 Welcome back to our lecture series, Math 1050, College Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first video in lecture four for series entitled Section 1.3 Combining Functions. It turns out we've already had some practice combining functions already. Previously, we talked about how one can add, subtract, multiply, and divide functions. When you combine functions together using the four arithmetic operations, you are creating a new function and that function is produced by just combining the y-coordinates of said functions using the appropriate operation of addition, subtraction, multiplication, and division. We did this numerically. One could do it graphically as well. We're going to add together the y-coordinates for some common x value inside their domains. In this situation, if we take the function f of x to be 4x squared and g of x to be 4x minus 3, adding the functions just means adding together the y-coordinates f of x plus g of x. The fact that the functions down given algebraically to us doesn't really change much. f of x, I want to be aware, it's not the function, it's the y-coordinate associated to the function here. The function is the rule. It assigns to each x-coordinate, f will assign to each x-coordinate the corresponding number four times that number squared. And similarly, g will associate to any number x the number four times that number minus 3, which we express in this algebraic formula. So when you'll see f of x, that means the y-coordinate. g of x is the associated y-coordinate to the function g. And so we're adding those things together, which then you'll take 4x squared plus 4x minus 3. And in a polynomial expression like this, we would add together any like terms, combined like terms as often the phrase we use here. Now f and g don't actually have any common terms between them, so the sum is just going to be 4x squared plus 4x minus 3. Now when it comes to the difference, the only thing that's not the same here is that we replace the plus with a minus sign, right? And that then affects that we're gonna subtract g of x from f of x. And so the only difference here is that we have to distribute the negative one on the 4x minus 3, pun intended there. And so we're gonna get 4x squared minus 4x plus 3. The plus 3 comes from a negative negative x, it's a double negative. And so when it comes to subtraction, the most common mistake is we forget to distribute the negative sign in all pieces in the second group. So as long as you're careful about the distribution of the negative sign right there, subtraction's no different than addition really. Subtraction we could think of as just adding the negative. Speaking of the distributive rule, when it comes to multiplying the functions, f times g, this just means f of x times g of x, so we're gonna get 4x squared times 4x minus 3. And which case since g has two terms in it, we need to distribute the 4x squared on the two pieces. The first piece will be 4x squared times 4x, for which case 4 times 4 gives us 16, the coefficient there. And x squared times x, we're gonna add together the exponents x squared times x will be x to the 2 plus 1, which is x cubed, as we can see right here. And so the product of the first two will be a 16x cubed. When you multiply the second two together, you get 4x squared times negative three, the coefficients multiply together, you get negative 12 right there, and then you've got an x squared. So again, if we can do the distributive property and we know how to just multiply things together like x to the a times x to the b equals x to the a plus b, we can handle basic multiplication of functions like this. And then so finally with division, well to divide f by g, we'll take the formula for f of x, we'll divide it by the formula of g of x, and then simplify if appropriate. Now at this stage of the game, turns out no simplification is necessary, you got 4x squared on top, you get 4x minus three in the bottom, there's no common divisors amongst the two, and so that's its simplified form right there. This gives us the formula for the four combinations of these two functions, f and g. But one thing we have to talk about also is their domains. When it comes to our functions, f of x for example, its domain is gonna be all real numbers. There is no restriction on multiplication, you can multiply any numbers together, so we can square any number, we can times it by four. The domain of g is likewise gonna be all real numbers, because there's no restriction on multiplication, there's no restriction on subtraction, so it's gonna be all real numbers. As a consequence, the domain of f plus g, f minus g and f times g, in general this is just the intersection of the domains of f and g, and so since f and g have no restrictions, the domains of f plus g, f minus g and f times g will be all real numbers as well. Now things get a little bit more hairy when it comes to f divided by g. Although f and g have no restrictions, division does have a restriction, we can't divide by zero. So looking at the denominator here, the 4x minus three, we have to determine when does 4x minus three equals zero. In this linear equation we can solve, we add three to both sides, we get 4x equals three, we divide both sides by four, we're gonna get x equals three-fourths. So this is what makes the denominator equals zero, and so therefore to find the domain of f divided by g, we look for all numbers not equal to three-fourths, that is x should not equal three-fourths, or if you write this in interval notation, you're gonna get negative infinity to three-fourths, union three-fourths to infinity. So we want every number except for three-fourths. And so we can very easily combine functions by addition, subtraction, multiplication, division. These were polynomial functions, so it wasn't too difficult. Let's turn up the heat a little bit. And what if this time the two functions are themselves these algebraic fractions that we would call rational functions? f of x is one over x plus two, and g of x is x over x minus one. Now one thing I should mention about this thing is that the domains of these things are not all real numbers, right? So for the first function, f, because it's a rational function, we can't let the denominator go to zero. Now, the denominator being x plus two right there, we want that not to be zero. If you solve for zero, you're gonna get that x should not equal negative two. So we want all numbers x such that x does not equal negative two. Similarly for g, right, the denominator is x minus one, so we have to