Analyzing Range of Function with and without Calculus

Date: 08/19/2007 at 10:24:03
From: Kieh
Subject: Show (x^2+x+1)/(x+1) does not lie between -3 and 1.

Show that for all real values of x, (x^2+x+1)/(x+1) does not lie 
between -3 and 1 on the y-axis.

I really do not know how to start it!  I think that only plotting the
graph can show it but I am sure that is not the way to solve this
question.  Please help me!  Thanks!

Date: 08/19/2007 at 10:50:02
From: Doctor Jaffee
Subject: Re: Show (x^2+x+1)/(x+1) does not lie between -3 and 1.

Hi Kieh,

You are correct that plotting the graph can show that the function
does not lie between -3 and 1.  I am going to assume that you are a
calculus student and suggest a way to solve the problem algebraically.
If you haven't had calculus, let me know and I'll try to suggest 
another solution.

In any case, divide x^2 + x + 1 by x + 1 and the result will be 
x + 1/(x+1).  You can find the derivative of this function fairly
easily and set it to 0.  Solving this equation will give you two 
values of x, one of which is at a local maximum and the other is at a
local minimum.  If you substitute these two numbers into the original
function, you will obtain the desired result.

Give it a try and if you want to check your answer with me or if you
want some clarification about this problem, write back and we'll
discuss it some more.

Thanks for writing to Dr. Math and Good Luck.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/21/2007 at 09:52:04
From: Kieh
Subject: Show (x^2+x+1)/(x+1) does not lie between -3 and 1.

Thanks for your reply.  I'm not a calculus student, I'm a student at
pre-university level. This is a question from my reference book in 
Equations, Inequalities and Absolute Values.  I'm just starting to
study limits, differentiation, and integration.  Can you suggest a
third way to do the problem besides looking at the graph and using
calculus?  Thanks.

Date: 08/21/2007 at 10:51:55
From: Doctor Jaffee
Subject: Re: Show (x^2+x+1)/(x+1) does not lie between -3 and 1.

Hi Kieh,

Here is a method that doesn't require any knowledge of calculus and
the graph is not necessary, either.  However, if you look at the graph
this method might be easier to understand.

Let's start with y = 1.  According to the directions, the range has to
be greater than or equal to 1 or less than or equal to -3.  What value
of x would make y exactly equal to 1?  We can find out by solving the
equation 1 = (x^2+x+1)/(x+1).  If you multiply both sides by x + 1,
you quickly come to the solution x = 0.  But, if you substitute any
number larger than 0 for x the numerator becomes larger than the
denominator, so the value of the fraction is positive and must be
larger than 1.  Furthermore, if x is less than 0 but larger than -1,
the numerator is larger than the denominator, the fraction is still
positive, so the value of the fraction is still greater than 1.

Next, answer the question:  How can (x^2+x+1)/(x+1) be equal to -3? 
Use a method just like I did above to show that as long as x is
smaller than -1, the value of (x^2+x+1)/(x+1) must be -3 or smaller.

Give it a try and if you want to check your  answer with me or if you
want some clarification about this problem, write back and we'll
discuss it some more.

Thanks for writing to Dr. Math and good luck.


- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/