Analyzing Range of Function with and without CalculusDate: 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/ |
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