make sure that the denominator is never zero, so that means x is forbidden to be the number one. So we can identify the domains of these functions very quickly. What about there's some difference multiplication in products, I should say, in quotients? Well, when you add together the functions f plus g, you take f of x plus g of x. Now in this situation, we have two fractions. f is one over x plus two, and g is x over x minus one. If we wanna add these things together, because they are fractions, a common denominator has to be found. So take one over x plus two, and then take g, which is x over x minus one. In this situation, our least common denominator would just be the product of the two. There's no common factors between x plus two and x minus one. So it's the least common denominator would just be the product of these two things. In order to add the fractions, we have to reproportion them, multiply the fraction by the factor and the denominator it's missing. So for example, the first one's missing x minus one, so we times top and bottom by x minus one. Notice that x minus one divided by x minus one, this is just the number one, a strategic number one, but number one nonetheless. So times is something by one doesn't change its quantity, but it could change its representation like we did in this case. The fraction is still proportional to what it started off with, but it'll now have the common denominator we desire. The second one we need to have x plus two over x plus two, like so, and when we do that, then our fractions are gonna look like one times x minus one over x plus two times x minus one, like so. I'm gonna erase this LCD I wrote over here. And then for the next one, we're gonna end up with x times x plus two over x plus two times x minus one. Now, as you're adding these fractions together, one thing I wanna mention to you is you are gonna wanna distribute out the numerator, right? So you're gonna take the one times x minus one, you're gonna take the x times x plus two, but with the denominator, you're gonna leave the denominator factored. There rarely is ever a benefit of multiplying out denominator. Sometimes people feel like, well, I have to multiply it out to be simplified, but what does simplified actually mean? Simplified is a term that's used in mathematics a lot and never really defined. I think the reason that is, is that people kinda expect that the definition's clear, but yet they don't use it according to the clear obvious definition. To simplify something would mean to make it simpler, but simple is dependent on context. When we simplify something, what we mean is we're trying to put it in a simpler form for the next calculation. But the problem is if you don't know what the next calculation is, that sometimes can be somewhat difficult to predict what the simplest form is. So what I can tell you with rational functions that it's nearly always the simplest thing to do to leave the denominator factored. And why I say that is for two reasons. One, by not factoring it, that means we have less work for us to do, which definitely I would say is simpler, but then two, we actually learn more about the function when the denominator's factored than when it's multiplied out. We'll see that in just a moment when we calculate the denominator, that is the domain of this function. So when you multiply out the numerators, you end up with an x minus one. You're gonna end up with an x squared plus two x from the first and second parts there. Leave the denominator factored, you get x plus two times x minus one, like so. Now we combine like terms, which we weren't able to do in the last example. You'll get x squared plus, well, the x plus two x will add together to get three x minus one. That's our numerator. And then the denominator is x plus two and x minus one. And I didn't deviate from what I said, leave the denominator factored when you do these type of calculations right here. x squared plus x minus one, you can leave. You do wanna expand the numerator, but leave the denominator factored. At this moment, we could try to factor the numerator, x squared plus three x minus one, and see if it cancels with any term in the denominator whatsoever. Problem is, as we try to factor a negative one, that only factors as one and negative one and those do not add it to be three. So it turns out the numerator is irreducible when this fraction is reduced. When it comes to subtraction, it's gonna be basically the same thing. You need to find a common denominator. So you're gonna get x minus one minus x times x plus two. I took some liberty of going through this one a little bit faster here. Adding and subtracting fractions is really just identical. You just have to make sure you have a negative sign there and you distribute that appropriately. So the common denominator here would again be x plus two times x minus one. So we times the first fraction by x minus one over x minus one. The second fraction by x plus two over x plus two. Make sure you distribute the negative sign when you distribute the x here as well. And so this would then give us, we're gonna have a negative x squared minus two x, but we have a plus x minus one right here over the denominator, which isn't gonna change for the rest of the exercise right here. And then when we combine like terms this time, things will be a little bit different, but we get negative x squared. We're gonna get a minus x and we're gonna get a minus one as our numerator there. x squared plus, or x plus two times x minus one. Now, because everything in the numerator is negative, you might actually consider just factoring the negative sign out of the entire expression. And we sometimes like to write this in front of the fraction, that's perfectly fine. And I recommend you do such a thing. So addition, subtraction of fractions is gonna work out basically the same way. We have to find a common denominator and then combine things together. Now, moving on to multiplication, you're gonna see that multiplication actually is super easy when it comes to fractions. When it comes to fractions, you just multiply the tops and you multiply the denominators. But remember, you were forbidden to multiply out the denominators. I told you to leave a factor. So you get x plus two and x minus one is the denominator, so nothing to do there. You're getting a one times x, which is just an x, and that's it. That's all there is to the multiplication there. And so something that's quite phenomenal that maybe you've never noticed before, but when it comes to fractions, multiplication