document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=41949>Question 41949</A>: <I><SPAN STYLE=\"word-break: break-all;\"> consider the function f(x) = (cx)/ (2x + 3) , where x is not ewual to -3/2.  Find all values of c (if any) for which f(f(x)) = x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27109 by </B><A HREF=/tutors/aboutme.mpl?userid=psbhowmick><B>psbhowmick(878)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=psbhowmick><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=psbhowmick><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=27109\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+c%2Ax%2F+%282x+%2B+3%29+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+c%2Ax%2F+%282x+%2B+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28f%28x%29%29+=+c%2Af%28x%29%2F+%282%2Af%28x%29+%2B+3%29+ ALIGN=MIDDLE  ALT=\"f%28f%28x%29%29+=+c%2Af%28x%29%2F+%282%2Af%28x%29+%2B+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "[Just replace 'x' with f(x)]\r<BR>\n" );
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document.write( "Thus, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28f%28x%29%29+=+c%2A%28c%2Ax%2F%282x+%2B+3%29%29%2F%282%2A%28c%2Ax%2F+%282x+%2B+3%29%29+%2B+3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28f%28x%29%29+=+%28c%5E2%2Ax%2F%282x+%2B+3%29%29%2F%28%282%2Ac%2Ax%2B6x%2B9%29%2F+%282x+%2B+3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28f%28x%29%29+=+c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29 ALIGN=MIDDLE  ALT=\"f%28f%28x%29%29+=+c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, if f(f(x)) = x then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29+=+x ALIGN=MIDDLE  ALT=\"c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2%2F%282%2Ac%2Ax%2B6x%2B9%29+=+1 ALIGN=MIDDLE  ALT=\"c%5E2%2F%282%2Ac%2Ax%2B6x%2B9%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+2%2Ac%2Ax%2B6x%2B9 ALIGN=MIDDLE  ALT=\"c%5E2+=+2%2Ac%2Ax%2B6x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-9=2cx%2B6x ALIGN=MIDDLE  ALT=\"c%5E2-9=2cx%2B6x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28c%5E2-9%29%2F%282c%2B6%29 ALIGN=MIDDLE  ALT=\"x=%28c%5E2-9%29%2F%282c%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "As x cannot be equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c%5E2-9%29%2F%282c%2B6%29 ALIGN=MIDDLE  ALT=\"%28c%5E2-9%29%2F%282c%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can also not be equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "But, however, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c%5E2-9%29%2F%282c%2B6%29+=+-3%2F2 ALIGN=MIDDLE  ALT=\"%28c%5E2-9%29%2F%282c%2B6%29+=+-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28c%5E2-9%29%2B3%282c%2B6%29+=+0 ALIGN=MIDDLE  ALT=\"2%28c%5E2-9%29%2B3%282c%2B6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c%5E2%2B6c+=+0 ALIGN=MIDDLE  ALT=\"2c%5E2%2B6c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%28c%2B3%29=0 ALIGN=MIDDLE  ALT=\"c%28c%2B3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Hence either c = 0 or c = -3.<BR>\n" );
document.write( "Thus f(f(x)) = x is valid for all values of 'x' except c = 0 and c = -3. </SPAN>\n" );
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