document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=128507>Question 128507</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of the following\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7%2F%28x%2B9%29 ALIGN=MIDDLE  ALT=\"y=7%2F%28x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #94022 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=94022\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "d)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%287%29%2F%28x%2B9%29 ALIGN=MIDDLE  ALT=\"y=%287%29%2F%28x%2B9%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%2B9=0 ALIGN=MIDDLE  ALT=\"x%2B9=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=x=0-9 ALIGN=MIDDLE  ALT=\"x=0-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract 9 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9 ALIGN=MIDDLE  ALT=\"x=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9 ALIGN=MIDDLE  ALT=\"x=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes the denominator equal to zero, this means we must exclude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9 ALIGN=MIDDLE  ALT=\"x=-9\"   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\in\mathbb{R} x\neq-9\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-9 ALIGN=MIDDLE  ALT=\"x%3C%3E-9\"   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, -9\right)\cup\left(-9,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> -9 from the domain\r<BR>\n" );
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document.write( "Now to find the range, simply graph the function to get\r<BR>\n" );
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document.write( "note: ignore the vertical lines, they are not part of the graph and are asymptotes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%287%29%2F%28x%2B9%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%287%29%2F%28x%2B9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at the graph, we can see that y can be any number <b>except</b> 0. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So our range is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{y|y\in\mathbb{R} y\neq0\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: y is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our range 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, 0\right)\cup\left(0,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> 0 from the range\r<BR>\n" );
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document.write( "<hr>\r<BR>\n" );
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document.write( "e)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"y=%283%29%2F%28x%5E2-4%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%5E2-4=0 ALIGN=MIDDLE  ALT=\"x%5E2-4=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-2%29%28x%2B2%29=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%2B2%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-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2=0 ALIGN=MIDDLE  ALT=\"x%2B2=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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   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\in\mathbb{R} x\neq-2 and x\neq2\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-2 ALIGN=MIDDLE  ALT=\"x%3C%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E2 ALIGN=MIDDLE  ALT=\"x%3C%3E2\"   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, -2\right)\cup\left(-2,2 \right)\cup\left(2,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> -2 and 2 from the domain\r<BR>\n" );
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document.write( "Now to find the range, simply graph the function to get\r<BR>\n" );
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document.write( "note: ignore the vertical lines, they are not part of the graph and are asymptotes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%283%29%2F%28x%5E2-4%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-15%2C5%2C-10%2C10%2C%283%29%2F%28x%5E2-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at the graph, we can see that y can be any number <b>except</b> 0. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So our range is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{y|y\in\mathbb{R} y\neq0\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: y is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our range 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, 0\right)\cup\left(0,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> 0 from the range </SPAN>\n" );
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