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If $2 f left parenthesis x right parenthesis plus x f open parentheses fraction numerator 1 over denominator x end fraction close parentheses minus 2 f open parentheses open vertical bar square root of 2 sin invisible function application pi open parentheses x plus fraction numerator 1 over denominator 4 end fraction close parentheses close vertical bar close parentheses equals 4 c o s to the power of 2 end exponent invisible function application open parentheses fraction numerator pi x over denominator 2 end fraction close parentheses plus x c o s invisible function application fraction numerator pi over denominator x end fraction comma for all x element of R minus left curly bracket 0 right curly bracket$ then which of the following statement(s) is/are true?

$f (2) + f (\frac{1}{2}) = 1$
$f (2) + f (1) = 0$
$f (2) + f (1) = f (\frac{1}{2})$
$f (1) f (\frac{1}{2}) f (2) = 1$

The correct answer is: $f left parenthesis 2 right parenthesis plus f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses equals 1$

$table row cell text end text text R end text text e end text text p end text text l end text text a end text text c end text text e end text text x end text text b end text text y end text text 2 end text text , end text text end text 2 f left parenthesis 2 right parenthesis plus 2 f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses minus 2 f left parenthesis 1 right parenthesis equals 4 rightwards double arrow f left parenthesis 2 right parenthesis plus f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses equals 2 plus f left parenthesis 1 right parenthesis minus negative negative negative negative left parenthesis 1 right parenthesis end cell row cell text end text text R end text text e end text text p end text text l end text text a end text text c end text text e end text text x end text text b end text text y end text text end text 1 comma f left parenthesis 1 right parenthesis equals negative 1 minus negative negative negative negative left parenthesis 2 right parenthesis end cell row cell text end text text R end text text e end text text p end text text l end text text a end text text c end text text e end text text x end text text b end text text y end text text end text fraction numerator 1 over denominator 2 end fraction comma 2 f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses plus fraction numerator 1 over denominator 2 end fraction f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses plus 2 equals fraction numerator 5 over denominator 2 end fraction minus negative negative negative negative left parenthesis 3 right parenthesis end cell row cell text end text text S end text text o end text text l end text text v end text text e end text text end text open parentheses 1 close parentheses a n d left parenthesis 3 right parenthesis rightwards double arrow f open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses equals 0 semicolon f left parenthesis 2 right parenthesis equals 1 end cell end table$

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