In Exercises 89–94, the equation for f is given by the simplified...

Question

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x^2−9)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to graph the given expression. After performing the indicated operation, our expression is we have two fractions in the first fraction. The numerator is X minus eight. That's divided by nine X plus six. That fraction is divided by the second fraction's numerator is X squared minus 16 X plus 64. And that's divided by nine X squared minus four. And for our answer traces, we have four graphs. We'll go over the differences between those graphs in more detail in a little bit. But for now let's move this expression to a part of our screen where there's a little bit more space to write and we will begin our graphing process. So step one for graphing is that we want to simplify this expression. So I'm going to begin by factoring where it's possible. So in the first fraction, our numerator X minus eight, that one's good to go. We can't factor that further. The denominator nine X plus six, we can factor out of three. That's three, multiplied by the quantity of three X plus two. Now, in the second fraction, we've got our numerator, we can factor that. I'm going to factor it very quickly. That would factor to be the quantity of X minus eight squared. And in the denominator nine X squared minus four will factor to be the quantity of three X plus two multiplied by the quantity of three X minus two. And now let's perform the operation. So we're dividing fractions, which means we're going to multiply by the reciprocal of the second fraction copying over the first fraction, we have X minus eight divided by three, multiplied by the quantity of three X plus two. Now that fraction is multiplied by the reciprocal of the second, which we now have in the numerator, the quantity of three X plus two multiplied by the quantity of three X minus two. And that is divided by the quantity of X minus eight squared. And I'm actually just going to write X minus eight squared as the quantity of X minus eight multiplied by the quantity of X minus eight. And now we want to see if any of our factors cancel each other out. So in our numerator and denominator, we have a factor of X minus eight. So those can cancel out in the denominator, only one of the X minus eight factors cancels out. Those become ones we also have in both the numerator and the denominator. A three X plus two factor, those cancel out and become ones. But we note that with that X minus eight factor there was still another X minus eight. But with three X plus two, when we're canceling that out, it's gone. But there's a potential problem there. If we were to set three X plus two equal to zero and then solve for X, we can do that quickly subtracting two from both sides. Three X equals negative two, dividing both sides by three, we get X equals a negative two thirds. If we were to go in and plug negative two thirds in. For X, in our expression, we would have a zero in the denominator. And that would make it undefined. We cannot have that. So X cannot equal a negative two thirds. This is something we need to keep in mind for when we're graphing. OK. Let's get back to simplifying this. In our numerator, we have a one multiplied by a one multiplied by three X minus two. That's just three X minus two. Now, in our denominator, we have three multiplied by one multiplied by one multiplied by X minus eight. That's three multiplied by the quantity of X minus eight. And now sometimes it's more helpful to have it factored. Sometimes it's more helpful to have it multiplied out. So we'll distribute that denominator copy over three, X minus two in the numerator divided by, in the denominator three multiplied by X. The quantity of X minus eight is three X minus 24. And now we're finally ready to begin graphing this. So we can draw a rough sketch of a coordinate plane. We have a vertical y axis, a horizontal X axis, they come together at the origin in the middle. Now, step one of graphing, we are going to look for symmetry, see if there's anything symmetrical about this. We check for symmetry by plugging in an F of negative X into our function and we'll see what we get. So in our numerator, we'd have three multiplied not by X, but now by negative X minus two. And then the denominator three multiplied not by X, but by negative X minus 24. And simplifying in the numerator will have a negative three, X minus two divided by in the denominator negative three X minus 24. And now here's where we're checking for symmetry. If we get exactly what we started with F of X, all the signs remain what we start with. When, after we've done the simplification, then we'll have symmetry about the Y axis. But the signs have changed here. The three XS are both now negative. So we don't have symmetry about the Y axis. If we have the negative F of X as our result, if all the signs have changed, then we would have symmetry about the origin. But not all the signs have changed the negative, the negative two and the negative 24 have not changed their signs. So we don't have symmetry about the origin either. We don't have any symmetry All right. Step one is done. Now, step two, we want to find our why intercept, we find our Y intercept by plugging in zero for X F of zero. So in the numerator, we'll have three multiplied not by X but by zero minus two, divided by three multiplied not by X but by zero minus 24. And in the numerator and the denominator three multiplied by zero is zero. So we have a negative two divided by a negative 24. While simplifying that, that's a positive 1/12. So when X equals zero, Y equals 1 12, that's our Y intercept. We can graph that from the origin. We'll go up 1/12 of a unit and mark that 01 12 as our why intercept? All right, step two is done. Now step three, we want to find our X intercept. We find the X intercept by setting the numerator equal to zero. Our numerator here three, X minus