document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=103627>Question 103627</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain of the function f(x)= (x+2)/(x^2-9)\r<BR>\n" );
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document.write( "(write your solution in interval notation) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #75347 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=75347\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%2B2%29%2F%28x%5E2-9%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%2B2%29%2F%28x%5E2-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-9=0 ALIGN=MIDDLE  ALT=\"x%5E2-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%28x%2B3%29=0 ALIGN=MIDDLE  ALT=\"%28x-3%29%28x%2B3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3=0 ALIGN=MIDDLE  ALT=\"x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<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;\" > or <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;\" >  Now solve for x in each case\r<BR>\n" );
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document.write( "So our solutions are <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;\" > or <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;\" >\r<BR>\n" );
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document.write( "Since <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;\" > 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;\" > make the denominator equal to zero, this means we must exclude <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;\" > 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;\" > from our domain\r<BR>\n" );
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document.write( "So our domain is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{x|x\epsilon\mathbb{R} x\neq-3 and x\neq3\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: x is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-3 ALIGN=MIDDLE  ALT=\"x%3C%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E3 ALIGN=MIDDLE  ALT=\"x%3C%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our domain looks like this in interval notation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, -3\right)\cup\left(-3,3 \right)\cup\left(3,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> -3 and 3 from the domain\r<BR>\n" );
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document.write( "If we wanted to graph the domain on a number line, we would get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C50%2C-10%2C10%2C-10%2C10%2C%0D%0Anumber_line%28+500%2C+-10%2C+10%29%2C%0D%0Ablue%28line%28-2.5%2C-7%2C2.65%2C-7%29%29%2C%0D%0Ablue%28line%28-2.5%2C-6%2C2.65%2C-6%29%29%2C%0D%0Ablue%28line%28-2.5%2C-5%2C2.65%2C-5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-7%2C10%2C-7%29%29%2C%0D%0Ablue%28arrow%283.5%2C-6.5%2C10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-6%2C10%2C-6%29%29%2C%0D%0Ablue%28arrow%283.5%2C-5.5%2C10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-5%2C10%2C-5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-7%2C-10%2C-7%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-6.5%2C-10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-6%2C-10%2C-6%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-5.5%2C-10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-5%2C-10%2C-5%29%29%2C%0D%0A%0D%0Acircle%28-3%2C-5.8%2C0.35%29%2C%0D%0Acircle%28-3%2C-5.8%2C0.4%29%2C%0D%0Acircle%28-3%2C-5.8%2C0.45%29%2C%0D%0A%0D%0A%0D%0Acircle%283%2C-5.8%2C0.35%29%2C%0D%0Acircle%283%2C-5.8%2C0.4%29%2C%0D%0Acircle%283%2C-5.8%2C0.45%29%0D%0A%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of the domain in blue and the excluded values represented by open circles\r<BR>\n" );
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document.write( "Notice we have a continuous line until we get to the holes 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;\" > 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;\" > (which is represented by the open circles).<BR>\n" );
document.write( "This graphically represents our domain in which x can be any number except x cannot equal -3 or 3\r<BR>\n" );
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document.write( " </SPAN>\n" );
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