Evaluating a Complex Function

Algebra Level 4

Consider the function f ( x ) = 1 1 − x 2 f(x)=\frac {1} {\sqrt {1-x^2} } which is defined for all complex numbers apart from 1 and -1. f ( 2011 ) ( 1 5 ) f^{(2011)} ( \frac {1} {5} ) has the form a b c \frac {a\sqrt{b}} {c} , where a a and c c are coprime integers and b b is not divisible by the square of any prime. What is the value of a + b + c a + b +c ?

Details and assumptions

f ( n ) ( x ) f^{(n)} (x) means that f f is applied n n times, e.g. f ( 2 ) ( x ) = f ∘ f ( x ) = f ( f ( x ) ) f^{(2)} (x) = f \circ f (x) = f( f( x) )


The answer is 23.

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Calvin Lin by Calvin Lin

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1 solution

Nikhil Kashyap
Mar 18, 2014

f ( 2 ) ( x ) = 1 − x 2 − x f ( 3 ) ( x ) = x f ( 2011 ) ( x ) = f ( 2011 − 3 × 670 ) ( x ) = f ( x ) f ( 1 5 ) = 5 6 12 a = 5 , b = 6 , c = 12 a + b + c = 5 + 6 + 12 = 23 f^{ \left( 2 \right) }\left( x \right) =\sqrt { \frac { 1-{ x }^{ 2 } }{ -x } } \\ f^{ \left( 3 \right) }\left( x \right) =x\\ f^{ \left( 2011 \right) }\left( x \right) =f^{ \left( 2011-3\times 670 \right) }\left( x \right) =f\left( x \right) \\ f\left( \frac { 1 }{ 5 } \right) =\frac { 5\sqrt { 6 } }{ 12 } \\ a=5,b=6,c=12\\ a+b+c=5+6+12=\boxed { 23 }

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can you explain me the first line?? f2x= sqrt 1-x^2/-x??

madhav gupta - 6 years, 4 months ago

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