SOLUTION: Let f(x) = {x - sqrt{3}}/{xsqrt{3} + 1}. What is f^{2012}(x), where the function is being applied 2012 times? This notation indicates repeated composition of functions, not exp

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Question 1178101: Let
f(x) = {x - sqrt{3}}/{xsqrt{3} + 1}. What is f^{2012}(x), where the function is being applied 2012 times?
This notation indicates repeated composition of functions, not exponentiation of functions. For example,f^2(x)=f(f(x))and not f(x) * f(x). Similarly,f^3(x)=f(f(f(x))).
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

https://www.quora.com/If-f-x-frac-x-sqrt-3-x-sqrt-3-+-1-What-is-f-2012-x
Answer by ikleyn(52879)   (Show Source): You can put this solution on YOUR website!
.
Let f(x) = {x - sqrt{3}}/{xsqrt{3} + 1}.
What is f^{2012}(x), where the function is being applied 2012 times?
This notation indicates repeated composition of functions, not exponentiation of functions.
For example,f^2(x)=f(f(x)) and not f(x) * f(x). Similarly,f^3(x)=f(f(f(x))).
~~~~~~~~~~~~~


            This problem is very original --- I never saw such or similar problems before in my practice.

            It allows absolutely unexpected and elegant solution. 


Let  x = tan(a)  (which means "let a = arctan(x)").  Then


    f(x) =  =  = .


It means that taking f(x) for x= tan(a) returns .


Obviously, that taking the composition (fof)(x) = f(f(x))      will return  ;

                taking the composition (fofof)(x) = f(f(f(x))) will return  ;


                . . . . and so on . . . 


                taking the composition f^{2012}(x) = f(f(f...f(x)))...)  will return  .


It easy to calculate:   =  + ;   THEREFORE  


     =  =  =  =  = .    ANSWER



ANSWER.  f^{2012}(x) = .


Solved.


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Without any doubts,  it is a  FULL  SCALE  Math  OLYMPIAD  level problem  (!)



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