document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=356329>Question 356329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> G(t)= (T^3+8)/(T^3-1)\r<BR>\n" );
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document.write( "1. Find G^-1(t)<BR>\n" );
document.write( "2. What is the domain and range of G. in each case, show or explain how you determine you answer.\r<BR>\n" );
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document.write( "Thanks in advance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #254731 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=254731\">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=G%28T%29=%28T%5E3%2B8%29%2F%28T%5E3-1%29+ ALIGN=MIDDLE  ALT=\"G%28T%29=%28T%5E3%2B8%29%2F%28T%5E3-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the inverse, interchange the positions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G ALIGN=MIDDLE  ALT=\"G\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for the new <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G ALIGN=MIDDLE  ALT=\"G\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "The new <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G ALIGN=MIDDLE  ALT=\"G\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the inverse. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28T%29=+%28%28highlight%28G%29%29%5E3%2B8%29%2F%28%28highlight%28G%29%29%5E3-1%29 ALIGN=MIDDLE  ALT=\"highlight%28T%29=+%28%28highlight%28G%29%29%5E3%2B8%29%2F%28%28highlight%28G%29%29%5E3-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28G%5E3-1%29=G%5E3%2B8 ALIGN=MIDDLE  ALT=\"T%28G%5E3-1%29=G%5E3%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TG%5E3-T=G%5E3%2B8 ALIGN=MIDDLE  ALT=\"TG%5E3-T=G%5E3%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TG%5E3-G%5E3=T%2B8 ALIGN=MIDDLE  ALT=\"TG%5E3-G%5E3=T%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%5E3%28T-1%29=T%2B8 ALIGN=MIDDLE  ALT=\"G%5E3%28T-1%29=T%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%5E3=%28T%2B8%29%2F%28T-1%29 ALIGN=MIDDLE  ALT=\"G%5E3=%28T%2B8%29%2F%28T-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28G%5Binv%5D%28T%29=root%283%2C%28T%2B8%29%2F%28T-1%29%29%29 ALIGN=MIDDLE  ALT=\"highlight%28G%5Binv%5D%28T%29=root%283%2C%28T%2B8%29%2F%28T-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".\r<BR>\n" );
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document.write( "The domain is all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28T%29 ALIGN=MIDDLE  ALT=\"G%28T%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is defined. The only problem is when the denominator equals zero since division by zero is undefined.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5E3-1=0 ALIGN=MIDDLE  ALT=\"T%5E3-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=1 ALIGN=MIDDLE  ALT=\"T=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the domain is all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" > except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=1 ALIGN=MIDDLE  ALT=\"T=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or in interval notation <BR>\n" );
document.write( "Domain:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) U (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%28x%5E3%2B8%29%2F%28x%5E3-1%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%28x%5E3%2B8%29%2F%28x%5E3-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The range is all values that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28T%29 ALIGN=MIDDLE  ALT=\"G%28T%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can have.<BR>\n" );
document.write( "As you see from the graph, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28T%29 ALIGN=MIDDLE  ALT=\"G%28T%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can take on pretty much any value except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28T%29=1 ALIGN=MIDDLE  ALT=\"G%28T%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Algrebraically you can see why it can't ever exactly equal 1.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28T%5E3%2B8%29%2F%28T%5E3-1%29=1+ ALIGN=MIDDLE  ALT=\"%28T%5E3%2B8%29%2F%28T%5E3-1%29=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5E3%2B8=T%5E3-1+ ALIGN=MIDDLE  ALT=\"T%5E3%2B8=T%5E3-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8=-1 ALIGN=MIDDLE  ALT=\"8=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the range can take on all values except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28T%29=1 ALIGN=MIDDLE  ALT=\"G%28T%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or in interval notation.<BR>\n" );
document.write( "Range:(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) U (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >) </SPAN>\n" );
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