is easier than addition. We kind of think of this counterintuitively because we usually learn how to do addition, then subtraction, then multiplication, then division of whole numbers. And so we think addition is the easiest of all the operations because we learned it first. And maybe that is true for whole numbers. But when it comes to numbers in general, with fractions, for the rational numbers, multiplication is an easier operation than addition because no common denominator is necessary. And the same thing can also be kind of set about division. When it comes to division, f of x divided by g of x, you get one over x plus two and you get x over x minus one. When it comes to division, we have to multiply by the reciprocal. So you take the first fraction, one over x plus two, which is f of x, you do nothing to it. And then the second fraction will be divided, we're just gonna flip it upside down, x minus one over x. And then we just put it together. It's now a multiplication problem. One times x minus one is gonna be x minus one. And the denominator, I told you to leave it factored. So you get x times x plus two. And that's all there is to division. So when it comes to fractions, rational functions, multiplication, division is super easy. Addition, subtraction actually were the hard parts. Now, before we end this video, let's talk about the domains of these functions a little bit more. We mentioned already that the domain of f is everything except for negative two and the domain of g is everything except for negative or positive one, excuse me. And as such, the domain, the domain of f plus g is actually gonna just be the intersection of those things. Look at the denominator here. The denominator, which we can very well see when it's factored, if x was negative two, the x plus two would go to zero, the denominator would be zero, that's not allowed. And then the, if x was one, then one minus one would be zero, the denominator would be zero, that wouldn't be allowed as well. And so the sum of the two fractions inherits the same restrictions that its parents had. Because f is undefined at negative two, the sum is undefined at negative two. And because g is undefined at one, the sum will be undefined at one as well. And so we see that the domain here will be everything, all real numbers x such that x does not equal one, one or negative two. And so the domain of the sum will just inherit the restrictions that the parents have. And that's also gonna be true for subtraction, right? If you take the difference of the two functions, then the denominator is gonna, you can't have negative two, you can't have one. So it inherits those restrictions as well. And if we do one more of these little cases, you look at the denominator for the product, you have x plus two on the bottom, you have x minus one on the bottom. And so the denominator cannot allow for x to be negative two or one. So the domain of f times g is again the same set, x is everything except for negative two and one. When it comes to division, it's a little bit more hairy because for division, we can't have any of the, we inherit all the restrictions that the parents have, but we also, there's other issues that could lead to division by zero. So let's kind of explain that for a second. If we look at the domain of f divided by g, by inspecting the final form, there's a few things you can see here. For example, x cannot equal zero because that would make this guy go to zero. x cannot equal negative two because that would make this guy go to zero. But it turns out that x cannot equal one either. So the domain of f divided by g here is all numbers except for zero, negative one, negative two, excuse me, and one. Now, while zero and negative two are typically obvious to students, sometimes we get a little bit confused about one. Why can't one be in there? One just makes the numerator go to zero, not the denominator. Now we have to remember that f divided by g is this function right here. This is where equality holds. And so we are then saying that this function circled in red is equal to this as well. And is that exactly true? Well, this right here is the simplified version of this thing right here. And so the question that really should be coming out here are these things genuinely equal to each other? Because when you look at that, I want you to consider the domains of these things. This thing is undefined at negative two, so is this one, because you divided by x plus two. This one is undefined when x equals zero. This is also undefined because that fraction, if you have a zero right here, you get zero and you divide by zero, right? But notice that this guy right here as a fraction is undefined at x equals one. While this one right here doesn't seem like to have that problem. So how can these two things be equal if they have the same domain? So when we write things like these two functions are equal to each other, what we mean is they're equal on their common domains, but the domain potentially could shift if we're not careful. And so when it comes to the domain of these functions when we combine them together, we need to investigate the original expression. And the original expression has three fractions in it. You have one over x plus two, and then you divide that by x over x minus one. And so you have one domain restriction which comes from this baby fraction, x can't equal negative two. This is inherited from f. You have a domain problem from this baby fraction, x over x minus one. This is just g of x and g cannot allow x to equal one. And but then there's this mommy fraction that the big fraction which this can equal zero, which happens when x equals zero. And so basically what I'm saying here is when you divide fractions and simplify it, look at all of the restrictions that come from the new denominator, but also go back and look at the restrictions of the parents. And you need to make sure those are included as well because this final form is not equal to f divided by g for all numbers. The problem is this function right here is not defined at one because this function is not defined at one. Even though this formula kind of allows for it. And in the future, when we talk about graphing rational functions, we'll see that the fact that you think one should be part of the domain, but it's not really, actually has an impact on the graph of the function. But that's a topic for another day. And we'll conclude this video about adding, subtracting, multiplying, dividing algebraic functions and calculating their domains.