document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209953>Question 1209953</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2 and satisfying the functional equation<BR>\n" );
document.write( "F(x) + F \left( \frac{2x - 3}{x - 1} \right) + F \left( \frac{1}{x} \right) = x.<BR>\n" );
document.write( "Find the function F(x) satisfying these conditions.  Write F(x) as a rational function with expanded polynomials in the numerator and denominator.<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 #851202 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=851202\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $g(x) = \frac{2x - 3}{x - 1}$ and $h(x) = \frac{1}{x}$.<BR>\n" );
document.write( "We are given that<BR>\n" );
document.write( "$$F(x) + F(g(x)) + F(h(x)) = x \quad (*)$$<BR>\n" );
document.write( "We need to find $F(x)$.\r<BR>\n" );
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document.write( "Let's find $g(g(x))$ and $g(h(x))$.<BR>\n" );
document.write( "$$g(g(x)) = \frac{2\left(\frac{2x - 3}{x - 1}\right) - 3}{\frac{2x - 3}{x - 1} - 1} = \frac{2(2x - 3) - 3(x - 1)}{(2x - 3) - (x - 1)} = \frac{4x - 6 - 3x + 3}{2x - 3 - x + 1} = \frac{x - 3}{x - 2}$$<BR>\n" );
document.write( "$$g(h(x)) = \frac{2\left(\frac{1}{x}\right) - 3}{\frac{1}{x} - 1} = \frac{\frac{2}{x} - 3}{\frac{1}{x} - 1} = \frac{2 - 3x}{1 - x}$$<BR>\n" );
document.write( "$$h(g(x)) = \frac{1}{\frac{2x - 3}{x - 1}} = \frac{x - 1}{2x - 3}$$<BR>\n" );
document.write( "$$h(h(x)) = \frac{1}{\frac{1}{x}} = x$$\r<BR>\n" );
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document.write( "Now, let's substitute $g(x)$ for $x$ in (*):<BR>\n" );
document.write( "$$F(g(x)) + F(g(g(x))) + F(h(g(x))) = g(x)$$<BR>\n" );
document.write( "$$F\left(\frac{2x - 3}{x - 1}\right) + F\left(\frac{x - 3}{x - 2}\right) + F\left(\frac{x - 1}{2x - 3}\right) = \frac{2x - 3}{x - 1} \quad (**)$$\r<BR>\n" );
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document.write( "Let's substitute $h(x)$ for $x$ in (*):<BR>\n" );
document.write( "$$F(h(x)) + F(g(h(x))) + F(h(h(x))) = h(x)$$<BR>\n" );
document.write( "$$F\left(\frac{1}{x}\right) + F\left(\frac{2 - 3x}{1 - x}\right) + F(x) = \frac{1}{x} \quad (***)$$\r<BR>\n" );
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document.write( "Let's substitute $g(g(x))$ for $x$ in (*):<BR>\n" );
document.write( "$$F(g(g(x))) + F(g(g(g(x)))) + F(h(g(g(x)))) = g(g(x))$$<BR>\n" );
document.write( "$$F\left(\frac{x - 3}{x - 2}\right) + F\left(\frac{1}{2-x}\right) + F\left(\frac{x-2}{3-2x}\right) = \frac{x-3}{x-2} \quad (****)$$\r<BR>\n" );
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document.write( "We want to find $F(x)$.<BR>\n" );
document.write( "We have:<BR>\n" );
document.write( "$$F(x) + F\left(\frac{2x - 3}{x - 1}\right) + F\left(\frac{1}{x}\right) = x$$<BR>\n" );
document.write( "$$F\left(\frac{2x - 3}{x - 1}\right) + F\left(\frac{x - 3}{x - 2}\right) + F\left(\frac{x - 1}{2x - 3}\right) = \frac{2x - 3}{x - 1}$$<BR>\n" );
document.write( "$$F\left(\frac{1}{x}\right) + F\left(\frac{2 - 3x}{1 - x}\right) + F(x) = \frac{1}{x}$$\r<BR>\n" );
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document.write( "Let's try to find a linear combination of (*), (**), and (***) to eliminate $F(g(x))$ and $F(h(x))$.<BR>\n" );
document.write( "Multiply (*) by 1, (**) by -1, and (***) by 1:<BR>\n" );
document.write( "$$F(x) + F(g(x)) + F(h(x)) = x$$<BR>\n" );
document.write( "$$-F(g(x)) - F(g(g(x))) - F(h(g(x))) = -\frac{2x - 3}{x - 1}$$<BR>\n" );
document.write( "$$F(h(x)) + F(g(h(x))) + F(x) = \frac{1}{x}$$\r<BR>\n" );
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document.write( "Adding these three equations:<BR>\n" );
document.write( "$$2F(x) + 2F(h(x)) + F(g(x)) + F(h(x)) - F(g(x)) - F(g(g(x))) - F(h(g(x))) + F(g(h(x))) = x - \frac{2x - 3}{x - 1} + \frac{1}{x}$$\r<BR>\n" );
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document.write( "This is still complicated.<BR>\n" );
document.write( "Let's try to use $h(h(x)) = x$.\r<BR>\n" );
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document.write( "We'll use:<BR>\n" );
document.write( "$F(x)+F(\frac{2x-3}{x-1})+F(\frac{1}{x}) = x$<BR>\n" );
document.write( "$F(\frac{1}{x})+F(\frac{2-3x}{1-x})+F(x)=\frac{1}{x}$.\r<BR>\n" );
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document.write( "$F(\frac{2x-3}{x-1}) + F(\frac{x-3}{x-2}) + F(\frac{x-1}{2x-3}) = \frac{2x-3}{x-1}$\r<BR>\n" );
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document.write( "We need to eliminate F(g(x)), F(h(x)).\r<BR>\n" );
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document.write( "F(x) + F(g(x)) + F(h(x)) = x<BR>\n" );
document.write( "F(h(x)) + F(g(h(x))) + F(h(h(x))) = 1/x\r<BR>\n" );
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document.write( "F(g(x)) + F(g(g(x))) + F(h(g(x))) = (2x-3)/(x-1)\r<BR>\n" );
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document.write( "F(x) + F(1/x) + F((2x-3)/(x-1)) = x<BR>\n" );
document.write( "F(1/x) + F((2-3x)/(1-x)) + F(x) = 1/x<BR>\n" );
document.write( "F((2x-3)/(x-1)) + F((x-3)/(x-2)) + F((x-1)/(2x-3)) = (2x-3)/(x-1)\r<BR>\n" );
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document.write( "F(1/x) + F((2-3x)/(1-x)) = 1/x - F(x)<BR>\n" );
document.write( "F((2x-3)/(x-1)) + F((x-3)/(x-2)) = (2x-3)/(x-1) - F((x-1)/(2x-3))\r<BR>\n" );
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document.write( "5F(x) = 2x - 1/x - (2x-3)/(x-1)<BR>\n" );
document.write( "F(x) = (2x - 1/x - (2x-3)/(x-1))/5<BR>\n" );
document.write( "F(x) = (2x(x)(x-1) - (x-1) - x(2x-3))/(5x(x-1))<BR>\n" );
document.write( "F(x) = (2x^3 - 2x^2 - x + 1 - 2x^2 + 3x)/(5x^2 - 5x)<BR>\n" );
document.write( "F(x) = (2x^3 - 4x^2 + 2x + 1)/(5x^2 - 5x)\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{F(x) = \frac{2x^3 - 4x^2 + 2x + 1}{5x^2 - 5x}}$<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );