document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=990624>Question 990624</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the domain and range of the function:<BR>\n" );
document.write( "f(x)=x^2+3x+10/x^2+6x+5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #610644 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52790)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=610644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Find the domain and range of the function: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3x%2B10%29%2F%28x%5E2%2B6x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3x%2B10%29%2F%28x%5E2%2B6x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "The denominator is a quadratic polynomial. &nbsp;It has the roots &nbsp;-1 &nbsp;and &nbsp;-5. &nbsp;(You can find them using the quadratic formula or the Viete's theorem). \r<BR>\n" );
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document.write( "At these values of &nbsp;<B>x</B>&nbsp; the denominator is zero, &nbsp;so the function is not determined in these points. &nbsp;Therefore the domain is the entire number line except &nbsp;x=-1 &nbsp;and &nbsp;x=-5.\r<BR>\n" );
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document.write( "The numerator is again a quadratic polynomial, &nbsp;and it is always positive, &nbsp;since its discriminant &nbsp;d = b^2 - 4ac = 3^2 -4*1*10 = 9 - 40 = -31 &nbsp;is negative. \r<BR>\n" );
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document.write( "The denominator is negative inside the interval &nbsp;(-5, -1) &nbsp;and positive outside the segment &nbsp;[-5, -1].\r<BR>\n" );
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document.write( "From this, we can conclude that the given rational function is negative inside the interval &nbsp;(-5, -1) &nbsp;and is positive outside the segment &nbsp;[-5, -1].\r<BR>\n" );
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document.write( "Besides of it, &nbsp;the given rational function tends to &nbsp;&nbsp;-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;when &nbsp;x ---> (-5)+ &nbsp;and &nbsp;x ---> (-1)-. \r<BR>\n" );
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document.write( "At the same time the given rational function tends to &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > when &nbsp;x ---> (-5)- &nbsp;and &nbsp;x ---> (-1)+. \r<BR>\n" );
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document.write( "The plot of the function is shown in the &nbsp;<B>Figure</B> &nbsp;below. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+330%2C+330%2C+-10.5%2C+5.5%2C+-10.5%2C+10.5%2C%0D%0A++++++++++%28x%5E2%2B3x%2B10%29%2F%28x%5E2%2B6x%2B5%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure</B>. \r<BR>\n" );
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document.write( "It is predictable that the positive branches of the given rational function go from &nbsp;&nbsp;+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;to &nbsp;1&nbsp; when &nbsp;&nbsp;x ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;and &nbsp;&nbsp;x ---> -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Thus the range includes the semi-infinite segment &nbsp;[1, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "It also includes the semi-infinite segment &nbsp;[alpha, -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >) for some negative &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "To find the value of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;we need to calculate the derivative &nbsp;f'(x) &nbsp;of the function and solve the equation &nbsp;f'(x) = 0. \r<BR>\n" );
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document.write( "It will be, &nbsp;actually, &nbsp;the equation for the numerator of the derivative, &nbsp;which would be again the quadratic polynomial. \r<BR>\n" );
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document.write( "But it is just too long way for me.&nbsp; Would you complete the solution and find the value of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?\r<BR>\n" );
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