document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104188>Question 1104188</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with this question: If f(x) = x^2-x-1 and f[g(x)]=4x^2 + 10x + 5, find g(x).<BR>\n" );
document.write( "Any help will be greatly appreciated!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #718924 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=718924\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There has to be a better way, but here it goes.<BR>\n" );
document.write( "If f[g(x)] =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+10x+%2B+5 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+10x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1 ALIGN=MIDDLE  ALT=\"f%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >{ f[g(x)] } =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%284x%5E2+%2B+10x+%2B+5%29 ALIGN=MIDDLE  ALT=\"f%5E-1%284x%5E2+%2B+10x+%2B+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the inverse function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-x-1 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-2%2C8%2Cx%5E2-x-1%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-2%2C8%2Cx%5E2-x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we exchange <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-x-1 ALIGN=MIDDLE  ALT=\"y=x%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=y%5E2-y-1 ALIGN=MIDDLE  ALT=\"x=y%5E2-y-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281+%2B-+sqrt%284x%2B5%29%29%2F2 ALIGN=MIDDLE  ALT=\"y=%281+%2B-+sqrt%284x%2B5%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C%281%2Bsqrt%284x%2B5%29%29%2F2%2C%281-sqrt%284x%2B5%29%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is really two functions,<BR>\n" );
document.write( "and when working with real numbers,<BR>\n" );
document.write( "it is only defined for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=-5%2F4 ALIGN=MIDDLE  ALT=\"x%3E=-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we continue, to see what we can get.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%28x%29=%281+%2B+sqrt%284x%2B5%29%29%2F2 ALIGN=MIDDLE  ALT=\"f%5E-1%28x%29=%281+%2B+sqrt%284x%2B5%29%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=f%5E-1%284x%5E2+%2B+10x+%2B+5%29=%281+%2B+sqrt%284%284x%5E2+%2B+10x+%2B+5%29+%2B5%29%29%2F2%0D%0A=%281%2Bsqrt%2816x%5E2%2B40x%2B20%2B5%29%29%2F2=%281%2Bsqrt%2816x%5E2%2B40x%2B25%29%29%2F2=%281%2B%284x%2B5%29%29%2F2=%284x%2B6%29%2F2=2x%2B3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28g%28x%29=2x%2B3%29 ALIGN=MIDDLE  ALT=\"highlight%28g%28x%29=2x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTES:<BR>\n" );
document.write( "1) There was no problem with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%28x%29=%281+%2B+sqrt%284x%2B5%29%29%2F2 ALIGN=MIDDLE  ALT=\"f%5E-1%28x%29=%281+%2B+sqrt%284x%2B5%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  being defined only for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=-5%2F4 ALIGN=MIDDLE  ALT=\"x%3E=-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because we are going to apply it to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+10x+%2B+5=%282x%2B5%2F2%29%5E2-5%2F4%3E=-5%2F4 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+10x+%2B+5=%282x%2B5%2F2%29%5E2-5%2F4%3E=-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "2)<BR>\n" );
document.write( "Would we get another solution out of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%28x%29=%281-sqrt%284x%2B5%29%29%2F2 ALIGN=MIDDLE  ALT=\"f%5E-1%28x%29=%281-sqrt%284x%2B5%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "Let's check.<BR>\n" );
document.write( "In that case, we would have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%284x%5E2+%2B+10x+%2B+5%29=%281-sqrt%284%284x%5E2+%2B+10x+%2B+5%29+%2B5%29%29%2F2%0D%0A=%281-sqrt%2816x%5E2%2B40x%2B20%2B5%29%29%2F2=%281-sqrt%2816x%5E2%2B40x%2B25%29%29%2F2=%281-%284x%2B5%29%29%2F2=%28-4x-4%29%2F2=-2x-2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2x-2%29=4x%5E2+%2B+10x+%2B+5 ALIGN=MIDDLE  ALT=\"f%28-2x-2%29=4x%5E2+%2B+10x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E-1%284x%5E2+%2B+10x+%2B+5%29=-2x-2 ALIGN=MIDDLE  ALT=\"f%5E-1%284x%5E2+%2B+10x+%2B+5%29=-2x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > could  be another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Let's see.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2x-2%29=%28-2x-2%29%5E2-%28-2x-2%29-1=4x%5E2%2B4x%2B1%2B2x%2B2-1=4x%5E2%2B6x%2B1 ALIGN=MIDDLE  ALT=\"f%28-2x-2%29=%28-2x-2%29%5E2-%28-2x-2%29-1=4x%5E2%2B4x%2B1%2B2x%2B2-1=4x%5E2%2B6x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "No. It does not work. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );