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

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. (1 − 3/(x+2)) / (1 + 1/(x−2))

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the given expression. After performing the indicated operation, our expression is in the numerator, we have two plus one divided by the quantity of X minus three. And that's all divided by, in the denominator, we have two minus the divided by the quantity of X plus three. And uh for our answer choices, we have four graphs. All right. So let's move this on the screen where we've got a little bit more room to work. We're going to perform the operation. And the given operation here is division. We get to divide our numerator and our denominator. And so in order to divide, we need to first combine the terms in our numerator in the terms in our denominators. So we have addition in the numerator that we need to do. In order to add when one thing is a fraction, we need to have a common denominator. Our common denominator will be X minus three. So we need to multiply the two by X minus three divided by X minus three in the numerator. Now, in the denominator, we have subtraction and again, we have one term being a fraction. So we need to have a common denominator. Again, the common denominator here will be X plus three. So we multiply this two in the denominator by X plus three divided by X plus three. So now this will read two multiplied by the quantity of X minus three divided by X minus three plus one, divided by X minus three, all divided by two, multiplied by the quantity of X plus three divided by X plus three minus 11 divided by X plus three. Now, we have common denominators so we can add and subtract across our numerators. So in the numerator, the numerator of the numerator, we have two multiplied by the quantity of X minus three plus one all divided by X minus three, all divided by. Now, in our denominator, our denominator's numerator is two multiplied by the quantity of X plus three minus 11 all divided by X plus three. Now let's simplify our two numerators. We can distribute these twos to what is in parentheses. So we will have two multiplied by X is two, X, two, multiplied by negative three is negative six and then plus one divided by X minus three. And that's our numerator and we divide all of that by two, multiplied by X is two, X two multiplied by positive three is positive six, then minus 11 all divided by X plus three. And we can combine like terms in our two numerators, this negative six plus one. And in the denominators, numerator, we have plus six minus 11. So for our numerator, we've got two X, the negative six plus one is minus five. That's all divided by X minus three. Now, instead of writing this as fractions upon fractions, I'm gonna put a division symbol because this big division bar in our fraction means division. So writing it with the division symbol, our fraction from the denominator. The numerator of that will be two X, then we have a positive six minus 11. That's minus five divided by X plus three. And now we know that when we are dividing fractions, that's the same as multiplying by the reciprocal of the second one. So this is the same as two X minus five divided by X minus three, multiplied by. Now the reciprocal of the second fraction X plus three divided by two X minus five. And we know that there are unwritten parentheses around our numerators and denominators. These are factors. And if there's a factor that occurs in both the numerator and the denominator, those can cancel out. We've got the two X minus five in the numerator and two X minus five factor in the denominator, those can cancel out. So multiplying straight across those become ones and we've got one multiplied by X plus three. Well, that is just X plus three, that's divided by X minus three, multiplied by one, that's just X minus three. So that whole thing simplified two X plus three divided by X minus three. Now, we need to graph that. There's a little bit of work we can do to make it easier to graph. We can get it looking like a familiar fact or excuse me, a familiar function. The familiar function that we want to have it looking like is one divided by X and we can achieve that by doing the division here. So further division, we can use long division, which won't take very long in this case. So we take our numerator X plus three divided by the denominator X minus three. We ask ourselves, how many times does X go into X? Well, that's just one time. So the one goes above the three, they are like terms. We take one multiplied by the X and one multiplied by the negative three, one multiplied by X is X, one multiplied by negative three is negative three. Now we're going to take X plus three minus the quantity of X minus three or in other words, be sure to subtract both terms, X minus X is nothing. There are no XS left three minus negative three is a positive six. So we can rewrite this as one plus six divided by X minus three. As we recall from previous lessons. And taking that just one step further to get this looking like our one divided by X function. We don't have a one in our numerator. So we can factor that right out. So this is the same as one plus six multiplied by the quantity of one divided by X minus three. And here now we can graph our function of one divided by X and do the transformations indicated here. So I'll draw a rough sketch of a coordinate plane. We have a horizontal X axis, a vertical y axis. They come together at the origin in the middle. We recall from previous lessons that the graph of one divided by X is going to come from positive infinity for Y it's heading toward positive infinity. And then we can draw going down through the 10.11 and then heading along the X axis toward positive infinity all within the first quadrant. And then also in the third quadrant of our coordinate plane, it's heading toward negative infinity for a but then we can draw it coming to the right all the way going through the point negative one negative one and then turning and heading down along the Y axis toward negative infinity for the Y axis. And if you ever don't recall exactly what a graph looks like, you can always make an X Y chart and just pick some points, you could pick negative 10, positive one and then plug those in to the function one divided by X. So that'd be one divided by negative one. Well, that's gonna be an one divided by negative one which is negative one, if you put zero in there, that will be undefined because you can't have zero in the denominator and you put one in there, that's going to be a one. That's how we get our points. Negative one, negative one and 11 noting that where we put zero in for X, that's undefined. We've got an asom tote along the Y axis where XX equals zero, X can't equal zero. We also have an asom tote along the X axis. We're going our fractions will get infinitely smaller, but we never actually hit the X axis where Y equals zero. So we have two asom totes which are important to note for our transformations. OK. What are the transformations taking place here? So let's look at our one plus six multiplied by the quantity of one divided by X minus three. So first, we look for horizontal shifts. Do we have any horizontal shifts? Yes, this happening directly to this X it's having three subtracted from it because that three is negative. It indicates a horizontal shift to the right because it's negative three units. So we are going to shift to the right three. All right. What else do we have? We now look for stretching and shrinking is anything being multiplied by our X? Well, not directly. So no horizontal stretching or shrinking, but that six is being multiplied by the parentheses of our function, that one divided by X function. So there is going to be a vertical stretching or shrinking. So which one? Well, six is greater than one. It's not between zero and one. So it's going to be vertical stretching. So it all stretch vertically. And next, we look for any reflections is anything being multiplied by negative. No, there are no reflections in this one. So everything's going to stay in their quadrants and then finally any vertical shifts, yes, we've got a vertical shift. We've got this one being added outside of the parentheses not happening directly to the X. So that indicates a vertical shift and because it is positive, that indicates it's going to shift up one unit. So we're going up one. So the important thing to note is what's happening to our Asom totes. So our new Asom tote is going to be, we're shifting to the right three. So we'll have a vertical ascent tote at X equals three and then we're going up one. So we're going to have a horizontal ASOM tote, not at Y equals zero, but now at Y equals one, everything stays in their quadrants. So we can draw sort of like this. I got a lot going on here. All right, they're going to stay along the asom totes, but this is approximately how our function will look. So now it's just a matter of finding the matching one back at the beginning. So let's see if I can grab this. We'll see if we can find our matching graph. OK. So essentially it's shifting to the right and up but it's still along the axis. So we look at our different graphs and we note that graph A has a vertical asom tode at X equals three. A horizontal asom tote at Y equals one. And it has our same function where A of X is X plus three divided by X minus three and this one works. So answer choice A 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. (1 − 3/(x+2)) / (1 + 1/(x−2))

Key Concepts

Simplifying Complex Rational Expressions

Domain of a Function

Graphing Rational Functions

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