Okay for problem 34. They give us a rational function. So a polynomial divided by a polynomial. Um, and in this case, it's a linear function divided by a quadratic. They want us to find the vertical assam totes. Now, vertical Assam totes occur when we have division by zero. Um, and there's one extra stipulation where, um, this does not occur. So where you cannot cancel any terms, Um, or I'm gonna say remove any variable factors. So what that essentially means is if I can factor the top and bottom and cancel a factor So something like X plus or minus the number divided by X plus or minus that same number. If I have the same factor divided by itself, I can cancel that out. Right? The division by itself always equals one which would make whatever this number is the plus or minus. That would be a removable dis continuity, otherwise known as a missing point or a whole dis continuity eso hole and or missing point, depending on what terminology you were taught. So I always referred to it as a whole. Um So what we need to do for this particular example? I'm going to factor. The top is already is factored is it could get but the bottom. I'm gonna I'm gonna factor um two x squared and see if there's any terms that will cancel. So two X squared minus six X minus eight. Well, first of all, I see that there's 26 and eight, so I can factor to out right away. Um, that's my greatest common factor. So I have two times x squared, minus three X minus four, and then I can see right away. I like to think about factoring. Is unf oiling? So this is the product of foiling the result. Right? So this is first times first, this is last times last. This is outer plus inner. So I need to figure out what numbers when they're multiplied together, they give me the last term. And when they're added together, they give me that middle term of negative three eso That's gonna be negative. Four and one. So negative for positive. One negative for a positive one. Um, which means that this factors into X minus four and X plus one right there is the minus four in the plus one and then don't forget about that to you. So I'm gonna rewrite f of X. Now I've factored the numerator, which nominator? I'm sorry. The top stays the same. It's already and fully factored form. So X plus two and then the numerator. The denominator is the two times X minus four times X plus one. Now I can see that there are no factors that will cancel, which means that both of these terms x minus four. These variable expressions and X plus one are non removable. So non removable tells me as, um tote vertical aspecto which hopefully we know that because of, uh, it's X and not why. So all I need to do now is solved for where I have division by zero. So where two times X minus four X plus one equals zero. The two doesn't really matter. I could divide to on both sides, and I you know, the cancel out. I just need to know where X minus four equals zero affordable sides. We get X equals four and where does X plus one equals zero? Subtract one. We get X equals negative one. So X equals for an X equals negative one. Our equations of vertical lines, those air the equations of our vertical Assam totes. And we know that for a negative one are not in the domain of this function because they give us division by zero at at those points, um, eso we know to exclude them from our domain. Um, but that is also what tells us that they're the vertical ass and totes of this function F of X.