, rationalizing the denominator gives 

, simplifying and rationalizing the denominator gives 

, simplifying and rationalizing the denominator gives 

And we're back where we started, at 

So we conclude:

When there are 0 f's, the answer is 
When there is 1 f, the answer is 
When there are 2 f's, the answer is 
When there are 3 f's, the answer is 
When there is 4 f's, the answer is 
When there are 5 f's, the answer is 
When there are 6 f's, the answer is 

It keeps cycling around through those 3 values.

So we conclude that when there is a multiple of 3 f's, the
answer is 

Since 45 is a multiple of 3, the answer is .

Edwin

If f(x) = ,  


then  f(f(x)) =  =  = ,


then  f(f(f(x))) =  =  = 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,   = ;   = ;   = ;   = ;  

                   = ;   = ;   = ;   = ,  and so on.



Since 45 is a multiple of 3,  f applied to    45 times is  ;

                              f applied to    45 times is  ;

                              f applied to    45 times is  ;

                              f applied to    45 times is  ;


                              f applied to    45 times is  ;

                              f applied to    45 times is  ;

                              f applied to    45 times is  ;

                              f applied to    45 times is  ,

and so on.