document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=155654>Question 155654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having touble with this F of G and G of F.<BR>\n" );
document.write( "A) Determine the domain and range of a) F(x)= 2/(x^2+1) and b) g(x)=Square root of x-3\r<BR>\n" );
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document.write( "B) Find the formula for f(g(x)) and the domain of f(g(x))\r<BR>\n" );
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document.write( "C) Find the formula for g(f(x)) and the domain of g(f(x)).\r<BR>\n" );
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document.write( "Thankyou so much... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #114685 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=114685\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to get you started\r<BR>\n" );
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document.write( "1)\r<BR>\n" );
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document.write( "a) Domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%29%2F%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%29%2F%28x%5E2%2B1%29\"   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%5E2%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero. Remember, you cannot divide by zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-1 ALIGN=MIDDLE  ALT=\"x%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 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=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "Since the square root of negative 1 is <b>not</b> a real number, this means that no real number will make the denominator equal to zero. So there are no domain restrictions. This means that the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282%29%2F%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%29%2F%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is all real numbers.\r<BR>\n" );
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document.write( "So the domain of f(x) in set-builder notation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{x\|x \in \mathbb{R}\right\}\"> which is <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\infty,\infty\"><font size=\"6\">)</font> in interval notation\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x-3%29 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x-3%29\"   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%3E=0 ALIGN=MIDDLE  ALT=\"x-3%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the radicand <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;\" > greater than or 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%3E=3 ALIGN=MIDDLE  ALT=\"x%3E=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 3 to both sides\r<BR>\n" );
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document.write( "So any number greater than or equal to 3 is in the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x-3%29 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the domain of g(x) in set-builder notation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{x\|x \ge 3\right\}\"> which is <font size=\"6\">[</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3,\infty\"><font size=\"6\">)</font> in interval notation\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=f%28x%29=%282%29%2F%28x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282%29%2F%28x%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28g%28x%29%29=%282%29%2F%28%28sqrt%28x-3%29%29%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29=%282%29%2F%28%28sqrt%28x-3%29%29%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=sqrt%28x-3%29 ALIGN=MIDDLE  ALT=\"g%28x%29=sqrt%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, replace each \"x\" with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29 ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29\"   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=f%28g%28x%29%29=%282%29%2F%28%28x-3%29%2B1%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29=%282%29%2F%28%28x-3%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29 ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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;\" >. Note: the value of <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;\" > is now positive. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=3 ALIGN=MIDDLE  ALT=\"x%3E=3\"   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=f%28g%28x%29%29=%282%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28g%28x%29%29=%282%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "Domain of f(g(x)):\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;\" > Set the denominator equal to zero. Remember, you cannot divide by zero.\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;\" > Add 2 to both sides\r<BR>\n" );
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document.write( "So if <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;\" >, then the whole denominator is zero. So this means that we must exclude the value <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 the domain. However, we specified earlier that x <b>must</b> be greater than or equal to 3. Since 2 is not greater than or equal to 3, we don't have to worry about restricting this value (as it has been already done)\r<BR>\n" );
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document.write( "So the domain of f(g(x)) in set-builder notation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{x\| x \ge 3 \right\}\"> which is <font size=\"6\">[</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3,\infty\"><font size=\"6\">)</font> in interval notation\r<BR>\n" );
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document.write( " </SPAN>\n" );
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