document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=125480>Question 125480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi. Can I have some help finding the domain of this function?\r<BR>\n" );
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document.write( "g(x)=&#8730;x/2x^2+x-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #91926 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=91926\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think you meant the function to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x%29%2F2x%5E2%2Bx-1 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x%29%2F2x%5E2%2Bx-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > rather than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x%2F2x%5E2%2Bx-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x%2F2x%5E2%2Bx-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The domain of a function is the set of all real values that x can take on such that the function is defined.  I'm also presuming that you are mapping the function to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2 ALIGN=MIDDLE  ALT=\"R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is undefined if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%2F2x%5E2 ALIGN=MIDDLE  ALT=\"sqrt%28x%29%2F2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is undefined at zero, the domain range is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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;\" >)\r<BR>\n" );
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document.write( "If, on the other hand you really meant the alternative function above with the entire expression under the radical, you could immediately say that zero is excluded from the domain, but you would have to work out the value for x that would cause <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2x%5E2%2Bx-1 ALIGN=MIDDLE  ALT=\"x%2F2x%5E2%2Bx-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be less than zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2x%5E2%2Bx-1%3C0 ALIGN=MIDDLE  ALT=\"x%2F2x%5E2%2Bx-1%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2x%5E3-2x%5E2%3C0 ALIGN=MIDDLE  ALT=\"x%2B2x%5E3-2x%5E2%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-2x%5E2%2Bx%3C0 ALIGN=MIDDLE  ALT=\"2x%5E3-2x%5E2%2Bx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, if we make a graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E3-2x%5E2%2Bx ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E3-2x%5E2%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we should be able to see where the function is less than zero, at least in an approximate sense.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-3%2C3%2C-3%2C3%2C2x%5E3-2x%5E2%2Bx%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-3%2C3%2C-3%2C3%2C2x%5E3-2x%5E2%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at this graph, we see that the only real number zero of the function is at zero, so the interval for which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E3-2x%5E2%2Bx%3C0 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E3-2x%5E2%2Bx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "That means that the domain of <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;\" > is the same regardless of whether <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x%29%2F2x%5E2%2Bx-1 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x%29%2F2x%5E2%2Bx-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x%2F2x%5E2%2Bx-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x%2F2x%5E2%2Bx-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, namely: (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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;\" >)\r<BR>\n" );
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document.write( "The following graphs are of both original functions.  The green one is the first way, and the red one is the one with the entire expression under the radical.  You can see that the graph visually supports the analysis of the domain interval.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-2%2C8%2C-2%2C8%2Csqrt%28x%2F2x%5E2%2Bx-1%29%2Csqrt%28x%29%2F2x%5E2%2Bx-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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