Question 1208001
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If f (x) = 1/(1 - x) , then (f(f(f(f...f)(sqrt2),(45 times) = ...., 

A) 0, 

B) (2 - sqrt2)/2, 

C) (2 + sqrt2)/2, 

D) 1, 

E) sqrt2.
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<pre>
If f(x) = {{{1/(1-x)}}},  


then  f(f(x)) = {{{1/(1-1/(1-x))}}} = {{{(1-x)/((1-x)-1)}}} = {{{(x-1)/x}}},


then  f(f(f(x))) = {{{1/(1-(x-1)/x)}}} = {{{x/(x-x+1)}}} = x.


Thus, applying function f to any real number x =/= 1,  x =/= 0  three times, we get x again.



In other words,  f(f(f(x))) == x identically, for all real x =/= 1,  x =/= 0.



So, for example,  {{{f(f(f(sqrt(2))))}}} = {{{sqrt(2)}}};  {{{f(f(f(sqrt(3))))}}} = {{{sqrt(3)}}};  {{{f(f(f(sqrt(5))))}}} = {{{sqrt(5)}}};  {{{f(f(f(sqrt(7))))}}} = {{{sqrt(7)}}};  

                  {{{f(f(f(root(3,2))))}}} = {{{root(3,2)}}};  {{{f(f(f(root(3,3))))}}} = {{{root(3,3)}}};  {{{f(f(f(root(3,5))))}}} = {{{root(3,5)}}};  {{{f(f(f(root(3,7))))}}} = {{{root(3,7)}}},  and so on.



Since 45 is a multiple of 3,  f applied to  {{{sqrt(2)}}}  45 times is  {{{sqrt(2)}}};

                              f applied to  {{{sqrt(3)}}}  45 times is  {{{sqrt(3)}}};

                              f applied to  {{{sqrt(5)}}}  45 times is  {{{sqrt(5)}}};

                              f applied to  {{{sqrt(7)}}}  45 times is  {{{sqrt(7)}}};


                              f applied to  {{{root(3,2)}}}  45 times is  {{{root(3,2))}}};

                              f applied to  {{{root(3,3)}}}  45 times is  {{{root(3,3))}}};

                              f applied to  {{{root(3,5)}}}  45 times is  {{{root(3,5))}}};

                              f applied to  {{{root(3,7)}}}  45 times is  {{{root(3,7))}}},

and so on.
</pre>

Solved and significantly expanded.



For example,  &nbsp;f applied &nbsp;2025 &nbsp;times to the number &nbsp;{{{root(2025,2025))}}} &nbsp;is &nbsp;{{{root(2025,2025)}}}.



Similarly,  &nbsp;f &nbsp;applied &nbsp;2025 &nbsp;times to the number &nbsp;{{{2025^2025}}} &nbsp;is &nbsp;{{{2025^2025}}}.



As well as &nbsp;f &nbsp;applied &nbsp;2025 &nbsp;times to the number &nbsp;2025! &nbsp;is  &nbsp;2025! , again.



You can easily construct a million other examples.