document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209940>Question 1209940</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f has the following properties:<BR>\n" );
document.write( "* f(x) is defined for x > 0<BR>\n" );
document.write( "* f(x) > 0 for all x > 0<BR>\n" );
document.write( "* f(x - y) = \sqrt{f(xy) + 4x + 4y + 8} for all x > y > 0\r<BR>\n" );
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document.write( "Determine f(1).<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 #851133 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=851133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given properties of the function $f(x)$ be:<BR>\n" );
document.write( "\begin{enumerate}<BR>\n" );
document.write( "    \item $f(x)$ is defined for $x > 0$<BR>\n" );
document.write( "    \item $f(x) > 0$ for all $x > 0$<BR>\n" );
document.write( "    \item $f(x - y) = \sqrt{f(xy) + 4x + 4y + 8}$ for all $x > y > 0$<BR>\n" );
document.write( "\end{enumerate}\r<BR>\n" );
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document.write( "We want to determine $f(1)$.\r<BR>\n" );
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document.write( "Let $x - y = 1$, so $x = y + 1$. We have $x > y > 0$.<BR>\n" );
document.write( "Substituting $x = y + 1$ into the third property, we get:<BR>\n" );
document.write( "$$f(1) = \sqrt{f(y(y+1)) + 4(y+1) + 4y + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(y^2+y) + 4y + 4 + 4y + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(y^2+y) + 8y + 12}$$\r<BR>\n" );
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document.write( "Since $f(1)$ is a constant, we must have $f(y^2+y) + 8y + 12$ be a constant. Let's try to find a particular value of $y$ that makes $y^2+y$ easy to work with.\r<BR>\n" );
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document.write( "Let $y = 1$. Then $x = 2$, and $x > y > 0$.<BR>\n" );
document.write( "$$f(1) = \sqrt{f(1(1+1)) + 4(2) + 4(1) + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(2) + 8 + 4 + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(2) + 20}$$\r<BR>\n" );
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document.write( "Let $y = 2$. Then $x = 3$, and $x > y > 0$.<BR>\n" );
document.write( "$$f(1) = \sqrt{f(2(2+1)) + 4(3) + 4(2) + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(6) + 12 + 8 + 8}$$<BR>\n" );
document.write( "$$f(1) = \sqrt{f(6) + 28}$$\r<BR>\n" );
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document.write( "From $f(1) = \sqrt{f(y^2+y) + 8y + 12}$, we have $f(1)^2 = f(y^2+y) + 8y + 12$.<BR>\n" );
document.write( "So, $f(y^2+y) = f(1)^2 - 8y - 12$.\r<BR>\n" );
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document.write( "Let $y^2+y = 1$. Then $y^2+y-1=0$.<BR>\n" );
document.write( "$y = \frac{-1 \pm \sqrt{1+4}}{2} = \frac{-1 \pm \sqrt{5}}{2}$.<BR>\n" );
document.write( "Since $y>0$, $y = \frac{-1 + \sqrt{5}}{2}$.<BR>\n" );
document.write( "Then<BR>\n" );
document.write( "$$f(1) = \sqrt{f(1) + 8\left(\frac{-1 + \sqrt{5}}{2}\right) + 12}$$<BR>\n" );
document.write( "$$f(1)^2 = f(1) + 4(-1 + \sqrt{5}) + 12$$<BR>\n" );
