 This video is going to talk about rational functions. A rational function is a function that's made up of f of x divided by g of x. They're both functions, and because you can't divide by zero, we have to make sure that g of x is not equal to zero, which brings us to the domain. A domain of a rational function is those values in the denominator that would cause it to be zero, so we have to restrict those values. And this can be done either with a table or with symbols, and we'll do both. So when we look at this function, when we're trying to find a domain, we really care about what's in the denominator. But at this particular time, I want to show you how to use your calculator to help you find that. So if we do that, we need to put the whole function in our calculator. So I have more than one term in the numerator, so I have to put parentheses x plus five, and then close the parentheses, divided by, and then parentheses again around my denominator, x squared minus x minus 20, and then close my parentheses. And when I go look at my table, I'm really looking for values that are going to give me this error, okay? So I have an error at x equal negative four. And if I bring my calculator back up and go the other direction, in fact. If we start at two, we can see the other error here at five. So we also have an error at x equal five. And it's the error that is the important part. That tells us that x cannot be negative four, and it cannot be five, or else we will have something undefined. So the domain is all reals except for these values. And I'm fine if you tell me the domain is just x can't be those things. Then we understand that x is everything else. So for this one, we're going to do it symbolically. So we only care about the denominator when we're speaking about the domain. For doing other things, the whole fraction will be important. But for the domain, this is the important part. And we're really saying 3x squared plus 2x minus five cannot be equal to zero, or it would be undefined. So we have a quadratic that is sort of equal to zero. So we can factor it, and I'm just going to do trial and error here. So 3x and x, and I need factors of five that will give me two. And it's a negative five. So that tells me that I need to have three and five, the difference would be two. So I need a positive five and a negative three. There's my 3x squared minus 3x plus 5x is 2x minus five. So I set these now equal to not zero. And 3x will be equal, not equal to negative five. And x is not equal to negative five-thirds. And over here, the easy one, if I, I'll start it this way. X minus one cannot be equal to zero. So if I add one to both sides, I find out that x cannot be equal to positive one. So my domain is all values except for these two. Some rationales can be simplified. And we are looking for common factors when we simplify these between the numerator and denominator, because remember if you have a factor on the top and the bottom, they cancel each other out because they've just become one. So when you look at 32x to the 12th and 24x to the ninth, I'm looking at that and I'm saying, well, 24 and 32 both have eight in common. This would be eight times four and then I've got my x to the 12th. And 24 would be eight times three and I've got x to the ninth. So my eighths cancel each other out. That leaves me number wise with a four and a three. And then remember that all of these are going to cancel out nine of those. So take nine of them off and we find out that we have x cubed. Now when we look at this one, we have polynomials in both parts of our fraction. But that's all right, we just factor everything. So x plus 5x minus 5 would be x squared minus 25. Remember that's a difference of squares. And on the bottom, factors of 15, I just need the factors of 15 because it's an x squared first. So factors of negative 15 that will add up to negative 2 sound like again 3 and 5, but this time we need it to be a negative 5 and a positive 3. And that'll give us negative 2. So x minus 5s are both ones on the top, ones on the bottom. And we're left with x plus 5 on the top and x plus 3 on the bottom. And finally, we have something a little bit bigger. So what can we do here? Well, let's try our factoring. And I'm gonna start with the top one and it's rather big. So I'm gonna come and do my x factoring. 12 times negative 3 is negative 36. And we need to add up to 5. And I happen to know that that would be a negative 4 and a positive 9. Just know that one, I don't know why. To 12x squared then remember you rewrite your middle term with those two facts, so minus 4x and plus 9x and then we're back to our minus 3. And we take the greatest common factor of 4x. So it leaves us with 3x minus 1. And then we come over to the second set and 9x and negative 3 have a positive 3 in common. And then that leaves us with 3x minus 1. So on the top here, we're gonna have that common factor of 3x minus 1, as well as the factor of 4x and red down there, plus 3, the leftovers. And then this one, we're gonna factor. And I think those are small enough that I probably could trial and error this one. So 3x and x and then 2 is the only possibilities of 2 and 1 would just have to decide which one's negative. And if I'm gonna get up to a 5, I'd better combine my 2 with my 3 here to get 6x and it would be the big one. So I want that to be a positive 2, that makes this a negative 1 here. And you can see that the 3x minus 1 and the 3x minus 1's cancel. Now, you have to do a lot of factoring with these kinds of problems. If you come back to this problem, this wasn't too difficult to factor, but I just want to introduce another way to factor and that's using the quadratic formula. So if we use the quadratic formula, remember that we had to say what A was and what B was and what C was. And if I look at this top one here, A is 12, B is 5, and C is negative 3. And I'm trying to get to my factors. Well, let's try this in the calculator. So put in your program. Mine will be a little bit different because I have an abbreviated version here. But A is 12, and B is 5, and C is negative 3. And it tells us that we get x is equal to 0.33 repeating. And you may or may not know what that is. And x is equal to negative 0.75. Most of us know negative 0.75 is actually negative 3 fourths. When we were solving quadratics, we started out with something that looked like a factor equal to zero. And we worked our way down until we had x equals something. Well, we have the x equals something and we want to work our way back up to the x equals zero. So we want to zero out the right-hand side. So the first thing I'm going to do is multiply both sides by 4 so that I can clear my fraction. And that'll give me 4x is equal to negative 3. And again, remember, I'm trying to get 0 over here. So I'm going to add 3 to both sides. And that'll give me my factor of 4x plus 3. And that being equal to 0, I know it's my factor. Now, the 0.33, you may know what it is, you may not know what it is. But if you don't, we can still use this 4x plus 3. So we want to remember that the first terms of our binomials actually came from the 12x squared. Well, we know we have 4x. So we look at it and say, oh, well, 3 times 4 would be 12, and x times x would be x squared. So it must be that my first term is 3x. And we know that the last terms of my binomials, if I multiplied them, would actually give me that negative 3. And I already know that I have a positive 3 because that's in my factor. So what do I have to multiply positive 3 by to get to negative 3? That would be negative 1. So we put a minus 1 here. And now we have the numerator factored. And we can do the same thing with the second one. We can come down here and say A is 3. Remember, this is my A. And B is 5. And C is negative 2. And that tells us that we have that 0.33 again. Now we know what that one is. And negative 2. So x equal negative 2, just reminding you. We'll do the easy one. If I want to bring this to the other side so I can make 0 over here, it'll be x plus 2. So again, x plus 2 is the easy factor. And the other one is it's 0.33. If you don't know, x is equal to 1 third would be that 0.33. So we multiply both sides by 3 to clear the fraction. And we have 3x is equal to 1 and 3x minus 1, again, is our factor. All I really wanted to do there was show you how to factor them. Let's look at one more problem. This one, if you look at the denominator, it has kind of a do-if-you-look-in denominator because the variable isn't first. So what do we do about that? Well, let's go ahead and start factoring. At the top, we take our 5 times the x minus 4 and then we look at this one and it looks like we have at least a 2 in common. And if we have 2 in common, then that gives us 4 minus x. And if we look closely at these two factors, this is positive x and a negative 4 and this is a positive 4 and a negative x. So they're opposites of each other. So all I really need to do is factor out a negative and that'll change the signs on everything. So I make this a negative 2. This becomes a negative 4 and a positive x and if that bothers you, you could rewrite it as x minus 4. And now we see that we have a common factor on the top of x minus 4 with the 1 on the bottom of x minus 4. And we're just left with 5 over negative 2.