document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165361>Question 1165361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a function f of a real variable which satisfies the function defined by x*f(2-x)-2*f(x)=x^2+1 for all real x, determine f(x) in terms of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #789880 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=789880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=f%28x%29%2C+b=f%282-x%29%29 ALIGN=MIDDLE  ALT=\"system%28a=f%28x%29%2C+b=f%282-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When the variable is x, the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Af%282-x%29-2%2Af%28x%29=x%5E2%2B1 ALIGN=MIDDLE  ALT=\"x%2Af%282-x%29-2%2Af%28x%29=x%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which we write in abbreviated form as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ab-2a=x%5E2%2B1 ALIGN=MIDDLE  ALT=\"x%2Ab-2a=x%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1)<BR>\n" );
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document.write( "When the variable is 2-x, the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29%2Af%282-%282-x%29%29-2%2Af%282-x%29=%282-x%29%5E2%2B1 ALIGN=MIDDLE  ALT=\"%282-x%29%2Af%282-%282-x%29%29-2%2Af%282-x%29=%282-x%29%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which we want to simplify as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29%2Af%28x%29-2%2Af%282-x%29=4-4x%2Bx%5E2%2B1 ALIGN=MIDDLE  ALT=\"%282-x%29%2Af%28x%29-2%2Af%282-x%29=4-4x%2Bx%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29%2Af%28x%29-2%2Af%282-x%29=x%5E2-4x%2B5 ALIGN=MIDDLE  ALT=\"%282-x%29%2Af%28x%29-2%2Af%282-x%29=x%5E2-4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and abbreviate as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-x%29%2Aa-2b=x%5E2-4x%2B5 ALIGN=MIDDLE  ALT=\"%282-x%29%2Aa-2b=x%5E2-4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2)<BR>\n" );
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document.write( "In (1) and (2), we have a system of equations on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2Aa-2b=x%5E2-4x%2B5%2C-2a%2Bx%2Ab=x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"system%28x%2Aa-2b=x%5E2-4x%2B5%2C-2a%2Bx%2Ab=x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . Let's solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "Multiplying (2) times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282x-x%5E2%29%2F2%29a-%28x%2F2%292b=%28x%2F2%29%28x%5E2-4x%2B5%29 ALIGN=MIDDLE  ALT=\"%28%282x-x%5E2%29%2F2%29a-%28x%2F2%292b=%28x%2F2%29%28x%5E2-4x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which we simplify to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%5E2%2B2x%29%2F2%29a-x%2Ab=%28x%5E3-4x%5E2%2B5x%29%2F2 ALIGN=MIDDLE  ALT=\"%28%28-x%5E2%2B2x%29%2F2%29a-x%2Ab=%28x%5E3-4x%5E2%2B5x%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and add it to (1) to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%5E2%2B2x%29%2F2%29a-2a=%28x%5E3-4x%5E2%2B5x%29%2F2%2Bx%5E2%2B1 ALIGN=MIDDLE  ALT=\"%28%28-x%5E2%2B2x%29%2F2%29a-2a=%28x%5E3-4x%5E2%2B5x%29%2F2%2Bx%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-x%5E2%2B2x-4%29%2F2%29a=%28x%5E3-4x%5E2%2B5x%2B2x%5E2%2B2%29%2F2 ALIGN=MIDDLE  ALT=\"%28%28-x%5E2%2B2x-4%29%2F2%29a=%28x%5E3-4x%5E2%2B5x%2B2x%5E2%2B2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2B2x-4=-%28x%5E2-2x%2B4%29 ALIGN=MIDDLE  ALT=\"-x%5E2%2B2x-4=-%28x%5E2-2x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not zero for any real number,<BR>\n" );
document.write( "so we can divide by that expression to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%5E2%2B2x-4%29a=%28x%5E3-2x%5E2%2B5x%2B2%29 ALIGN=MIDDLE  ALT=\"%28-x%5E2%2B2x-4%29a=%28x%5E3-2x%5E2%2B5x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-%28x%5E3-2x%5E2%2B5x%2B2%29%2F%28x%5E2-2x%2B4%29 ALIGN=MIDDLE  ALT=\"a=-%28x%5E3-2x%5E2%2B5x%2B2%29%2F%28x%5E2-2x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28f%28x%29=-%28x%5E3-2x%5E2%2B5x%2B2%29%2F%28x%5E2-2x%2B4%29%29 ALIGN=MIDDLE  ALT=\"highlight%28f%28x%29=-%28x%5E3-2x%5E2%2B5x%2B2%29%2F%28x%5E2-2x%2B4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );