document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87830>Question 87830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a question that states:  find the inverse of the function f(x)=3x^2+12x+6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63879 by </B><A HREF=/tutors/aboutme.mpl?userid=tutor_paul><B>tutor_paul(519)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tutor_paul><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=tutor_paul><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=63879\">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=3x%5E2%2B12x%2B6 ALIGN=MIDDLE  ALT=\"f%28x%29=3x%5E2%2B12x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rename f(x) as y:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2%2B12x%2B6 ALIGN=MIDDLE  ALT=\"y=3x%5E2%2B12x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for x:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B12x%2B6-y=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B12x%2B6-y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that you can use the quadratic formula, with the 'c' term = 6-y:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12%2B-sqrt%28144-%284%2A3%2A%286-y%29%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"x=%28-12%2B-sqrt%28144-%284%2A3%2A%286-y%29%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12%2B-sqrt%28144-%2812%2A%286-y%29%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"x=%28-12%2B-sqrt%28144-%2812%2A%286-y%29%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12%2B-sqrt%28144-%2872-12y%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"x=%28-12%2B-sqrt%28144-%2872-12y%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12%2B-sqrt%2872%2B12y%29%29%2F6 ALIGN=MIDDLE  ALT=\"x=%28-12%2B-sqrt%2872%2B12y%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So now that I have simplified this, I see that I have a +/- in my result.<BR>\n" );
document.write( "This immediately tells me that any given 'x' corresponds to 2 different y's.<BR>\n" );
document.write( "Hence the inverse of this IS NOT A FUNCTION. So the answer is that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=3x%5E2%2B12x%2B6 ALIGN=MIDDLE  ALT=\"f%28x%29=3x%5E2%2B12x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > has no inverse.\r<BR>\n" );
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document.write( "-----------------------------------------------------------------\r<BR>\n" );
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document.write( "Another way to do this is to apply the \"Horizontal Line Test\" to determine if<BR>\n" );
document.write( "a function has an inverse. This test states: If any horizontal line intersects the graph of f more than once, then f does not have an inverse.\r<BR>\n" );
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document.write( "Here is the graph of your function - you can see that this does not pass the Horizontal Line Test.\r<BR>\n" );
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document.write( " <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B12x%2B6+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B12x%2B6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2812%29%5E2-4%2A3%2A6=72 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2812%29%5E2-4%2A3%2A6=72\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=72 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-12%2B-sqrt%28+72+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-12%2B-sqrt%28+72+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+72+%29%29%2F2%5C3+=+-0.585786437626905 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+72+%29%29%2F2%5C3+=+-0.585786437626905\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%2812%29-sqrt%28+72+%29%29%2F2%5C3+=+-3.41421356237309 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2812%29-sqrt%28+72+%29%29%2F2%5C3+=+-3.41421356237309\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B12x%2B6 ALIGN=MIDDLE  ALT=\"3x%5E2%2B12x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B12x%2B6+=+3%28x--0.585786437626905%29%2A%28x--3.41421356237309%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B12x%2B6+=+3%28x--0.585786437626905%29%2A%28x--3.41421356237309%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.585786437626905, -3.41421356237309.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B12%2Ax%2B6+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B12%2Ax%2B6+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "<BR>\n" );
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document.write( "Good Luck,<BR>\n" );
document.write( "tutor_paul@yahoo.com\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );