Range of rational functions
Date: Thu, 01 Dec 1994 09:48:40 -0500 (EST) From: Anonymous Subject: pre calc question Hello, this is our first attempt to use your service. We are a senior class in upstate NY who is struggling with rational functions. Given the following rational function: f(x)= (x-1)/(x^2+3x-1) How do you find the range? Our book gives the answer of: (neg infinity to 0.12] U [0.65 to pos infinity) Thank You in advance. John Mehrman and Danielle Nowotenski Keene Central School, Keene Valley New York
Date: Thu, 1 Dec 1994 15:58:37 -0500 (EST) From: Dr. Ethan Subject: Re: pre calc question Well your book is right. That is the range. I think that you said that this was a pre calc. class so I don't know how much calculus I can use but I will try to use as little as possible. First of all do you have an idea what this graph looks like? I guess that is kinda what we are trying to do. Okay. To start with, we know that some of the points we need to watch out for are the roots of the equation on the bottom. Those are the points where we get the indeterminant form k/0 where k is a constant. You probably know that those points are problems. So at those points on both sides we need to figure out if it is going to positive and negative infinity. Next take the limits as x goes to positive and negative infinity. You hopefully will notice that since the power of x is bigger on the top than the bottom, then the limit will be zero in both directions. However in the positive direction it is coming in from the positive that means that it will have to have a maximum point between the root (x=1) and positive infinity (that point is where they get the .12). How to find this point without calculus is very hard but using the first derivative it is relatively straightforward (the calculations are a bit nasty on this one). Similarly, the thing in the middle of the two asymptotic lines (do you know what these are?) looks like a parabola and you can see that it is going to have a mininimum. Again to find this min is pretty straightforward using calculus but hard otherwise. Then you have a pretty clear drawing of the thing and the range falls right out. I hope that helps; if not then right back and somebody else can take a shot. Ethan, Doctor On Call
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