Relations and Functions

Relations and Functions

Question

Find the number of non-zero integral solutions of the equation.

space space space space space space space space space space space space open vertical bar 1 minus straight i close vertical bar to the power of straight x space equals 2 to the power of straight x space end exponent

Answer

Given equation is
  space space space space space space space space open vertical bar 1 minus straight i close vertical bar to the power of straight x space equals space 2 to the power of straight x
or,       space space space space space space space space space space space space open parentheses square root of open parentheses 1 close parentheses squared plus left parenthesis negative 1 right parenthesis squared end root close parentheses to the power of straight x equals 2 to the power of straight x space space space space space space space space space space space space space space space space space space space space space or space space space space space space space space space space space space space space open parentheses square root of 2 close parentheses to the power of straight x space equals space 2 to the power of straight x
or,                             space space space space space space straight x over 2 squared equals 2 to the power of straight x space space space space space space space space space space space space space space space space space space space space space space or space space space space space space space space space space space space space space space space space straight x over 2 equals straight x
or,                space space space space space space space space space space space space space space space space space space space space straight x over 2 minus straight x space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space or space space space space space space space space space space space space space space fraction numerator negative straight x over denominator 2 end fraction equals 0
or,                                 x = 0
This equation has no non-zero integral solution.

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