Maths-
General
Easy

Question

f colon R not stretchy rightwards arrow R text  is given by  end text f left parenthesis x right parenthesis equals fraction numerator a to the power of x over denominator a to the power of x plus square root of a end fraction straight for all x element of R comma text  then  end text f open parentheses 1 over 1997 close parentheses plus f open parentheses 2 over 1997 close parentheses plus midline horizontal ellipsis plus f open parentheses 1995 over 1997 close parentheses plus f open parentheses 1996 over 1997 close parentheses equals

  1. 997
  2. 998
  3. 1997
  4. 1998

hintHint:

f : R rightwards arrow R signifies a function f(f(x)) having a domain as a set of real numbers and having a codomain is a set of real numbers.

The correct answer is: 998


    G i v e n space colon space f colon R not stretchy rightwards arrow R text  is given by  end text f left parenthesis x right parenthesis equals fraction numerator a to the power of x over denominator a to the power of x plus square root of a end fraction straight for all x element of R comma space......... space left parenthesis 1 right parenthesis
f left parenthesis 1 minus x right parenthesis space equals space fraction numerator a to the power of 1 minus x end exponent over denominator a to the power of 1 minus x end exponent space plus space square root of a end fraction space equals space fraction numerator begin display style a over a to the power of x end style over denominator begin display style a over a to the power of x end style space plus space square root of a end fraction equals space fraction numerator a over denominator a space plus space a to the power of x square root of a end fraction equals space fraction numerator square root of a over denominator a to the power of x space plus space square root of a end fraction space...... space left parenthesis 2 right parenthesis
A d d i n g space e q u a t i o n space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis. space w e space g e t comma
f left parenthesis x right parenthesis space plus space f left parenthesis 1 minus x right parenthesis space equals space fraction numerator a to the power of x over denominator a to the power of x space plus space square root of a end fraction plus space fraction numerator square root of a over denominator a to the power of x space plus square root of a end fraction space equals space 1
T h u s comma space f left parenthesis 1 over 1997 right parenthesis space plus space f open parentheses 1 minus 1 over 1997 close parentheses space equals space 1
rightwards double arrow f left parenthesis 1 over 1997 right parenthesis space plus space f left parenthesis 1996 over 1997 right parenthesis space equals space 1
S i m i l a r l y comma space f left parenthesis 2 over 1997 right parenthesis space plus space f left parenthesis 1 minus 2 over 1997 right parenthesis space equals space 1
N o w comma space text end text f open parentheses 1 over 1997 close parentheses plus f open parentheses 2 over 1997 close parentheses plus midline horizontal ellipsis plus f open parentheses 1995 over 1997 close parentheses plus f open parentheses 1996 over 1997 close parentheses space equals space
open curly brackets f left parenthesis 1 over 1997 right parenthesis space plus space f left parenthesis 1996 over 1997 right parenthesis space close curly brackets space plus open curly brackets f left parenthesis 2 over 1997 right parenthesis space plus space f left parenthesis 1995 over 1997 right parenthesis close curly brackets space plus space........... space open curly brackets f left parenthesis 998 over 1997 right parenthesis space plus space f left parenthesis 999 over 1997 right parenthesis close curly brackets space equals space 998

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