document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1003122>Question 1003122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of the following function in interval form.\r<BR>\n" );
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document.write( "f(x)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28+x%5E2%2B3x-4+%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28+x%5E2%2B3x-4+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >* <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28+2-x-x%5E2+%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28+2-x-x%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619931 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=619931\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> without graphing these equations, you wouldn't have been able to see what is going on.\r<BR>\n" );
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document.write( "first i graphed sqrt(x^2 + 3x - 4)\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111111.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "then i graphed sqrt(2 - x - x^2)\r<BR>\n" );
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document.write( "that graph is shwon below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111112.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "then i graphed both together on the same graph.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111113.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "then i graphed sqrt(x^2 + 3x - 4) * (2 - x - x^2)\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/111114.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "what happens is that the last graph shows nothing.\r<BR>\n" );
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document.write( "the reason for this is that the only valid interval for the graph of (2 - x - x^2) is within the invalid interval of (x^2 + 3x - 4) so it disappears.\r<BR>\n" );
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document.write( "the exclusion zone for the first expression is -4 < x < 1.\r<BR>\n" );
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document.write( "the exclusion zone for the second expression is x < -2 and x > 1.\r<BR>\n" );
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document.write( "put those 2 exclusion zones together and you have the entire domain of x being excluded which results in a null graph.\r<BR>\n" );
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document.write( "it took a little bit of analysis to figure this out algebraically.\r<BR>\n" );
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document.write( "first the expressions needed to be factored.\r<BR>\n" );
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document.write( "then the factors were analyzed to determine when the value under the square root sign became negative.\r<BR>\n" );
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document.write( "in order for sqrt((x+4)*(x-1)) to be negative, (x+4) had to be positive while (x-1) was negative and (x+4) had to be negative when (x-1) was positive.\r<BR>\n" );
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document.write( "in order for sqrt((2+x)*(1-x)) to be negative, (2+x) has to be positive while (1-x) was negative and (2+x) had to be negative when (1-x) was positive.\r<BR>\n" );
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document.write( "the analysis confirmed what was shown on the graph.\r<BR>\n" );
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document.write( "sqrt((x+4)*(x-1)) was negative when -4 < x < 1\r<BR>\n" );
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document.write( "sqrt((2+x)*(1-x)) was negative when x < -2 and x > 1.\r<BR>\n" );
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document.write( "sqrt((x+4)*(x-1)) was positive when x <= -4 and x >= 1\r<BR>\n" );
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document.write( "sqrt((2+x)*(1-x)) was positive when -2 <= x < 1\r<BR>\n" );
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document.write( "when you put them together on the same grpah, their exclusion zones together comprised all values of x so there were not values of x that could be graphed.\r<BR>\n" );
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document.write( "the domain was therefore null.\r<BR>\n" );
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document.write( "that's what i believe is happening.\r<BR>\n" );
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document.write( "the graphs confirmed it.\r<BR>\n" );
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document.write( "the analysis was fairly length but and a little bit complex, but it confirmed what the graphs showed.\r<BR>\n" );
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document.write( "i didn't show the analysis  because it would have complicate the results of the analysis unnecessarily and i didn't have enough time to finish it.\r<BR>\n" );
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document.write( "if you need a better explanation, then write and i'll take the time tomorrow or the next day to work it out for you.\r<BR>\n" );
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document.write( "you should, however, be able to work it out for yourself, now that you know what's happening and how you need to proceed.\r<BR>\n" );
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document.write( "the graphing software i used is at <a  rel=nofollow HREF= \"http://www.desmos.com/calculator\" target = \"_blank\">link to calculator used</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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