 Welcome back to our lecture series, Math 31-20, Transition to Advanced Mathematics for students at Southern Utah University. As usual, be your professor today, Dr. Angie Missildine. Lecture 30 represents the start of the final unit of this lecture series. So it's the beginning of the end, and which case we are gonna study the important topic of functions. One of the most important topics you can discuss in a mathematical setting. I mean, classes like College Algebra are nothing but discussion of functions here. So what is a function? We have to make sure we understand the proper meaning of a function, and therefore we begin this video with a definition. So a function is actually defined as a special type of relation. So let's suppose we have two sets A and B, then F is a relation from A to B. Now, we originally defined relations so that these two sets potentially could be different, but then we focus on things like partial order, equivalence relations. We always seem to be focused on when the sets were the same. Functions are a key example of relations for which the two sets need not be the same and oftentimes they are not. So F is a relation on the set A to B, which let's note here that a relation is a subset of the Cartesian product A times B. So we should think of F as a collection of ordered pairs. Now, the ordering of a function matters a lot. So when we denote a function, it'll typically be denoted as F colon A arrow B. So the direction here matters a lot because you don't expect these two sets to be the same. So you have your first set A, your second set B. So we could say that F is a function from A to B. Sometimes you call this as a mapping or it's a map from A to B. In advanced mathematics, functions are often referred to as maps. You could call it a transformation, but function and mapping. These are some of the most common terms used here. Sometimes people will denote the mapping here as an arrow A to B and you draw the little F as above the arrow, sometimes below the arrow. And so that indicates the direction here. The sets A and B are not interchangeable in this situation. Okay, so we have a function, but what makes it a function as opposed to just any other relation? So this is what actually makes it a function. It satisfies the property that if each element, so it's a function if each element A in the first set A occurs exactly once in ordered pairs of the form A comma B. That is to say that for every element in the first set A, there is exactly one element that's related to in the second set. This is exactly one. This means there's at least one, but there's also at most one. So every element in the first set is related to exactly one element in the second set. This is what we refer to as a functional relation. This property here where every element in the first set is related to exactly one in the second set. We're gonna see this actually translates to the so-called vertical line test that's often used in a college algebra setting, but we'll talk about that in just a second here. Let me introduce some more vocabulary associated to this. Now, since every element in the first set is related to exactly one element in the second set, given an element of the first set called A, this unique element in the second set called B, we refer to it as the image of A with respect to the function F. And this is, again, a unique relationship. Everything in A is connected to a unique element in B. Not, I mean, different elements of A can be connected to different elements of B, but if you have a fixed element in A, it's related to a unique element of B. We call that the image of A with respect to F, and we'll denote this as F of A is equal to B, or sometimes you'll see this arrow here, A maps to B. The latex symbol here is actually backslash maps to. Again, this is idea of the mapping. You transform A into B, and so F maps A into B. You see this notation all the time in advanced mathematics. You often see this one as well, which you saw this in the elementary mathematical setting as well, calculus called algebra. They use this notation all the time, but this one is used commonly, not more commonly, but it's used commonly here. Now these sets A and B, and to keep talking awkward like the set A, the set B, we'd be needing names, because we're talking about them all the time here. So the set A, this is the set of where the things from the function are coming from. We commonly refer to this as the domain of the function. If you think of your function as like a machine for which it transforms objects, you have A coming in over here into the machine. The domain is then the set of all possible input that goes into the machine. Well, the things that pop out of the machine is the image, right? This second set is commonly referred to as the co-domain. Co here is short for the complement. So it's the complement of the domain. Now, be aware that your previous knowledge of functions might be doing you into service right now. The co-domain is not the same thing as the range. The co-domain is what we like to often refer to as the target space. It's the target set. These are what we're mapping in two. So this element F of A, it belongs to the set capital B, but not necessarily everything in B is hit by this function relation. We'll get to more of that in a second. So let me, for the moment, introduce the vocabulary here. We do have a notion of the image or the range. Range is not commonly used in advanced mathematics like it is in elementary mathematics. The word image here is often used here. So like the image of an element, but then the image itself is the set of all things that belong to the functions output there. So take, for example, the image here by definition, it's the collection of all images. So as A ranges over the elements of the domain, if you take the set of images, F of A's, this is called the image of the function. So F of A is the image of A. The collection of all of the F of A's is the image of the function. So this will be called F of capital A, the set. Some people might call it the image of F right here, because like this domain set is sometimes called the domain of F. When that's clear, this would be the co-domain. The image of F is a very important set there. Sometimes it's called the range, but again, not very common in more advanced mathematical settings. Another set that we wanna introduce right now that is rarely introduced before this moment is what we call the pre-image of some set here. So let X be a subset of B. Then we can define the pre-image, which is called F inverse of X. And this is gonna be the collection of all elements of the domain. So A's inside of the domain A, such that the image of A is contained inside of this set X here for a moment. Let me actually back up for a moment before we understand this pre-image a little bit better. When we define the image of a function, we could actually define this for any subset of the domain. Like if you take some subset Y that lives inside of A, we can talk about the image of Y