document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209971>Question 1209971</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies<BR>\n" );
document.write( "xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13<BR>\n" );
document.write( "for all real x.  Find f(x). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851282 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=851282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's solve this functional equation.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "We're given a functional equation that relates f(x) and f(1 - x). Our goal is to find an explicit expression for f(x).\r<BR>\n" );
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document.write( "**Solving the Functional Equation**\r<BR>\n" );
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document.write( "1.  **Original Equation:**\r<BR>\n" );
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document.write( "    $$xf(x) + \frac{f(1 - x)}{x} = x^3 + 3x^2 + 14x - 13 \quad (*)$$\r<BR>\n" );
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document.write( "2.  **Substitute (1 - x) for x:**\r<BR>\n" );
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document.write( "    Replace every instance of 'x' with '(1 - x)' in the original equation:\r<BR>\n" );
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document.write( "    $$(1 - x)f(1 - x) + \frac{f(1 - (1 - x))}{1 - x} = (1 - x)^3 + 3(1 - x)^2 + 14(1 - x) - 13$$\r<BR>\n" );
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document.write( "    Simplify:\r<BR>\n" );
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document.write( "    $$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = (1 - 3x + 3x^2 - x^3) + 3(1 - 2x + x^2) + 14 - 14x - 13$$\r<BR>\n" );
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document.write( "    $$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = 1 - 3x + 3x^2 - x^3 + 3 - 6x + 3x^2 + 14 - 14x - 13$$\r<BR>\n" );
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document.write( "    $$(1 - x)f(1 - x) + \frac{f(x)}{1 - x} = -x^3 + 6x^2 - 23x + 5 \quad (**)$$\r<BR>\n" );
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document.write( "3.  **Multiply (*) by (1 - x):**\r<BR>\n" );
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document.write( "    Multiply the original equation (*) by (1 - x):\r<BR>\n" );
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document.write( "    $$x(1 - x)f(x) + \frac{(1 - x)f(1 - x)}{x} = (1 - x)(x^3 + 3x^2 + 14x - 13) \quad (***)$$\r<BR>\n" );
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document.write( "4.  **Multiply (**) by x:**\r<BR>\n" );
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document.write( "    Multiply the substituted equation (**) by x:\r<BR>\n" );
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document.write( "    $$x(1 - x)f(1 - x) + f(x) = x(-x^3 + 6x^2 - 23x + 5) \quad (****)$$\r<BR>\n" );
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document.write( "5.  **Solve for f(1-x) in (***):**\r<BR>\n" );
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document.write( "    $$(1-x)f(1-x) = x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)$$\r<BR>\n" );
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document.write( "    Divide by (1-x):\r<BR>\n" );
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document.write( "    $$f(1-x) = \frac{x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)}{1-x}$$\r<BR>\n" );
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document.write( "6.  **Substitute f(1-x) into (****):**\r<BR>\n" );
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document.write( "    $$x(1 - x)\left(\frac{x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x)}{1-x}\right) + f(x) = x(-x^3 + 6x^2 - 23x + 5)$$\r<BR>\n" );
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document.write( "    $$x(x^3 + 3x^2 + 14x - 13) - x^2(1-x)f(x) + f(x) = x(-x^3 + 6x^2 - 23x + 5)$$\r<BR>\n" );
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document.write( "    $$x^4 + 3x^3 + 14x^2 - 13x - x^2(1-x)f(x) + f(x) = -x^4 + 6x^3 - 23x^2 + 5x$$\r<BR>\n" );
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document.write( "    $$f(x)(1 - x^2(1-x)) = -x^4 + 6x^3 - 23x^2 + 5x - x^4 - 3x^3 - 14x^2 + 13x$$\r<BR>\n" );
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document.write( "    $$f(x)(1 - x^2 + x^3) = -2x^4 + 3x^3 - 37x^2 + 18x$$\r<BR>\n" );
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document.write( "    $$f(x) = \frac{-2x^4 + 3x^3 - 37x^2 + 18x}{x^3 - x^2 + 1}$$\r<BR>\n" );
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document.write( "7. **Polynomial Long Division**<BR>\n" );
document.write( "Perform long division to simplify the rational function:\r<BR>\n" );
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document.write( "$$f(x) = -2x - 1 + \frac{-36x^2 + 20x +1}{x^3-x^2+1}$$\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "$$f(x) = -2x-1 + \frac{-36x^2 + 20x}{x^3-x^2+1}$$<BR>\n" );
document.write( " </SPAN>\n" );
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