Finding a Viewing Window

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Question

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

f(x) = (x² − 1)/(x² + 1)

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says which of the following is the graph of the function below, and we're given the function F of X, which is equal to the quantity of X2 minus 9 in quantity, divided by the quantity of X2 minus 4 in quantity. Below the problem we're given 4 answer choices, and each answer choice consists of a graph, and we'll actually come back to these graphs here in a minute. Now, before we look at the graphs, we're actually going to want to factor both the numerator and denominator of a function F of X since we have the difference of two squares in both cases. So that means that F of X is going to be equal to, and in the numerator I'll have the quantity of X plus 3 in quantity multiplied by the quantity of X minus 3 in quantity, and that is going to be divided by in the denominator. I'll have the quantity of X plus 2 in quantity, multiplied by the quantity of X minus 2 in quantity. So the first thing we're going to look for are intercepts on the X and Y axis. Now for our X intercepts, that's going to occur whenever a function F of X is equal to 0, and that's only going to occur when the numerator is equal to 0. So we're going to set the quantity of X + 3 in quantity multiplied by the quantity of X minus 3 in quantity equal to 0 and solve for x. To do this, we'll set each factor equal to 0 and then solve for x. So for our first factor, we'll have X + 3 is equal to 0. And solving for X we'll get X is equal to -3. And for the second factor, we'll have X minus 3 is equal to 0, and therefore X is equal to 3. So our X intercepts occur at x equal to -3 and X equal to 3. Now we can also find our Y intercept, and that's going to occur when X is equal to 0. So we need to calculate f of 0, and this is going to be equal to, and here we'll use the original form of our equation. So we'll have the quantity of 02 minus 9 in quantity divided by the quantity of 02 minus 4. And simplify this expression, this is going to be equal to 9 divided by 4, which is equal to 2.25. So just based upon the intercepts, we can eliminate some of our answer choices. So, if we look at answer choice A, and we look at the graph, we see that it has X intercepts at X equal to plus minus 2, and it has a Y intercept of Y equal to 2. None of those actually match with the X intercepts or Y intercepts that we calculated. So that means that choice A cannot be correct. If we look at choice C, Again, we see that we have X intercepts occurring at X equal to -2 and X equal to + 2, and we have a Y intercept of 4. So none of those are the correct values that we're looking for, so that means that choice C cannot be correct. If we look at choice D, we have a Y intercept of Y equal to 3, which is not what we calculated. Also, the X intercepts are not correct. They occur at X equals to + or minus 1.75. So again, since the intercepts do not match, that means that choice D is not correct. So that leaves us choice B. And if we look at choice B, we see that it does have the correct intercepts. So, the X intercepts occur at X equal to -3 and X equal to 3, and the Y intercepts occurs at Y equal to 2.25. Now we're just going to check some other features of our function F of X in order to ensure that choice B is the correct answer. And we're actually going to look for the asymptotes. So we call that for vertical asintodes, that's going to occur whenever our denominator is equal to 0. So we're going to set the quantity of X plus 2 in quantity multiplied by the quantity of X minus 2 in quantity equal to 0 and solved for x. To do this, we'll set each factor equal to 0 and solve for x. So for the first factor we'll have X + 2 is equal to 0, so that means that X is going to be equal to -2. And for the second factor, we'll have X minus 2 is equal to 0, and therefore X is equal to 2. And if we look at our graph for choice B, we see that we do have vertical asymptotes at X equal to + or minus 2. Now, the other thing that we can check for are horizontal asymptotes, and we're gonna have a horizontal asptodes since the degree of a polynomial in the numerator is the same as the degree of the polynomial in the denominator. So, we need to check the limit. As x approaches infinity. And here we're going to take the original expression for function F of X and divide both the numerator and denominator by x2. So that's going to give me the quantity of 1 minus 9 divided by X2 in quantity, divided by the quantity of 1 minus 4 divided by x2 in quantity. Now when we take the limit as x approaches infinity, the quantities of 9 divided by X2 and 4 divided by X2 will approach 0. So that means that this is going to be equal to the quantity of 1 minus 0 in quantity divided by the quantity of 1 minus 0, which is equal to 1. And we will actually get the same thing when we take the limit as X approaches minus infinity. So that means that we have a vertical, or sorry, a horizontal asymptote at Y is equal to 1. And if we look at our graph, we see that we do have a horizontal asmatote at Y equal to 1. So based on this information, we can see that the correct answer to this problem is actually choice B. So that is the graph that corresponds to F of X is equal to the quantity of X2 minus 9 in quantity divided by the quantity of X2 minus 4 in quantity. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Function Behavior Analysis

Graphing Software Viewing Window

Rational Functions

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