document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172442>Question 1172442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of the function<BR>\n" );
document.write( "f(x) = {x^2 + 14x + 9}/{x^2 + 2x + 3}, as x varies over all real numbers.\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #797470 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=797470\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "f(x) = (x^2 + 14x + 9)/(x^2 + 2x + 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <BR>\n" );
document.write( "g(x) = x^2 + 14x + 9<BR>\n" );
document.write( "h(x) = x^2 + 2x + 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those two other functions are set up such that<BR>\n" );
document.write( "f(x) = g(x)/h(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This way we can apply the quotient rule<BR>\n" );
document.write( "f(x) = g(x)/h(x)<BR>\n" );
document.write( "f ' (x) = [ g'(x)*h(x) - g(x)*h'(x) ]/[ (h(x))^2 ]\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's compute each derivative piece<BR>\n" );
document.write( "g(x) = x^2 + 14x + 9<BR>\n" );
document.write( "g'(x) = 2x + 14<BR>\n" );
document.write( "and<BR>\n" );
document.write( "h(x) = x^2 + 2x + 3<BR>\n" );
document.write( "h'(x) = 2x + 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Allowing us to get <BR>\n" );
document.write( "f ' (x) = [ g'(x)*h(x) - g(x)*h'(x) ]/[ (h(x))^2 ]<BR>\n" );
document.write( "f ' (x) = [ (2x+14)*(x^2+2x+3) - (x^2+14x+9)*(2x+2) ]/[ (x^2+2x+3)^2 ]<BR>\n" );
document.write( "f ' (x) = [ 2x^3+18x^2+34x+42 - (2x^3+30x^2+46x+18) ]/[ (x^2+2x+3)^2 ]<BR>\n" );
document.write( "f ' (x) = [ -12x^2 - 12x + 24 ]/[ (x^2+2x+3)^2 ]\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solve the derivative equal to 0 to solve for x<BR>\n" );
document.write( "Since the denominator cannot be zero, this means the numerator must be 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f ' (x) = 0<BR>\n" );
document.write( "[ -12x^2 - 12x + 24 ]/[ (x^2+2x+3)^2 ] = 0<BR>\n" );
document.write( "-12x^2 - 12x + 24 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Apply the quadratic formula to find the roots are <BR>\n" );
document.write( "x = -2 and x = 1<BR>\n" );
document.write( "I'll skip showing these steps\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now set up a sign chart. Draw a number line. Plot -2 and 1 on the number line. Mark the points as A and B.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll need to test three regions:<BR>\n" );
document.write( "1) The region to the left of A<BR>\n" );
document.write( "2) The region between A and B<BR>\n" );
document.write( "3) The region to the right of B<BR>\n" );
document.write( "by \"test\", I mean determine the sign of f ' (x) for these regions. As you can probably guess by now, I'm going to use the first derivative test.<BR>\n" );
document.write( "The second derivative test is an alternative option, but it requires calculating the second derivative which is going to be a bit of a pain. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's just stick with the first derivative test.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on the region to the left of point A, ie the region to the left of x = -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The value x = -3 is one possible test value. Plug this into f ' (x)<BR>\n" );
document.write( "f ' (x) = [ -12x^2 - 12x + 24 ]/[ (x^2+2x+3)^2 ]<BR>\n" );
document.write( "f ' (-3) = [ -12(-3)^2 - 12(-3) + 24 ]/[ ((-3)^2+2(-3)+3)^2 ]<BR>\n" );
document.write( "f ' (-3) = -1.33<BR>\n" );
document.write( "Which is approximate.<BR>\n" );
document.write( "The actual value doesn't matter. All we're after is whether f ' (x) is positive or negative.<BR>\n" );
document.write( "Since f ' (x) is negative, this means f(x) is decreasing on the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity+%3C+x+%3C+-2 ALIGN=MIDDLE  ALT=\"-infinity+%3C+x+%3C+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now test the region between A and B, so the region between x = -2 and x = 1. The value x = 0 is a good candidate.<BR>\n" );
document.write( "f ' (x) = [ -12x^2 - 12x + 24 ]/[ (x^2+2x+3)^2 ]<BR>\n" );
document.write( "f ' (0) = [ -12(0)^2 - 12(0) + 24 ]/[ ((0)^2+2(0)+3)^2 ]<BR>\n" );
document.write( "f ' (0) = 2.667<BR>\n" );
document.write( "Also approximate.<BR>\n" );
document.write( "The result here is positive.<BR>\n" );
document.write( "So f(x) is increasing on the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+%3C+x+%3C+1 ALIGN=MIDDLE  ALT=\"-2+%3C+x+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The change from decreasing to increasing through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > indicates we have a local minimum here. Check out the graph below and it visually confirms this.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last region to check is everything to the right of B, so when x > 1<BR>\n" );
document.write( "I'll pick x = 2<BR>\n" );
document.write( "f ' (x) = [ -12x^2 - 12x + 24 ]/[ (x^2+2x+3)^2 ]<BR>\n" );
document.write( "f ' (2) = [ -12(2)^2 - 12(2) + 24 ]/[ ((2)^2+2(2)+3)^2 ]<BR>\n" );
document.write( "f ' (2) = -0.397<BR>\n" );
document.write( "Approximately. <BR>\n" );
document.write( "We get a negative value. <BR>\n" );
document.write( "f(x) is decreasing on the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3C+x+%3C+infinity ALIGN=MIDDLE  ALT=\"1+%3C+x+%3C+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The change from increasing (previous interval) to decreasing shows we have a local max.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To summarize:<BR>\n" );
document.write( "x = -2 leads to a local min<BR>\n" );
document.write( "x = 1 leads to a local max\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug each of these values into the original f(x) function to find the actual local min and local max\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Local Min:<BR>\n" );
document.write( "f(x) = (x^2 + 14x + 9)/(x^2 + 2x + 3)<BR>\n" );
document.write( "f(-2) = ((-2)^2 + 14(-2) + 9)/((-2)^2 + 2(-2) + 3)<BR>\n" );
document.write( "f(-2) = -5<BR>\n" );
document.write( "The lowest f(x) can go is f(x) = -5. We can say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%3C=+y ALIGN=MIDDLE  ALT=\"-5+%3C=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > where y = f(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Local Max:<BR>\n" );
document.write( "f(x) = (x^2 + 14x + 9)/(x^2 + 2x + 3)<BR>\n" );
document.write( "f(1) = ((1)^2 + 14(1) + 9)/((1)^2 + 2(1) + 3)<BR>\n" );
document.write( "f(1) = 4<BR>\n" );
document.write( "The highest f(x) can go is f(x) = 4. We can say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+4 ALIGN=MIDDLE  ALT=\"y+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Combine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%3C=+y ALIGN=MIDDLE  ALT=\"-5+%3C=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+4 ALIGN=MIDDLE  ALT=\"y+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ie overlap the two intervals (use a number line if needed), and we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%3C=+y+%3C=+4 ALIGN=MIDDLE  ALT=\"-5+%3C=+y+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which represents the range\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In short, the range is any real number between -5 and 4, including both endpoints. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph below visually confirms this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/wJ1RD9h.png\"><BR>\n" );
document.write( "Side note: The horizontal asymptote is y = 1, which can be determined through applying a limit as x goes to either infinity.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=red>Answer: Range is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%3C=+y+%3C=+4 ALIGN=MIDDLE  ALT=\"-5+%3C=+y+%3C=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ></font></font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );