One of the most fundamental sentence from the English “phrasebook” that almost every beginner will learn is this sentence, using which one can ask another one’s nationality or country/region of origin. &nbsp;However, this sentence has always troubled me. &nbsp;  The word where is an adverb in this context, and it functions as an adverbial; from is a preposition, and it functions as part of the subjective complement in this sentence. &nbsp;Id est, when responding to this clause, people may say, for example, “I am from Milan.” &nbsp;Now, in that response, I is a pronoun functioning as the subject, and the rest of the sentence, am from Milan, is the predicate, which is comprised of the copula, or the linking verb, am, and the subjective complement, from Milan, which is a spatial adverbial. &nbsp;  What I have learned about English grammar prescribes that an interrogative/relative adverb, such as where or when, cannot follow a preposition, because an adverb alone functions as an adverbial, and the structure “preposition + noun/pronoun” ought to correspond to an adverb. &nbsp;Id est, the phrase from Milan (preposition + noun) can and should correspond to that adverb where, and structures such as from where are illegitimate. &nbsp;  I think it is fairly common that adverbs do not follow prepositions. &nbsp;People always say clauses such as where are you at is grammatically incorrect, because at is a preposition and where is an adverb, so it should simply be where are you. &nbsp;Therefore, what troubles me is that why the phrase where are you at is incorrect and criticized, while I have not seen people avoiding where are you from. &nbsp;  In my humble opinion, the canonical form of this clause should be from which country/region are you, or at least which country/region are you from, but it seems awfully pedantic and there is a highly probable chance that the utterer will get thrashed…
Like many others, I commonly find myself ending a sentence with a preposition.  Yes, it makes me cringe.  I usually rewrite the sentence, but sometimes (in emails) I just live with it.  To, with... you know who you are.  Should I keep fighting myself on this one, or is it okay in some circumstances?
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Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. 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Possible (unanswered) duplicate: Understanding Dirac's notation In The Matrix Reloaded, the Architect reveals to Neo that Trinity is about to die. I can't help but wonder if Neo would still choose the left door if the Architect had withheld that information.  The Architect could have designed a better choice mechanism so as to trick Neo into making a choice in the machine's favor rather than a highly risky one.   Did I miss anything important in the movie? What is the difference between <%# and <%= in ASP.NET MVC? When using a new Windows 8 search bar:    What is the syntax of the search expression? Is it possible to use wildcards like "*", "?" or any other special symbol?  For example, when searching with Google i can use:   '-' (minus) for excluding certain words; "quoted phrase" for exact phrase matching. Is there any way to get the ID of the element that fires an event?  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Sometimes pressing dead area, the phone takes it as menu button  As well as replacing the screen what other methods can I try? Is there any app, launcher or trick to resize the screen usable only to the working area?  P.S. a few hours after leaving it in airplane mode suddenly the touch had this problem but I do not remember if I inadvertently placed something heavy in that area of ​​the cell. Show inline math as if it were display math (and vice versa) Use of would in sentences Prove that if $a \mid b$ then $\gcd(a,b) = |a|$ How to copy files via terminal?