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Question

Letf left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator x minus 1 end fraction comma text  fof  end text left parenthesis x right parenthesis equals f squared left parenthesis x right parenthesis semicolon f o f o f left parenthesis x right parenthesis equals f cubed left parenthesis x right parenthesis comma text  fofofof  end text left parenthesis x right parenthesis equals f to the power of 4 left parenthesis x right parenthesis horizontal ellipsis text  then  end text f subscript left parenthesis x right parenthesis end subscript superscript 2008 equals

  1. x
  2. f left parenthesis x right parenthesis
  3. 0

  4. 1 over x

hintHint:

For the given function we have to find f to the power of 2008 space left parenthesis x right parenthesis , where f left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator x minus 1 end fraction comma text  fof  end text left parenthesis x right parenthesis equals f squared left parenthesis x right parenthesis semicolon f o f o f left parenthesis x right parenthesis equals f cubed left parenthesis x right parenthesis comma text  fofofof  end text left parenthesis x right parenthesis equals f to the power of 4 left parenthesis x right parenthesis.... first we will check the pattern of odd and even power through which we will get the required answer.

The correct answer is: x


    F r o m comma space f o f left parenthesis x right parenthesis equals f squared left parenthesis x right parenthesis
rightwards double arrow L. H. S space equals f o f left parenthesis x right parenthesis
space space space space space space space space space space space space space space space space space equals f left parenthesis f left parenthesis x right parenthesis right parenthesis
space space space space space space space space space space space space space space space space space equals f open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses space space space space space space space space space space space space space space open square brackets f left parenthesis x right parenthesis equals fraction numerator x plus x over denominator x minus 1 end fraction close square brackets
space space space space space space space space space space space space space space space space equals fraction numerator begin display style fraction numerator x plus 1 over denominator x minus 1 end fraction end style plus 1 over denominator begin display style fraction numerator x plus 1 over denominator x minus 1 end fraction end style minus 1 end fraction
space space space space space space space space space space space space space space space space equals fraction numerator begin display style fraction numerator x plus 1 plus x minus 1 over denominator x minus 1 end fraction end style over denominator begin display style fraction numerator x plus 1 minus x plus 1 over denominator x minus 1 end fraction end style end fraction
space space space space space space space space space space space space space space space space equals fraction numerator 2 x over denominator 2 end fraction
space space space space space space space space space space space space space space space space equals x
S o comma f squared left parenthesis x right parenthesis equals x space
    Similarly, for every f to the power of n left parenthesis x right parenthesis equals x, where n is even.
    F r o m comma space f o f o f left parenthesis x right parenthesis equals f cubed left parenthesis x right parenthesis
rightwards double arrow L. H. S equals space f left parenthesis f left parenthesis f left parenthesis x right parenthesis right parenthesis right parenthesis equals f left parenthesis f squared left parenthesis x right parenthesis right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals f left parenthesis x right parenthesis space space space space space space space space space space space space space space space space open square brackets f squared left parenthesis x right parenthesis equals x close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator x plus 1 over denominator x minus 1 end fraction
    Similarly, for every f to the power of n left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator x minus 1 end fraction, where n is odd.
    Now, as 2008 is an even number. So, f to the power of 2008 left parenthesis x right parenthesis equals x.

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