document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=739462>Question 739462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the range of :h(x)=2x+1/x-3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #451174 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=451174\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What you wrote, h(x)=2x+1/x-3, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"h%28x%29=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-4%2C4%2C-8%2C6%2C2x%2B1%2Fx-3%2C2x-3%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-4%2C4%2C-8%2C6%2C2x%2B1%2Fx-3%2C2x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But maybe you meant h(x)=(2x+1)/(x-3), which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=%282x%2B1%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"h%28x%29=%282x%2B1%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-9%2C15%2C-15%2C20%2C%282x%2B1%29%2F%28x-3%29%2C2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-9%2C15%2C-15%2C20%2C%282x%2B1%29%2F%28x-3%29%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both functions get as low as <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( "and as high as <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( "but we have to figure out if there are some in-between values that the functions cannot achieve. <BR>\n" );
document.write( "I will help you discover what each of those functions does.<BR>\n" );
document.write( " <BR>\n" );
document.write( "EASIEST FIRST:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=%282x%2B1%29%2F%28x-3%29=%282x-6%2B7%29%2F%28x-3%29=%282x-6%29%2F%28x-3%29%2B7%2F%28x-3%29=2%2B7%2F%28x-3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a very large absolute value, the last term gets very small and the value of the function gets very close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You can get as close as you want, by making that last term as small (in absolute value) as necessary.<BR>\n" );
document.write( "So the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a horizontal asymptote.<BR>\n" );
document.write( "However, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29=%282x%2B1%29%2F%28x-3%29=2%2B7%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x%29=%282x%2B1%29%2F%28x-3%29=2%2B7%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can not equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because the last term can never be zero.<BR>\n" );
document.write( "So you could say that the range of that function is all the real numbers, except 2.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE OTHER FUNCTION:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"h%28x%29=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > does something different.<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a very large absolute value, the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets very small and the value of the function gets very close to the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-3 ALIGN=MIDDLE  ALT=\"y=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You can get as close as you want, by making that term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > as small in absolute value as necessary.<BR>\n" );
document.write( "So the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-3 ALIGN=MIDDLE  ALT=\"y=2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an oblique asymptote.<BR>\n" );
document.write( "For <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29%3C-3 ALIGN=MIDDLE  ALT=\"h%28x%29%3C-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29%3E-3 ALIGN=MIDDLE  ALT=\"h%28x%29%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so the function never takes the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but there are other forbidden values.<BR>\n" );
document.write( "You can figure out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"h%28x%29=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot take any values between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"-3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"-3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that means that<BR>\n" );
document.write( "THE RANGE IS all the real numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3C=-3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=-3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3E=-3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You may want to express that in whatever notation is expected of you,<BR>\n" );
document.write( "maybe as a union of intervals:<BR>\n" );
document.write( "(-infinity,-3-2sqrt(2)] U [-3+2sqrt(2),infinity)<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you know about derivatives, you would find that the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29 ALIGN=MIDDLE  ALT=\"h%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "h'(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"2-1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x=-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means that the function will have a maximum with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29%3C-3 ALIGN=MIDDLE  ALT=\"h%28x%29%3C-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x=-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and a minimum with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29%3E-3 ALIGN=MIDDLE  ALT=\"h%28x%29%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You could calculate them as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-sqrt%282%29%2F2%29=2%28-sqrt%282%29%2F2%29%2B1%2F%28-sqrt%282%29%2F2%29-3=-sqrt%282%29-sqrt%282%29-3=-2sqrt%282%29-3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28sqrt%282%29%2F2%29=2%28sqrt%282%29%2F2%29%2B1%2F%28sqrt%282%29%2F2%29-3=sqrt%282%29%2Bsqrt%282%29-3=2sqrt%282%29-3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Without mentioning derivatives, you could figure out what values of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"y=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > are possible for some real number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by solving that equation for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"y=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=yx=2x%5E2%2B1-3x ALIGN=MIDDLE  ALT=\"yx=2x%5E2%2B1-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-3x-yx%2B1=0 ALIGN=MIDDLE  ALT=\"2x%5E2-3x-yx%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-%283%2By%29x%2B1=0 ALIGN=MIDDLE  ALT=\"2x%5E2-%283%2By%29x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You know that there will be one or two solutions wherever the discriminant is not negative, meaning when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2By%29%5E2-8%3E=0 ALIGN=MIDDLE  ALT=\"%283%2By%29%5E2-8%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B6y%2By%5E2-8%3E=0 ALIGN=MIDDLE  ALT=\"9%2B6y%2By%5E2-8%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B6y%2B1%3E=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B6y%2B1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B6y%2B1%3E=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B6y%2B1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3+%2B-+2sqrt%282%29 ALIGN=MIDDLE  ALT=\"y=-3+%2B-+2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "the solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B6y%2B1%3E=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B6y%2B1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the range of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=2x%2B1%2Fx-3 ALIGN=MIDDLE  ALT=\"h%28x%29=2x%2B1%2Fx-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3C=-3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=-3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"y%3E=-3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );