document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92797>Question 92797</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range for the function. Use interval notation.\r<BR>\n" );
document.write( "\n" );
document.write( "f(x)= (x^2 + x - 12)/[(x-3)(x+5)] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67530 by </B><A HREF=/tutors/aboutme.mpl?userid=mathispowerful><B>mathispowerful(115)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathispowerful><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=67530\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First simplify it:\r<BR>\n" );
document.write( "\n" );
document.write( "factor the numerator. we get\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-3%29%2F%28%28x-3%29%28x%2B5%29%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-3%29%2F%28%28x-3%29%28x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
document.write( "\n" );
document.write( "cancel the common factor x-3\r<BR>\n" );
document.write( "\n" );
document.write( "we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "for a rational expression, the denominator can not be 0,\r<BR>\n" );
document.write( "\n" );
document.write( "so x can not equal to -5\r<BR>\n" );
document.write( "\n" );
document.write( "the domain is any real number except -5.\r<BR>\n" );
document.write( "\n" );
document.write( "The range: Let's simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > further\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B5-1%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%2B5-1%29%2F%28x%2B5%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%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"1-1%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "let's take a look at the fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"1%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "when x is in the neighborhood of -5, the fraction can be positive infinity and <BR>\n" );
document.write( "negative infinity, and thus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-1%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"1-1%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be both positive infinity and<BR>\n" );
document.write( "negative infinity.\r<BR>\n" );
document.write( "\n" );
document.write( "so the range is from negative infinity to positive infinity.\r<BR>\n" );
document.write( "\n" );
document.write( "this conclusion can be verified by the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > below:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C+-5.05%2C-4.95%2C+-1000%2C+1000%2C++%281-1%2F%28x%2B5%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C+-5.05%2C-4.95%2C+-1000%2C+1000%2C++%281-1%2F%28x%2B5%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );