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text  If  end text f left parenthesis x right parenthesis equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator 2 open square brackets fraction numerator x plus 21 pi over denominator pi end fraction close square brackets minus 41 end fraction comma x not equal to n pi comma text end textthen f is

  1. Odd
  2. Even
  3. Both even and odd
  4. Neither even nor odd

The correct answer is: Odd


    f left parenthesis x right parenthesis equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator 2 open square brackets fraction numerator x plus 21 pi over denominator pi end fraction close square brackets minus 41 end fraction equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator 2 open square brackets 21 plus x over pi close square brackets minus 41 end fraction equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator 2 open square brackets x over pi close square brackets plus 1 end fraction
f left parenthesis negative x right parenthesis equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator 2 open square brackets negative x over pi close square brackets plus 1 end fraction equals fraction numerator 2 x left parenthesis sin x plus tan x right parenthesis over denominator negative 2 open square brackets x over pi close square brackets minus 1 end fraction equals negative f left parenthesis x right parenthesis left parenthesis because left square bracket x right square bracket plus left square bracket negative x right square bracket equals negative 1 right parenthesis
therefore f left parenthesis x right parenthesis text  is odd.  end text

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