document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=700998>Question 700998</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find any points of discontinuity for the rational function : <BR>\n" );
document.write( "(x^3+x^2-4x-4/x^2+2x-3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #432229 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=432229\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You probably mean f(x)=(x^3+x^2-4x-4)/(x^2+2x-3) ,<BR>\n" );
document.write( "which I can write as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E3%2Bx%5E2-4x-4%29%2F%28x%5E2%2B2x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E3%2Bx%5E2-4x-4%29%2F%28x%5E2%2B2x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Factoring polynomials never goes away. You have to factor numerator and denominator.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx%5E2-4x-4 ALIGN=MIDDLE  ALT=\"x%5E3%2Bx%5E2-4x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored \"by grouping\" as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx%5E2-4x-4=%28x%5E3-4x%29%2B%28x%5E2-4%29=x%28x%5E2-4%29%2B%28x%5E2-4%29=%28x%2B1%29%28x%5E2-4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It can be factored further because the difference of squares <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4=x%5E2-2%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-4=x%5E2-2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "factors as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%2B2%29\"   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%5E2%2B2x-3=%28x%2B3%29%28x-1%29 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x-3=%28x%2B3%29%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Putting it all together:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E3%2Bx%5E2-4x-4%29%2F%28x%5E2%2B2x-3%29=%28x%2B1%29%28x%2B2%29%28x-2%29%2F%28%28x%2B3%29%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The function is not defined (it does not exist) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the denominator is zero for those values of <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( "To be continuous, the function has to be defined. So at those points the function is not continuous.<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the function has a vertical asymptote.<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > from either side, the denominator approaches zero.<BR>\n" );
document.write( "At the same time, the numerator approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B1%29%281%2B2%29%281-2%29=-6 ALIGN=MIDDLE  ALT=\"%281%2B1%29%281%2B2%29%281-2%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "As a consequence the absolute value of the function grows without limits,<BR>\n" );
document.write( "and the graph hugs the vertical line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Something similar happens at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-7%2C5%2C-20%2C20%2C%28x%5E3%2Bx%5E2-4x-4%29%2F%28x%5E2%2B2x-3%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-7%2C5%2C-20%2C20%2C%28x%5E3%2Bx%5E2-4x-4%29%2F%28x%5E2%2B2x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: I would call those values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , points of discontinuity.<BR>\n" );
document.write( "However, names of different kinds of discontinuity are not universally agreed upon,<BR>\n" );
document.write( "and some may not like to call <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > points of discontinuity.<BR>\n" );
document.write( "They would say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > are not really points, with an x value and a y value, and the name \"points of discontinuity\" could be confused with \"point discontinuity,\" which is a different kind of discontinuity.<BR>\n" );
document.write( "A function may not be continuous at one point that is just a hole in the graph,<BR>\n" );
document.write( "as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28x%2B1%29%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=%28x%2B1%29%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That function has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=1 ALIGN=MIDDLE  ALT=\"g%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where g(x) is not defined.<BR>\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28x%2B1%29%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=%28x%2B1%29%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graphs as a horizontal line minus the point (-1,1) that is a hole in the graph.<BR>\n" );
document.write( "Some call that a \"point discontinuity\", and others call it a \"removable discontinuity\". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );