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 #851086 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step by step.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* f(xy) + x = xf(y) + f(x) + xy²<BR>\n" );
document.write( "* f(-1) = 3\r<BR>\n" );
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document.write( "**1. Find f(0):**\r<BR>\n" );
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document.write( "Let x = 0. Then:<BR>\n" );
document.write( "f(0) + 0 = 0*f(y) + f(0) + 0*y²<BR>\n" );
document.write( "f(0) = f(0)<BR>\n" );
document.write( "This equation doesn't give us a specific value for f(0).\r<BR>\n" );
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document.write( "Let y = 0. Then:<BR>\n" );
document.write( "f(0) + x = x*f(0) + f(x) + 0<BR>\n" );
document.write( "f(0) + x = x*f(0) + f(x)<BR>\n" );
document.write( "f(x) = x + f(0) - x*f(0)\r<BR>\n" );
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document.write( "Let x = 1 and y = 0.<BR>\n" );
document.write( "f(0) + 1 = 1*f(0) + f(1) + 0<BR>\n" );
document.write( "f(1) = 1\r<BR>\n" );
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document.write( "**2. Find f(-x):**\r<BR>\n" );
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document.write( "Let y = -1. Then:<BR>\n" );
document.write( "f(-x) + x = x*f(-1) + f(x) + x<BR>\n" );
document.write( "f(-x) + x = 3x + f(x) + x<BR>\n" );
document.write( "f(-x) = 4x + f(x)\r<BR>\n" );
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document.write( "**3. Find f(x):**\r<BR>\n" );
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document.write( "We know that f(x) = x + f(0) - xf(0).<BR>\n" );
document.write( "Let x = -1.<BR>\n" );
document.write( "f(-1) = -1 + f(0) - (-1)f(0)<BR>\n" );
document.write( "3 = -1 + f(0) + f(0)<BR>\n" );
document.write( "4 = 2f(0)<BR>\n" );
document.write( "f(0) = 2\r<BR>\n" );
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document.write( "Now, substitute f(0) = 2 into f(x) = x + f(0) - xf(0):<BR>\n" );
document.write( "f(x) = x + 2 - 2x<BR>\n" );
document.write( "f(x) = 2 - x\r<BR>\n" );
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document.write( "Let's check if this function satisfies the given equation:\r<BR>\n" );
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document.write( "f(xy) + x = 2 - xy + x<BR>\n" );
document.write( "xf(y) + f(x) + xy² = x(2 - y) + (2 - x) + xy² = 2x - xy + 2 - x + xy² = x - xy + 2 + xy²\r<BR>\n" );
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document.write( "These are not equal.\r<BR>\n" );
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document.write( "Let's use the given information f(-x) = 4x + f(x)\r<BR>\n" );
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document.write( "Let x = 1 and y = -1.<BR>\n" );
document.write( "f(-1) + 1 = 1*f(-1) + f(1) + 1<BR>\n" );
document.write( "3 + 1 = 3 + f(1) + 1<BR>\n" );
document.write( "4 = 4 + f(1)<BR>\n" );
document.write( "f(1) = 0\r<BR>\n" );
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document.write( "Let y = 1:<BR>\n" );
document.write( "f(x) + x = xf(1) + f(x) + x<BR>\n" );
document.write( "f(x) + x = x(0) + f(x) + x<BR>\n" );
document.write( "f(x) + x = f(x) + x<BR>\n" );
document.write( "This equation is true but doesn't give us a specific f(x).\r<BR>\n" );
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document.write( "Let x = -1:<BR>\n" );
document.write( "f(-y) - 1 = -f(y) + f(-1) - y²<BR>\n" );
document.write( "f(-y) - 1 = -f(y) + 3 - y²<BR>\n" );
document.write( "f(-y) = 4 - f(y) - y²\r<BR>\n" );
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document.write( "Let y = -x<BR>\n" );
document.write( "f(-x^2)+x = xf(-x)+f(x)+x^3<BR>\n" );
document.write( "f(-x^2)+x = x(4x+f(x))+f(x)+x^3<BR>\n" );
document.write( "f(-x^2) = 4x^2+xf(x)+f(x)+x^3-x\r<BR>\n" );
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document.write( "f(-x) = 4x + f(x)<BR>\n" );
document.write( "let x=1:<BR>\n" );
document.write( "f(-1) = 4+f(1)<BR>\n" );
document.write( "3 = 4+f(1)<BR>\n" );
document.write( "f(1) = -1\r<BR>\n" );
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document.write( "Let y=1:<BR>\n" );
document.write( "f(x)+x=xf(1)+f(x)+x<BR>\n" );
document.write( "f(x)+x=-x+f(x)+x<BR>\n" );
document.write( "0=0\r<BR>\n" );
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document.write( "Try f(x) = ax^2+bx+c<BR>\n" );
document.write( "f(-1)=a-b+c=3<BR>\n" );
document.write( "f(1)=a+b+c=-1<BR>\n" );
document.write( "2a+2c=2<BR>\n" );
document.write( "a+c=1<BR>\n" );
document.write( "c=1-a<BR>\n" );
document.write( "a-b+1-a=3<BR>\n" );
document.write( "-b=2<BR>\n" );
document.write( "b=-2<BR>\n" );
document.write( "a+b+c=-1<BR>\n" );
document.write( "a-2+1-a=-1<BR>\n" );
document.write( "-1=-1\r<BR>\n" );
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document.write( "f(x) = ax^2-2x+1-a<BR>\n" );
document.write( "f(xy) + x = a(xy)^2-2xy+1-a+x<BR>\n" );
document.write( "xf(y)+f(x)+xy^2 = x(ay^2-2y+1-a)+ax^2-2x+1-a+xy^2<BR>\n" );
document.write( "axy^2-2xy+1-a+x = axy^2-2xy+x-ax+ax^2-2x+1-a+xy^2<BR>\n" );
document.write( "0 = -ax+ax^2-x+xy^2\r<BR>\n" );
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document.write( "Let a=1<BR>\n" );
document.write( "0 = x^2-x-x+xy^2<BR>\n" );
document.write( "0 = x^2-2x+xy^2<BR>\n" );
document.write( "0 = x(x-2+y^2)<BR>\n" );
document.write( "This equation is not true for all x,y.\r<BR>\n" );
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document.write( "Try f(x)=x^2-2x.<BR>\n" );
document.write( "f(-1)=1+2=3<BR>\n" );
document.write( "Let y=0<BR>\n" );
document.write( "f(0)+x = xf(0)+f(x)+0<BR>\n" );
document.write( "x = f(x) = x^2-2x<BR>\n" );
document.write( "f(xy)+x=(xy)^2-2xy+x<BR>\n" );
document.write( "xf(y)+f(x)+xy^2 = x(y^2-2y)+x^2-2x+xy^2<BR>\n" );
document.write( "xy^2-2xy+x = xy^2-2xy+x^2-2x+x<BR>\n" );
document.write( "0 = x^2-2x<BR>\n" );
document.write( "x(x-2)=0<BR>\n" );
document.write( "x=0 or x=2\r<BR>\n" );
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document.write( "f(x) = x^2-2x<BR>\n" );
document.write( "f(100)=100^2-2*100=10000-200=9800\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{9800}$<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );