document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208404>Question 1208404</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Find the domain of the expression x/(x^2 - 9).\r<BR>\n" );
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document.write( "x^2 - 9 = 0\r<BR>\n" );
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document.write( "x^2 = 9\r<BR>\n" );
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document.write( "sqrtx^2} = sqrt{9}\r<BR>\n" );
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document.write( "x = -3, and x = 3.\r<BR>\n" );
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document.write( "Domain = {x | x cannot be -3 and 3}.\r<BR>\n" );
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document.write( "2. Find the domain of the expression<BR>\n" );
document.write( "(-9x^2 - x + 1)/(x^3 + x).\r<BR>\n" );
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document.write( "x^3 + x = \r<BR>\n" );
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document.write( "x(x^2 + 1) = 0\r<BR>\n" );
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document.write( "Setting x^2 + 1 to zero and solving for x leads to complex roots. I will disregard complex roots for now.  The only value that x cannot be is 0.\r<BR>\n" );
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document.write( "Domain = {x | x cannot be 0}\r<BR>\n" );
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document.write( "What do y I say?\r<BR>\n" );
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document.write( "Problem 1.\r<BR>\n" );
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document.write( "x^2-9 = 0 had been correctly solved to get x = -3 and x = 3.<BR>\n" );
document.write( "And you are correct that x cannot equal either of these values; otherwise, we get a division by zero error.\r<BR>\n" );
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document.write( "Another way to solve is:<BR>\n" );
document.write( "x^2-9 = 0<BR>\n" );
document.write( "x^2-3^2 = 0<BR>\n" );
document.write( "(x-3)(x+3) = 0 .... difference of squares rule<BR>\n" );
document.write( "x-3 = 0 or x+3 = 0<BR>\n" );
document.write( "x = 3 or x = -3\r<BR>\n" );
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document.write( "The domain in set-builder notation is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\{\text{x} | \text{x} \in \mathbb{R}, \ \text{x} \not= -3, \text{x} \not= 3 \right\}\"><BR>\n" );
document.write( "Or you can slightly condense things to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\{\text{x} | \text{x} \in \mathbb{R}, \ \text{x} \not= \pm3 \right\}\"><BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \text{x} \in \mathbb{R}\"> portion translates to \"x is in the set of real numbers\" i.e. \"x is a real number\".\r<BR>\n" );
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document.write( "The interval notation would look like this <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( "This is basically us starting with the entire number line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty,\infty\right) \"> and then poking two holes at x = -3 and x = 3 on the number line. <BR>\n" );
document.write( "Each \"U\" refers to a set union joining or gluing together the disjoint intervals.<BR>\n" );
document.write( "Curved parenthesis exclude each endpoint. \r<BR>\n" );
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document.write( "Problem 2.\r<BR>\n" );
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document.write( "You have the correct scratch work. I'll assume your second step should have read x^3+x=0.\r<BR>\n" );
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document.write( "The domain in set-builder notation is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\{\text{x} | \text{x} \in \mathbb{R}, \text{x} \not= 0 \right\}\"><BR>\n" );
document.write( "In words we can say \"x is any nonzero real number\".\r<BR>\n" );
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document.write( "The domain as interval notation would be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty,0\right) \cup  \left(0,\infty\right) \"><BR>\n" );
document.write( "It's the same idea as earlier, except we only have one hole this time. <BR>\n" );
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