document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166656>Question 166656</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I missed the day of class when we covered domains in my class, and now i am hopelessly lost.\r<BR>\n" );
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document.write( "Please help me solve the following problems, and explain how you did it. I am not very good at math, so the more you can break it down and explain it, the better.\r<BR>\n" );
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document.write( "find the domain of the rational function:<BR>\n" );
document.write( "h(t)=(t-5)/(t^2-25)\r<BR>\n" );
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document.write( "g(x)=(x^3-27)/(4x)\r<BR>\n" );
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document.write( "the problems are written in faction form, but I don't know how to do that with the computer, so I am not sure if what i have written means the same thing. It should read t-5 over t^2-25.\r<BR>\n" );
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document.write( "I would really appreciate the help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122840 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=122840\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember, the domain is simply the set of all x values that produce a y value. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=%28t-5%29%2F%28t%5E2-25%29 ALIGN=MIDDLE  ALT=\"h%28t%29=%28t-5%29%2F%28t%5E2-25%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=t%5E2-25=0 ALIGN=MIDDLE  ALT=\"t%5E2-25=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 \"t\" 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=%28t-5%29%28t%2B5%29=0 ALIGN=MIDDLE  ALT=\"%28t-5%29%28t%2B5%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=t-5=0 ALIGN=MIDDLE  ALT=\"t-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2B5=0 ALIGN=MIDDLE  ALT=\"t%2B5=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=t=5 ALIGN=MIDDLE  ALT=\"t=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=-5 ALIGN=MIDDLE  ALT=\"t=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now solve for t 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=t=5 ALIGN=MIDDLE  ALT=\"t=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=-5 ALIGN=MIDDLE  ALT=\"t=-5\"   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=t=-5 ALIGN=MIDDLE  ALT=\"t=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5 ALIGN=MIDDLE  ALT=\"t=5\"   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=t=-5 ALIGN=MIDDLE  ALT=\"t=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5 ALIGN=MIDDLE  ALT=\"t=5\"   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{t|t\in\mathbb{R} t\neq-5 and t\neq5\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: t is the set of all real numbers except t CANNOT equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or t CANNOT equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   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, -5\right)\cup\left(-5,5 \right)\cup\left(5,\infty \right)\">\r<BR>\n" );
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document.write( "note: remember, the parenthesis <font size=4><b>excludes</b></font> -5 and 5 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%28-4.5%2C-7%2C4.65%2C-7%29%29%2C%0D%0Ablue%28line%28-4.5%2C-6%2C4.65%2C-6%29%29%2C%0D%0Ablue%28line%28-4.5%2C-5%2C4.65%2C-5%29%29%2C%0D%0Ablue%28arrow%285.5%2C-7%2C10%2C-7%29%29%2C%0D%0Ablue%28arrow%285.5%2C-6.5%2C10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%285.5%2C-6%2C10%2C-6%29%29%2C%0D%0Ablue%28arrow%285.5%2C-5.5%2C10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%285.5%2C-5%2C10%2C-5%29%29%2C%0D%0Ablue%28arrow%28-5.5%2C-7%2C-10%2C-7%29%29%2C%0D%0Ablue%28arrow%28-5.5%2C-6.5%2C-10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%28-5.5%2C-6%2C-10%2C-6%29%29%2C%0D%0Ablue%28arrow%28-5.5%2C-5.5%2C-10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%28-5.5%2C-5%2C-10%2C-5%29%29%2C%0D%0A%0D%0Acircle%28-5%2C-5.8%2C0.35%29%2C%0D%0Acircle%28-5%2C-5.8%2C0.4%29%2C%0D%0Acircle%28-5%2C-5.8%2C0.45%29%2C%0D%0A%0D%0A%0D%0Acircle%285%2C-5.8%2C0.35%29%2C%0D%0Acircle%285%2C-5.8%2C0.4%29%2C%0D%0Acircle%285%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=t=-5 ALIGN=MIDDLE  ALT=\"t=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5 ALIGN=MIDDLE  ALT=\"t=5\"   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 t can be any number except t cannot equal -5 or 5\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28x%5E3-27%29%2F%284x%29 ALIGN=MIDDLE  ALT=\"g%28x%29=%28x%5E3-27%29%2F%284x%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=4x=0 ALIGN=MIDDLE  ALT=\"4x=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=%280%29%2F%284%29 ALIGN=MIDDLE  ALT=\"x=%280%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 4 to isolate x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   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=0 ALIGN=MIDDLE  ALT=\"x=0\"   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\neq0\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: x is the set of all real numbers except x CANNOT equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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, 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 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%28arrow%280.2%2C-7%2C10%2C-7%29%29%2C%0D%0Ablue%28arrow%280.2%2C-6.5%2C10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%280.2%2C-6%2C10%2C-6%29%29%2C%0D%0Ablue%28arrow%280.2%2C-5.5%2C10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%280.2%2C-5%2C10%2C-5%29%29%2C%0D%0Ablue%28arrow%28-0.2%2C-7%2C-10%2C-7%29%29%2C%0D%0Ablue%28arrow%28-0.2%2C-6.5%2C-10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%28-0.2%2C-6%2C-10%2C-6%29%29%2C%0D%0Ablue%28arrow%28-0.2%2C-5.5%2C-10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%28-0.2%2C-5%2C-10%2C-5%29%29%2C%0D%0A%0D%0Acircle%280%2C-5.8%2C0.35%29%2C%0D%0Acircle%280%2C-5.8%2C0.4%29%2C%0D%0Acircle%280%2C-5.8%2C0.45%29%2C%0D%0Acircle%280%2C-5.8%2C0.4%29%2C%0D%0Acircle%280%2C-5.8%2C0.45%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of the domain in blue and the excluded value represented by open circle\r<BR>\n" );
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document.write( "Notice we have a continuous line until we get to the hole at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is represented by the open circle).<BR>\n" );
document.write( "This graphically represents our domain in which x can be any number except x cannot equal 0<BR>\n" );
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