document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208157>Question 1208157</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range with algebra (no graphing) of f(x)/g(x) if \r<BR>\n" );
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document.write( "f(x) = sqrt{4 - x^2}\r<BR>\n" );
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document.write( "g(x) = sqrt{9 - x^2} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846377 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284-x%5E2%29%2Fsqrt%289-x%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%284-x%5E2%29%2Fsqrt%289-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use the property <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28A%29%2Fsqrt%28B%29=sqrt%28A%2FB%29 ALIGN=MIDDLE  ALT=\"sqrt%28A%29%2Fsqrt%28B%29=sqrt%28A%2FB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for A and B positive):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284-x%5E2%29%2F%289-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284-x%5E2%29%2F%289-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Rewrite the radicand:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%289-x%5E2%29-5%29%2F%289-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%289-x%5E2%29-5%29%2F%289-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281-5%2F%289-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%281-5%2F%289-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This expression is the square root of something less than 1.<br><BR>\n" );
document.write( "Its minimum value is 0, when the radicand is zero:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-5%2F%289-x%5E2%29=0 ALIGN=MIDDLE  ALT=\"1-5%2F%289-x%5E2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F%289-x%5E2%29=1 ALIGN=MIDDLE  ALT=\"5%2F%289-x%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-x%5E2=5 ALIGN=MIDDLE  ALT=\"9-x%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The lower end of the range of the given function is 0, when x is 2 or -2.<br><BR>\n" );
document.write( "The maximum value of the radicand is when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F%289-x%5E2%29 ALIGN=MIDDLE  ALT=\"5%2F%289-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a minimum -- i.e., when the smallest possible value is subtracted from 1.  That is when x is 0; the upper end of the range of the given function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281-%285%2F%289-0%29%29%29=sqrt%281-5%2F9%29=sqrt%284%2F9%29=2%2F3 ALIGN=MIDDLE  ALT=\"sqrt%281-%285%2F%289-0%29%29%29=sqrt%281-5%2F9%29=sqrt%284%2F9%29=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: the range of the given function is [0,2/3]<br><BR>\n" );
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