two said that equal to zero. Quickly solved. Adding two to both sides. We have three, X equals a positive two, dividing both sides by three, we get X equals two thirds. So when X is two thirds, Y is zero, that's our X intercept from the origin. We go to the right two thirds of a unit still very close to the origin label that two thirds zero, there's our X intercept three steps done. Now step four, we want to find our vertical asom totes, we find our vertical asom totes by setting the denominator equal to zero. Here, it's easier to do with the factored one. So we take three multiplied by the quantity of X minus eight equal to zero. So worth solving for X here dividing both sides by three, we've just got X minus eight equals zero, adding eight to both sides. We get X equals a positive eight. So we have a vertical asom tode to X equals eight from the origin, we go to the right eight unit. We draw a vertical asom tote and label that X equals eight. And this isn't terribly to scale. So just a rough sketch. All right, step five, step five is to find any horizontal or slant as some totes, we find that by comparing the degrees of our numerator and our denominator. So the degree in our numerator is one, we have X to the first power. The degree of our denominator is also one X to the first power. So in this case, the degrees of our numerator and denominator are equal. What does that mean? It means we are going to take the numerators leading coefficient and divided by the denominators leading coefficient. And that's going to be our horizontal ASOM tote. And so the numerator leading coefficient is three, the denominators leading coefficient is also 33 divided by three is one. So we have a horizontal ASOM tode at Y equals one from the origin. We go up one unit, we draw our horizontal asom tote and we can label that Y equals one. All right, this is looking good so far. The next thing is to plot some points so we can make an X Y chart. And we want to choose X values from each of our intervals defined by our X intercepts and our vertical asymptotes. So for the first X value, we want it to be between negative infinity and our X intercept of two thirds zero. I chose the X value of zero because we already figured that out. Now for the next interval between the X intercepted two third zero and the vertical Asymptote at X equals eight, I chose the X value of a positive one. And then for the last interval from the X, from the vertical Asymptote of X equals eight to positive infinity, I chose the X value of a positive nine. Now we plug those X values into our function and then we will get corresponding Y values. I've already done these ahead of time. You are absolutely welcome to pause the video and plug those in. But for X equals zero, we get that Y value of 1/12. That's our Y intercept that we already have. For the X value of one, I got a negative 1 20. 1st, we can plot that from the origin. We go to the right one unit and then go down one 21st of the unit. Which is still very close to the X axis. And then for the X value of nine, I got a Y value of 25 3rd. So plotting that point from the origin, we go to the right nine units and then go up 25 3rd of unit, which is close to eight. And so now we have plotted those points. But remember we have a whole, we have a spot where X cannot exist at negative two thirds. We plug negative two thirds in for our X value, we get a Y value of 2/13. So from the origin, we go to the left two thirds of unit and go up 2/13 of unit. And there we put a hole X cannot exist there or this would be undefined. And now we can go ahead and draw our graph. So looking at our graph to the left of our vertical Asymptote, X equals eight, it's going to be below the horizontal asom tode Y equals one. So we're going toward negative infinity for our X values just below our horizontal Asymptote Y equals one. It's going to continue close to the ASOM tote until it gets to the hole. And then it's going to have a hole and skip that part and then continue, it'll go through the Y intercept at 01 12. It'll go through the X intercept at two thirds zero. It'll go through our 00.1 negative one 21st and it'll just kind of continue along until it gets close to our vertical Asymptote X equals eight and then head down toward negative infinity for the Y values. Now to the right of our vertical asom tote what our graph line is doing is it's heading up toward positive infinity for the Y values just to the right of the vertical asom tote, we can draw it coming down through our 0.9 3rd And then it's going to turn and it's going to head toward positive infinity for the X values just above our horizontal Asymptote of Y equals one. And there we have it, let me label this whole really quick at negative two thirds, 2/13. And then we can move this graph to the part of our screen where all of the answers are and we can find the correct answer. Choice shrink this a little bit and now moving it right. No, let's look for answer choices that have a vertical Asymptote at X equals eight. Well, all four of our answer choices have a vertical Asymptote at X equals eight. Now let's look for a horizontal asom tote of Y equals one. Only answer. Choice B has a horizontal asom tode of Y equals one answer. Choice A it's at Y equals negative one answer choice C it's at Y equals negative X minus eight. That's a slant Asymptote and answer choice answer choice D has a slight asom tote at Y equals X plus eight. So answer choice B is looking good. Let's see if there's anything we might be missing. Answer. Choice B also has a hole at negative two thirds, 2/13. Well, that looks good. And all of the lines are drawn in the same quadrants in relation to our asom totes. So answer choice B is the correct answer. Well done. We'll catch you on the next one.

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x^2−9)

Key Concepts

Rational Functions

Simplifying Expressions

Graphing Functions

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