document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209937>Question 1209937</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f be a function such that<BR>\n" );
document.write( "f(xy) + x = xf(y) + f(x) + xy^2<BR>\n" );
document.write( "for all real numbers x and y.  If f(-1) = 3, then compute f(100).<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851141 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52788)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Let f be a function such that<BR>\n" );
document.write( "f(xy) + x = xf(y) + f(x) + xy^2<BR>\n" );
document.write( "for all real numbers x and y.  If f(-1) = 3, then compute f(100).<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below &nbsp;I &nbsp;will show, &nbsp;in couple of lines, &nbsp;that the posed &nbsp;\" problem \" \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is non-sensical and mathematically &nbsp;IMPOSSIBLE &nbsp;self-CONTRADICTORY &nbsp;gibberish.\r<BR>\n" );
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document.write( "Indeed, &nbsp;take &nbsp;&nbsp;x =1 &nbsp;&nbsp;in this given identity \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;f(xy) + x = xf(y) + f(x) + xy^2\r<BR>\n" );
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document.write( "You will get\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;f(y) + 1 = f(y) + f(1) + y^2.\r<BR>\n" );
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document.write( "Cancel &nbsp;f(y) &nbsp;in both sides and get\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 = f(1) + y^2,\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 - f(1) = y^2.\r<BR>\n" );
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document.write( "It says that the function &nbsp;&nbsp;y --> y^2 &nbsp;&nbsp;has a constant value of &nbsp;&nbsp;1 - f(1), \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;which is  &nbsp;&nbsp;I M P O S S I B L E.\r<BR>\n" );
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document.write( "Therefore, &nbsp;I call this &nbsp;\" problem \" &nbsp;as &nbsp;GIBBERISH: &nbsp;such  a function,<BR>\n" );
document.write( "as described in the post, &nbsp;does not exist and can not exist.\r<BR>\n" );
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document.write( "An immediate consequence from this my post that the &nbsp;\" solution \" &nbsp;by &nbsp;@CPhill in his post is &nbsp;GIBBERISH, &nbsp;too.\r<BR>\n" );
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