document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=937605>Question 937605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of this function f(x) defined over the largest possible domain.\r<BR>\n" );
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document.write( "x + 1/x\r<BR>\n" );
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document.write( "I tried to do this by completing the square by multiplying out the x at the bottom. So I got:<BR>\n" );
document.write( "x^2 + 1 = 0<BR>\n" );
document.write( "(x+1/2)^2 -1/4 = 0<BR>\n" );
document.write( "So f(x) > 1/4 \r<BR>\n" );
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document.write( "But the textbook says f(x) < or equal to -2<BR>\n" );
document.write( "or f(x) > or equal to 2.\r<BR>\n" );
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document.write( "I really need help on how to get to these answers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #570997 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39620)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=570997\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)=x+1/x  is ambiguous.  Further it is not a quadratic function.  You can get better help asking for what you exactly want and giving the full problem description.\r<BR>\n" );
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document.write( "Some clarification given:<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1%2Fx ALIGN=MIDDLE  ALT=\"y=x%2B1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which was exactly as you wrote.  Sometimes students want the x+1 grouped as a separate expression but omit the grouping symbols.\r<BR>\n" );
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document.write( "The allowed values for x must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so that y will be undefined for x at 0.\r<BR>\n" );
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document.write( "You wanted to find the range.  Think what happens to y very near to 0 on the left and on the right.  ....<BR>\n" );
document.write( "....  UNBOUNDED negative as x approaches 0 from the left, and unbounded positive as x approaches 0 from the right.\r<BR>\n" );
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document.write( "Without further algebra, although you might want it,...  I skip ahead to a graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-15%2C15%2C-15%2C15%2Cx%2B1%2Fx%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-15%2C15%2C-15%2C15%2Cx%2B1%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A derivative might be needed to find the rest of the range restriction.<BR>\n" );
document.write( "THIS MEANS YOU ARE  OR HAVE STUDIED CALCULUS 1.  ( I cannot think of another way).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fdy=d%28x%2B1%2Fx%29 ALIGN=MIDDLE  ALT=\"d%2Fdy=d%28x%2B1%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B%28-1%29x%5E%28-2%29 ALIGN=MIDDLE  ALT=\"1%2B%28-1%29x%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"1-1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%28x%5E2%2Fx%5E2%29-1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"1%28x%5E2%2Fx%5E2%29-1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-1%29%2Fx%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2-1%29%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "WHERE is this derivative equal to 0?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-1=0 ALIGN=MIDDLE  ALT=\"x%5E2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28x=-1%29 ALIGN=MIDDLE  ALT=\"highlight_green%28x=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28x=1%29 ALIGN=MIDDLE  ALT=\"highlight_green%28x=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note from the sketch of the graph you might make,  x=-1 will give a local max, and x=1 will give a local min.  Use your given function definition to find those function values.  The range must EXCLUDE the values between f(-1) and f(1).\r<BR>\n" );
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document.write( "To finish....<BR>\n" );
document.write( "Range is the set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%3C=-2 ALIGN=MIDDLE  ALT=\"f%28x%29%3C=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29%3E=2 ALIGN=MIDDLE  ALT=\"f%28x%29%3E=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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