document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195991>Question 1195991</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of f(x) = (x-3)/(sqrt(3-sqrt(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 #828654 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=828654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Find the range of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1) The numerator is monotonically increasing;<BR>\n" );
document.write( "(2) The denominator is monotonically decreasing<br><BR>\n" );
document.write( "Therefore, the function is monotonically increasing.<br><BR>\n" );
document.write( "Because of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the denominator, the minimum value for x is -2.  Then, since the function is monotonically increasing, the minimum value of the function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2%29=%28-5%29%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"f%28-2%29=%28-5%29%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -2.88675 to 5 decimal places.<br><BR>\n" );
document.write( "The denominator of the function is positive, because it is of the form sqrt(A).  The value of x that makes the denominator 0 is x=7; so x must be less than 7.<br><BR>\n" );
document.write( "But as x approaches 7 from the left, the denominator gets arbitrarily close to zero; and that means the value of the function gets arbitrarily large.<br><BR>\n" );
document.write( "So there is no maximum value of the function.<br><BR>\n" );
document.write( "ANSWER: The range of the function is from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"-5%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to infinity.<br><BR>\n" );
document.write( "Here is a graph of the function on the window [-3,8,-5,50], showing the minimum value of the function at (-2,-2.88675).  Note that the graph is so steep close to x=7 that the graphing utility can't graph the function past a certain point, making it appear that the function has a maximum value.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C400%2C-3%2C8%2C-5%2C50%2C%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C400%2C-3%2C8%2C-5%2C50%2C%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "But here is a graph of the function on the window [6.99,7,0,1000], showing that the function value is still increasing.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C400%2C6.99%2C7%2C0%2C1000%2C%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C400%2C6.99%2C7%2C0%2C1000%2C%28x-3%29%2F%28sqrt%283-sqrt%28x%2B2%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
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