document.write( "Question 1209940: The function f has the following properties:
\n" ); document.write( "* f(x) is defined for x > 0
\n" ); document.write( "* f(x) > 0 for all x > 0
\n" ); document.write( "* f(x - y) = \sqrt{f(xy) + 4x + 4y + 8} for all x > y > 0\r
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Algebra.Com's Answer #851140 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"f%28x-y%29+=+sqrt%28f%28x%2Ay%29%2B4%2Ax%2B4%2Ay-8%29\"\r\n" );
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document.write( "\"f%28x-y%29+=+sqrt%28f%28x%2Ay%29%2B4%2Ax%2B4%2Ay-8%29\"\r\n" );
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document.write( "If we can find a case of x and y where \"f%28x-y%29=f%28x%2Ay%29=f%281%29\", then we could\r\n" );
document.write( "solve for f(1).  That would require  \r\n" );
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document.write( "\"x-y=x%2Ay=1\"\r\n" );
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document.write( "\"y=1%2Fx\", \"x-1%2Fx=1\"\r\n" );
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document.write( "\"x-1%2Fx=1\"\r\n" );
document.write( "\"x%5E2-1=x\"\r\n" );
document.write( "\"x%5E2-x-1=0\"\r\n" );
document.write( "\"x=%281+%2B-+sqrt%281%2B4%29%29%2F2\" <--we can only use the + sign\r\n" );
document.write( "\"x=%281%2Bsqrt%285%29%29%2F2\"\r\n" );
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document.write( "Incidentally, that, or its reciprocal, is the golden ratio, famous in \r\n" );
document.write( "historical architecture. Let's call it \"G=%281%2Bsqrt%285%29%29%2F2\".\r\n" );
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document.write( "\"G=%281%2Bsqrt%285%29%29%2F2\" and \"G-1%2FG=1\"\r\n" );
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document.write( "\"f%28G-1%2FG%29+=+sqrt%28f%28G%2Aexpr%281%2FG%29%29%2B4%2AG%2B4%2Aexpr%281%2FG%29%29\"\r\n" );
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document.write( "\"f%281%29+=+sqrt%28f%281%29%2B4%2A%28G%2B1%2FG%29%29\"\r\n" );
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document.write( "Since \"G-1%2FG=1\", \"1%2FG=G-1\"\r\n" );
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document.write( "\"f%281%29+=+sqrt%28f%281%29%2B4%2A%28G%2BG-1%29%29\"\r\n" );
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document.write( "\"f%281%29+=+sqrt%28f%281%29%2B4%2A%282G-1%29%29\"\r\n" );
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document.write( "\"f%281%29+=+sqrt%28f%281%29%2B8G-4%29\"\r\n" );
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document.write( "\"%28f%281%29%5E%22%22%29%5E2+=+f%281%29%2B8G-4\"\r\n" );
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document.write( "\"%28f%281%29%5E%22%22%29%5E2+-+f%281%29-%288G-4%29=0\"\r\n" );
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document.write( "Solve for f(1)\r\n" );
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document.write( "\"f%281%29=%281+%2B-+sqrt%281%2B4%288G-4%29%29%29%2F2\"  <--- we can only use +\r\n" );
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document.write( "\"f%281%29=%281+%2B+sqrt%281%2B32G-16%29%29%2F2\"\r\n" );
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document.write( "\"f%281%29=%281+%2B+sqrt%2832G-15%29%29%2F2\"\r\n" );
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document.write( "Substituting \"G=%281%2Bsqrt%285%29%29%2F2\", and doing some algebra,\r\n" );
document.write( "it simplifies to\r\n" );
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document.write( "\"f%281%29=%281%2Bsqrt%281%2B16sqrt%285%29%29%29%2F2\"\r\n" );
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document.write( "\"f%281%29=3.532205783\" approximately.\r\n" );
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document.write( "Edwin
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