document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896344>Question 896344</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please find the domain of the following function:<BR>\n" );
document.write( " sqrt(2x^3-x^2-18x+9)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543429 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=543429\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want to find out the intervals where the square root is positive.<BR>\n" );
document.write( "those intervals will be your domain.\r<BR>\n" );
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document.write( "start with 2x^3 - x^2 - 18x + 9 = 0\r<BR>\n" );
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document.write( "fortunately this factors cleanly.\r<BR>\n" );
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document.write( "if not, then you probably would need to graph the equation to see where the intervals lie.\r<BR>\n" );
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document.write( "graphing is the easiest way if you have a graphing calculator handy.\r<BR>\n" );
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document.write( "i did both.\r<BR>\n" );
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document.write( "first the factoring method.\r<BR>\n" );
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document.write( "your equation to factor is:\r<BR>\n" );
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document.write( "2x^3 - x^2 - 18x + 9 = 0\r<BR>\n" );
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document.write( "break this up into 2 separate parts as shown below:\r<BR>\n" );
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document.write( "(2x^3 - x^2) - (18x - 9) = 0\r<BR>\n" );
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document.write( "factor the first part to get:\r<BR>\n" );
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document.write( "x^2 * (2x - 1) - (18x - 9) = 0\r<BR>\n" );
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document.write( "factor the second part so the expression in the parentheses will be equal to 2x - 1.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "x^2 * (2x - 1) - 9 * (2x - 1) = 0\r<BR>\n" );
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document.write( "factor out the common term of (2x - 1) to get:\r<BR>\n" );
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document.write( "(x^2 - 9) * (2x - 1) = 0\r<BR>\n" );
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document.write( "x^2 - 9 can further be factored into (x - 3) * (x + 3)\r<BR>\n" );
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document.write( "your equation becomes:\r<BR>\n" );
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document.write( "(x - 3) * (x + 3) * (2x - 1) = 0\r<BR>\n" );
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document.write( "the roots of this equation are:\r<BR>\n" );
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document.write( "x = 3<BR>\n" );
document.write( "x = -3<BR>\n" );
document.write( "x = 1/2\r<BR>\n" );
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document.write( "those are the values of x where the value of the equation is equal to 0.\r<BR>\n" );
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document.write( "in other words, the graph of the equations crosses the x-axis at those values of x.\r<BR>\n" );
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document.write( "you now need to test each interval to see whether the graph is negative or positive in those intervals.\r<BR>\n" );
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document.write( "first interval is x < -3<BR>\n" );
document.write( "graph is negative there.\r<BR>\n" );
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document.write( "second interval is x > -3 and < 1/2<BR>\n" );
document.write( "graph is positive there.\r<BR>\n" );
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document.write( "third interval is x > 1/2 and < 3<BR>\n" );
document.write( "graph is negative there.\r<BR>\n" );
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document.write( "fourth interval is x > 3<BR>\n" );
document.write( "graph is positive there.\r<BR>\n" );
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document.write( "your domain is the intervals where the square root of the equation is not negative.\r<BR>\n" );
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document.write( "those intervals will be:\r<BR>\n" );
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document.write( "-3 <= x <= 1/2\r<BR>\n" );
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document.write( "x >= 3\r<BR>\n" );
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document.write( "here's a graph of the equation within the square root sign where you wanted to find the intervals where it was positive and where it was negative.\r<BR>\n" );
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document.write( "the equation used is y = 2x^3 - x^2 - 18x + 9\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/083004.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "here's a graph of the equation you wanted to find the domain for.\r<BR>\n" );
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document.write( "that equation is y = sqrt(2x^3 - x^2 - 18x + 9)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/083005.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "since the intervals where the square root is negative are invalid because they don't result in any real values, the graph is missing in those intervals.\r<BR>\n" );
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