document.write( "$$f(1)^2 - f(1) - 4\sqrt{5} - 8 = 0$$<BR>\n" );
document.write( "This is a difficult equation to solve.\r<BR>\n" );
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document.write( "Let us try to find a function $f(x) = ax + b$ that satisfies the property.<BR>\n" );
document.write( "Then $a(x-y) + b = \sqrt{a(xy) + b + 4x + 4y + 8}$.<BR>\n" );
document.write( "Squaring both sides,<BR>\n" );
document.write( "$$a^2(x-y)^2 + 2ab(x-y) + b^2 = axy + b + 4x + 4y + 8$$<BR>\n" );
document.write( "$$a^2(x^2 - 2xy + y^2) + 2ab(x-y) + b^2 = axy + b + 4x + 4y + 8$$<BR>\n" );
document.write( "$$a^2x^2 - 2a^2xy + a^2y^2 + 2abx - 2aby + b^2 = axy + b + 4x + 4y + 8$$\r<BR>\n" );
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document.write( "Let $f(x) = 4x+c$. Then<BR>\n" );
document.write( "$4(x-y) + c = \sqrt{4xy + c + 4x + 4y + 8}$<BR>\n" );
document.write( "$16(x-y)^2 + 8c(x-y) + c^2 = 4xy + c + 4x + 4y + 8$<BR>\n" );
document.write( "$16x^2 - 32xy + 16y^2 + 8cx - 8cy + c^2 = 4xy + c + 4x + 4y + 8$<BR>\n" );
document.write( "$16x^2 - 36xy + 16y^2 + (8c-4)x + (-8c-4)y + c^2 - c - 8 = 0$<BR>\n" );
document.write( "For this to hold for all $x > y > 0$, we require the coefficients of $x^2$, $xy$, $y^2$, $x$, $y$ and the constant term to be zero.<BR>\n" );
document.write( "$16 = 0$, impossible.\r<BR>\n" );
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document.write( "Let $f(x) = 4x+4$.<BR>\n" );
document.write( "$4(x-y)+4 = \sqrt{4xy+4+4x+4y+8}$<BR>\n" );
document.write( "$4x-4y+4 = \sqrt{4xy+4x+4y+12}$<BR>\n" );
document.write( "$16(x-y)^2 + 32(x-y) + 16 = 4xy+4x+4y+12$<BR>\n" );
document.write( "$16x^2 - 32xy + 16y^2 + 32x - 32y + 16 = 4xy+4x+4y+12$<BR>\n" );
document.write( "$16x^2 - 36xy + 16y^2 + 28x - 36y + 4 = 0$\r<BR>\n" );
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document.write( "Let $f(x) = 4x+8$.<BR>\n" );
document.write( "$4(x-y)+8 = \sqrt{4xy+8+4x+4y+8}$<BR>\n" );
document.write( "$4x-4y+8 = \sqrt{4xy+4x+4y+16}$<BR>\n" );
document.write( "$16(x-y)^2 + 64(x-y) + 64 = 4xy+4x+4y+16$<BR>\n" );
document.write( "$16x^2 - 36xy + 16y^2 + 60x - 68y + 48 = 0$\r<BR>\n" );
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document.write( "Let $f(x) = 4x+4$.<BR>\n" );
document.write( "$f(1) = 8$.<BR>\n" );
document.write( "$f(1) = \sqrt{f(2) + 20} = \sqrt{12+20} = \sqrt{32} \ne 8$.<BR>\n" );
document.write( "Let $f(x) = 4x+8$.<BR>\n" );
document.write( "$f(1) = 12$.<BR>\n" );
document.write( "$f(1) = \sqrt{f(2) + 20} = \sqrt{16+20} = \sqrt{36} = 6$.<BR>\n" );
document.write( "$f(1) = \sqrt{f(6)+28} = \sqrt{32+28} = \sqrt{60} \neq 6$\r<BR>\n" );
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document.write( "Let $f(x) = 4x+c$.<BR>\n" );
document.write( "$f(1) = 4+c$.<BR>\n" );
document.write( "$f(1)^2 = f(2) + 20 = 8+c+20 = 28+c$.<BR>\n" );
document.write( "$(4+c)^2 = 28+c$.<BR>\n" );
document.write( "$16+8c+c^2 = 28+c$.<BR>\n" );
document.write( "$c^2+7c-12=0$.<BR>\n" );
document.write( "$c = \frac{-7 \pm \sqrt{49+48}}{2} = \frac{-7 \pm \sqrt{97}}{2}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{8}$<BR>\n" );
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