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Question

Let f colon R to the power of plus end exponent rightwards arrow R be a function which satisfies f left parenthesis x right parenthesis times f left parenthesis y right parenthesis equals f left parenthesis x y right parenthesis plus 2 open parentheses fraction numerator 1 over denominator x end fraction plus fraction numerator 1 over denominator y end fraction plus 1 close parentheses for x comma y greater than 0 can be

  1. fraction numerator 2 x plus 1 over denominator x end fraction    
  2. open parentheses fraction numerator x plus 1 over denominator x end fraction close parentheses    
  3. fraction numerator 5 x plus 2 over denominator 3 x end fraction    
  4. fraction numerator 2 left parenthesis 2 x plus left parenthesis a right parenthesis blank over denominator negative 3 x end fraction    

The correct answer is: fraction numerator 2 x plus 1 over denominator x end fraction


    f to the power of 2 end exponent left parenthesis a right parenthesis blank minus f left parenthesis a right parenthesis blank minus 6 equals 0 rightwards double arrow f left parenthesis a right parenthesis blank equals 3 text  or  end text f left parenthesis a right parenthesis blank equals negative 2
    text put  end text x equals 1 comma y equals 1
    Now put y equals 1
    f left parenthesis x right parenthesis times f left parenthesis a right parenthesis blank equals f left parenthesis x right parenthesis plus 2 open parentheses fraction numerator 1 over denominator x end fraction plus 2 close parentheses equals f left parenthesis x right parenthesis plus 2 open parentheses fraction numerator 2 x plus 1 over denominator x end fraction close parentheses rightwards double arrow f left parenthesis x right parenthesis open square brackets f left parenthesis a right parenthesis blank minus 1 equals fraction numerator 2 left parenthesis 2 x plus 1 right parenthesis blank over denominator x end fraction close
    f left parenthesis x right parenthesis equals fraction numerator 2 left parenthesis 2 x plus left parenthesis a right parenthesis blank over denominator x left square bracket f left parenthesis a right parenthesis blank minus 1 right square bracket end fraction put f left parenthesis a right parenthesis blank equals 3 or -2 to get (A)

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