DOES.*
What word comes before the infinitive to serve? live in this world/ LIVE
“Live” is what part of speech? A VERB*
What word is being described by the word phrase to serve? *LIVE
Therefore, “to serve” in the sentence is an adverb for it comes after the verb meaning IT MODIFIES THE VERB.
Hope it helps.