document.write( "Question 1209938: Let f be a function such that
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Algebra.Com's Answer #851981 by ikleyn(52797) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Let f be a function such that
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\n" ); document.write( "\n" ); document.write( "        Be aware: - the solution in the post by @CPhill is INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( "        Below I copy-paste the solution by @CPhill and point the places
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\n" ); document.write( "\n" ); document.write( "- - - - - - - - This is a part of the solution by @CPhill with my notices - - - - - - \r
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document.write( "Let's solve this functional equation problem step by step.\r\n" );
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document.write( "**1. Simplify the equation:**\r\n" );
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document.write( "f(x) + f(2x + y) + 5xy = f(4x - y) - x² + 5xy - 8x + 17y + 1\r\n" );
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document.write( "Subtract 5xy from both sides:\r\n" );
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document.write( "f(x) + f(2x + y) = f(4x - y) - x² - 8x + 17y + 1     (*)\r\n" );
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document.write( "**2. Strategically substitute values for x and y:**\r\n" );
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document.write( "* **Let x = 0:**\r\n" );
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document.write( "f(0) + f(y) = f(-y) + 17y + 1\r\n" );
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document.write( "* **Let y = 0:**\r\n" );
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document.write( "f(x) + f(2x) = f(4x) - x² - 8x + 1\r\n" );
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document.write( "* **Let y = 2x:**\r\n" );
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document.write( "f(x) + f(4x) = f(0) - x² + 34x + 1   <<<---=== the mistake is here: the term -8x in equation (*) is missed. \r\n" );
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document.write( "* **Let y = 4x:**\r\n" );
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document.write( "f(x) + f(6x) = f(0) - x² + 68x + 1   <<<---=== the mistake is here: the term -8x in equation (*) is missed. \r\n" );
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document.write( "* **Let y = -2x:**\r\n" );
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document.write( "f(x) + f(0) = f(6x) - x² - 8x - 34x + 1\r\n" );
document.write( "f(x) + f(0) = f(6x) - x² - 42x + 1\r\n" );
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document.write( "**3. Combine equations:**\r\n" );
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document.write( "From the equations with y=4x and y=-2x:\r\n" );
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document.write( "f(x) + f(6x) = f(0) - x² + 68x + 1   <<<---===  This equation is erroneous.\r\n" );
document.write( "f(x) + f(0) = f(6x) - x² - 42x + 1\r\n" );
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document.write( "Subtract the second equation from the first:\r\n" );
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document.write( "f(6x) - f(0) = f(0) - x² + 68x + 1\r\n" );
document.write( "f(6x) = 2f(0) - x² + 68x + 1         <<<---===  This equation is erroneous.\r\n" );
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document.write( "Subtract f(x) from f(x) + f(2x) = f(4x) - x² - 8x + 1 and f(x) + f(4x) = f(0) - x² + 34x + 1\r\n" );
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document.write( "f(2x) - f(4x) = f(4x) - f(0) - 42x\r\n" );
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document.write( "f(2x) -2f(4x)+f(0)=-42x\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, everything what follows further in the post by @CPhill is not trustworthy.\r
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