document.write( "Question 1209952: The function f(n) is defined for all integers n, such that\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
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Algebra.Com's Answer #851205 by ikleyn(52790)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "The function f(n) is defined for all integers n, such that\r
\n" ); document.write( "\n" ); document.write( "f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
\n" ); document.write( "for all integers x and y, and f(1) = 1. Find f(n).
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @CPhill,  giving the answer  f(n) = \"%283%2F2%29%2An%2A%28n-1%29%29\",  is  INCORRECT.\r
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document.write( "(a)  Substitute x=0, y=0 into the basic formula.  You will get\r\n" );
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document.write( "     f(0) + f(0) = f(0) - 0 - 1 + f(0) + f(0).\r\n" );
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document.write( "Cancel f(0) in both sides everywhere where possible.  You will get\r\n" );
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document.write( "     0 = f(0) - 1\r\n" );
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document.write( "     f(0) = 1.\r\n" );
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document.write( "(b)  Substitute y=0 into the basic formula. Now x is an arbitrary integer number.\r\n" );
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document.write( "        f(x) + f(0) = f(x+0) - 0 - 1 + f(x^2) + f(0),\r\n" );
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document.write( "        f(x) + 1 = f(x) - 0 - 1 + f(x^2) + 1,\r\n" );
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document.write( "        1 = f(x^2).\r\n" );
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document.write( "Thus, from the general basic formula, f(n^2) = 1 for any integer 'n'.\r\n" );
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document.write( "But it is not so from the formula  f(n) = \"%283%2F2%29%2An%2A%28n-1%29%29\"  by @CPhill.\r\n" );
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\n" ); document.write( "\n" ); document.write( "It disproves and kills the solution by @CPhill to the death.\r
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document.write( "Let's go further in analyzing the problem.\r\n" );
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document.write( "In the general equation, replace the terms f(x^2) and f(y^2) by 1, as we deduced it above.\r\n" );
document.write( "You will get\r\n" );
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document.write( "    f(x) + f(y) = f(x+y) - 4xy + 1.\r\n" );
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document.write( "Put  y= 1  in this equation\r\n" );
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document.write( "    f(x) + f(1) = f(x+1) - 4x + 1,\r\n" );
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document.write( "    f(x) + 1 = f(x+1) - 4x + 1,\r\n" );
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document.write( "    f(x+1) =  f(x) + 4x.\r\n" );
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document.write( "We get a nice recurrent formula and can calculate the values f(x) moving forward.\r\n" );
document.write( "We get\r\n" );
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document.write( "    f(2) = f(1) + 4*1 = 1 + 4 = 5;\r\n" );
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document.write( "    f(3) = f(2) + 4*2 = 5 + 8 = 13;\r\n" );
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document.write( "    f(4) = f(3) + 4*3 = 13 + 12 = 25.\r\n" );
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document.write( "But this equality f(4) = \"f%282%5E2%29\" = 25 CONTRADICTS to equality f(n^2) = 1, which we established earlier above.\r\n" );
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document.write( "It tells that the given general formula DOES NOT define a function f.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, the posted problem is  SELF-CONTRADICTORY.\r
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\n" ); document.write( "\n" ); document.write( "Its description  DECEIVES  the reader from the very beginning,
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\n" ); document.write( "\n" ); document.write( "                On  CONTRARY,  it  DOES  NOT  define.\r
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\n" ); document.write( "\n" ); document.write( "In the last several days,  I saw several similar  ABSOLUTELY  DEFECTIVE  \" problems \",  submitted to the forum.\r
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\n" ); document.write( "\n" ); document.write( "They all are created by unprofessional/(illiterate ?) Math composers.\r
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