Question 990624
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Find the domain and range of the function: {{{f(x)}}} = {{{(x^2+3x+10)/(x^2+6x+5)}}}.
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The denominator is a quadratic polynomial.  It has the roots  -1  and  -5.  (You can find them using the quadratic formula or the Viete's theorem). 

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.


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. 


The denominator is negative inside the interval &nbsp;(-5, -1) &nbsp;and positive outside the segment &nbsp;[-5, -1].


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].


Besides of it, &nbsp;the given rational function tends to &nbsp;&nbsp;-{{{infinity}}} &nbsp;when &nbsp;x ---> (-5)+ &nbsp;and &nbsp;x ---> (-1)-. 


At the same time the given rational function tends to &nbsp;&nbsp;{{{infinity}}} when &nbsp;x ---> (-5)- &nbsp;and &nbsp;x ---> (-1)+. 


The plot of the function is shown in the &nbsp;<B>Figure</B> &nbsp;below. 

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -10.5, 5.5, -10.5, 10.5,
          (x^2+3x+10)/(x^2+6x+5)
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure</B>. 

  </TD>
  </TR>
</TABLE>


It is predictable that the positive branches of the given rational function go from &nbsp;&nbsp;+{{{infinity}}} &nbsp;to &nbsp;1&nbsp; when &nbsp;&nbsp;x ---> {{{infinity}}} &nbsp;and &nbsp;&nbsp;x ---> -{{{infinity}}}. 


Thus the range includes the semi-infinite segment &nbsp;[1, {{{infinity}}}).


It also includes the semi-infinite segment &nbsp;[alpha, -{{{infinity}}}) for some negative &nbsp;{{{alpha}}}. 


To find the value of &nbsp;{{{alpha}}}, &nbsp;we need to calculate the derivative &nbsp;f'(x) &nbsp;of the function and solve the equation &nbsp;f'(x) = 0. 

It will be, &nbsp;actually, &nbsp;the equation for the numerator of the derivative, &nbsp;which would be again the quadratic polynomial. 


But it is just too long way for me.&nbsp; Would you complete the solution and find the value of &nbsp;{{{alpha}}} ?