Maths-
General
Easy

Question

If f left parenthesis x right parenthesis is a polynomial function such that f space left parenthesis x right parenthesis f space open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f space open parentheses 1 over x close parentheses text  and  end text f space left parenthesis 3 right parenthesis equals negative 80 text  then  end text f left parenthesis x right parenthesis minus f space open parentheses 1 over x close parentheses equals

  1. x to the power of 4 plus 1 over x to the power of 4
  2. x to the power of 4 minus 1 over x to the power of 4
  3. x to the power of 4 minus 1 over x to the power of 4
  4. negative 1 over x to the power of 4 minus x to the power of 4

Hint:

In this question, we have  f left parenthesis x right parenthesis is a polynomial function such that f space left parenthesis x right parenthesis f space open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f space open parentheses 1 over x close parentheses text  and  end text f space left parenthesis 3 right parenthesis equals negative 80 text   and we have to find  end text f left parenthesis x right parenthesis minus f space open parentheses 1 over x close parentheses equals. Firstly, we will use the given information to find the value of f(x) and fopen parentheses 1 over x close parentheses. Then, we will multiply the equations derived. And later assume g(x) to be f(x)-1 and gopen parentheses 1 over x close parentheses to be fopen parentheses 1 over x close parentheses-1. Further, we will get the value of f(x) and fopen parentheses 1 over x close parentheses and thus we can find the required sum.

The correct answer is: x to the power of 4 minus 1 over x to the power of 4


    G i v e n comma space
space space space space space space space space space f open parentheses x close parentheses f open parentheses 1 over x close parentheses equals f open parentheses x close parentheses plus f open parentheses 1 over x close parentheses
rightwards double arrow f open parentheses x close parentheses f open parentheses 1 over x close parentheses minus f open parentheses 1 over x close parentheses equals f open parentheses x close parentheses
rightwards double arrow f open parentheses 1 over x close parentheses open square brackets f open parentheses x close parentheses minus 1 close square brackets equals f open parentheses x close parentheses space space _ _ _ _ _ _ _ _ _ _ left parenthesis i right parenthesis

G i v e n comma space
space space space space space space space space space space f open parentheses x close parentheses f open parentheses 1 over x close parentheses equals f open parentheses x close parentheses plus f open parentheses 1 over x close parentheses
rightwards double arrow f open parentheses x close parentheses f open parentheses 1 over x close parentheses minus f open parentheses x close parentheses space equals f open parentheses 1 over x close parentheses
rightwards double arrow f open parentheses x close parentheses open square brackets f open parentheses 1 over x close parentheses minus 1 close square brackets equals f open parentheses 1 over x close parentheses space _ _ _ _ _ _ _ _ _ _ left parenthesis i i right parenthesis

M u l t i p l y i n g space e q u a t i o n space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis comma space w e space g e t colon
f open parentheses 1 over x close parentheses open square brackets f open parentheses x close parentheses minus 1 close square brackets. space f open parentheses x close parentheses open square brackets f open parentheses 1 over x close parentheses minus 1 close square brackets equals f open parentheses x close parentheses f open parentheses 1 over x close parentheses
rightwards double arrow open square brackets f open parentheses x close parentheses minus 1 close square brackets. open square brackets f open parentheses 1 over x close parentheses minus 1 close square brackets equals 1 space space _ _ _ _ _ _ _ _ left parenthesis i i i right parenthesis

N o w comma space l e t space g left parenthesis x right parenthesis space equals space f open parentheses x close parentheses minus 1 space a n d space g open parentheses 1 over x close parentheses equals space f open parentheses 1 over x close parentheses minus 1
S o comma space e q u a t i o n space left parenthesis i i i right parenthesis space i s comma space
rightwards double arrow g left parenthesis x right parenthesis. g open parentheses 1 over x close parentheses equals 1
T h u s comma space g left parenthesis x right parenthesis equals plus-or-minus x to the power of n space end exponent space space a n d space space g open parentheses 1 over x close parentheses equals plus-or-minus 1 over x to the power of n
S o comma space f left parenthesis x right parenthesis equals plus-or-minus x to the power of n plus 1 space a n d space f open parentheses 1 over x close parentheses equals plus-or-minus 1 over x to the power of n plus 1

f left parenthesis 3 right parenthesis equals plus-or-minus 3 to the power of n plus 1
rightwards double arrow negative 80 equals plus-or-minus 3 to the power of n plus 1
rightwards double arrow negative 81 equals plus-or-minus 3 to the power of n
rightwards double arrow negative 81 equals negative 3 to the power of n
rightwards double arrow 3 to the power of 4 equals 3 to the power of n
rightwards double arrow n space equals space 4

S o comma space f left parenthesis x right parenthesis equals negative x to the power of 4 plus 1 space a n d space f open parentheses 1 over x close parentheses equals fraction numerator negative 1 over denominator x to the power of 4 end fraction plus 1

f left parenthesis x right parenthesis space plus space f open parentheses 1 over x close parentheses space
equals space minus x to the power of 4 plus 1 minus open square brackets fraction numerator negative 1 over denominator x to the power of 4 end fraction plus 1 close square brackets
equals negative x to the power of 4 plus 1 plus 1 over x to the power of 4 minus 1
equals 1 over x to the power of 4 minus x to the power of 4

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