document.write( "Question 1120935: Let f:R->R be a function such that f(a+b)=f(a)+f(b) and that f(2008)=3012. What is f(2009)? \n" ); document.write( "

Algebra.Com's Answer #736636 by ikleyn(52835) $\"\"$ $\"About$

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document.write( "From this functional equation\r\n" );
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document.write( "    f(a+b) = f(a) + f(b)\r\n" );
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document.write( "you have\r\n" );
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document.write( "     f(0) = f(0) + f(0)  ====>  f(0) = 2*f(0)  ====>  f(0) = 0,\r\n" );
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document.write( "and also  \r\n" );
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document.write( "     f(2) = 2*f(1),  f(3) = 3*f(1),  f(4) = 4*f(1),   . . . . ,  f(2008) = 2008*f(1).\r\n" );
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document.write( "Since  f(2008) = 3012 by the condition,  it implies that  2008*f(1) = 3012,  and,  hence,  f(1) =  = .\r\n" );
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document.write( "Then  f(2009) = 2009*f(1) = .\r\n" );
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document.write( "Answer.  f(2009) =  = .\r\n" );
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