here. And this is gonna be the collection here of all of the F of Y's such that Y belongs to Y. I should mention that this is gonna be a subset of the co-domain. In particular, it's gonna be a subset of the entire image of F. But we don't have to take the image of the entire domain. We could take this image of any subset of the domain. And we're interested in what's the image of this subset. What we wanna do is be able to reverse this process. If I take a subset of the co-domain, call it X, I wanna ask what are the things that map onto X? Okay? And so this is gonna be a subset of the domain A. And so you're looking for all the things in the domain that map into X. This is called the pre-image. And we use this superscript notation, F to the negative one. It is resemblant of the inverse function, but our functions might not necessarily be invertible, but this pre-image is defined for all functions, whether they're invertible or not. We'll get into more details about this in future lectures, but I do wanna introduce this pre-image right now so that given any subset of the domain, we can talk about its image. And given any subset of the co-domain, we can also talk about its pre-image. So let's look at some examples to understand this better here. So let's take as our domain something very simple. A equals the set one, two, three, four. So our domain only contains four elements. And our co-domain will make it something similar. It'll be one, two, three, four, five. And so the co-domain will contain five elements of the situation. Then the function is then a collection of ordered pairs. And so for this one, let's define our function to be the set of pairs one, two, two, three, four and four, one. And so there's a couple ways you wanna think about this. So the first way you wanna think about it is each of these ordered pairs gives us a function evaluation. So the first one tells us that f of one is equal to two. The second one tells us that f of two is equal to two. The third one here tells us that f of three is equal to four. And then the last one, four common one there tells us that f of four is equal to one. So this is how you can turn this relation into the usual way we think about functions where we think of functions as evaluations. Another way to think about it is in terms of a graph. We often like to graph functional relations in a way like the following. Since the domain A and B and the co-domain B are potentially different sets, we're gonna partition the two into these two collections. The domain we typically put on the left and the co-domain we put on the right, typically. That's not required, that's usually how we do it. And then what we do is we're gonna, as we list the elements of the domain, we're gonna draw an arrow where the arrow then points from the element of the domain to its image in the co-domain. So since we had the ordered pair of one comma two, we're gonna draw an arrow from one to two and the arrow head then points towards the image. Since we had the ordered pair two, two is also gonna point over here to two. We have the ordered pair of three, four. So we draw an arrow from three to four like so. And then lastly, we have the ordered pair four, one for which it's gonna point over here. Now, the illustration over here, this graph, helps us see things we might not have seen previously. So some things to note, this is a function because there is an arrow coming out of each element of the domain, one and exactly one arrow is coming out of the elements of the domain. So this is a function relation. Now some things to note here is that in the domain and the co-domain, there are some elements that get missed. That is perfectly fine. A function relation does not require anything about the co-domain. It just says that everything in the domain is related to exactly one thing in the co-domain. We do not get that reverse relation. We'll actually talk about that a little bit more in the next lecture, but there are definitely things in the co-domain that could be missed. That's why we actually are interested in the range. When we look at this function here, we can see that the image of the function, the image of the function is none other than the elements one, two, and four. These are the things that are actually hit by the function. This is, of course, the domain of F. This is the co-domain of F. The image, you have to calculate this one. The image is gonna be one, two, and four, because three and five get missed. Some other things I want you to note is that it's very possible that two different elements in the domain could be assigned the same element in the co-domain, both one and two go to two. That is perfectly fine. Functions are allowed to do that. Now, we will see examples of functions in the future where different places will point to different things, but that gets us a little bit ahead of ourselves. I wanna also mention that when we talk about the image here, this is the image of the whole thing, right? This is F of A. If I were to introduce a set Y, where we take the set Y to be something like one and two, I want you to note here that the image of Y is actually just equal to the singleton two, because both the elements one and two will map to two. So the image of the set Y is just the singleton two there. Let's do another example. Let's look at the image this time of the subset one, two, three. Okay, the image of this set, we're gonna get the image of one, which is two, the image of two, which is also two, so we don't need to write it twice, and then the image of three is equal to four. So we can look at the image of any subset of the domain whatsoever. Let's think about pre-images for a moment. If I want to take the pre-image of a single element, we could ask, what's the pre-image of one? What elements map to one? Well, in this case, only four is gonna do that. We could ask ourselves, what's the pre-image of the element two? Well, what maps to two here? It's kind of reversing this question right here. If we know the things that map to two, then we know the pre-image is gonna be one and two. All right, what are the things that map to three? Well, it turns out that with regard to this function, nothing maps to three, so the pre-image in that case is the empty set. What are the things that map to four? Well, this is gonna be just three in this situation. And then lastly, what are the things that map to five? This is also an empty set. Nothing maps to five. And so this will be the pre-images of single tens. If you wanted to look at the pre-image of a collection, it's gonna be unions of these things, like you could ask yourself, what is the pre-image of the set one comma two? Well, we would grab everything that maps to one, sorry, everything that maps to one, which is four. Now that's what I meant to say. And then we also map everything, or we grab everything that maps to two, which would be one and two. So that would be the pre-image of one and two. We get the set one, two, four. All right, I want you to focus on this thing for a moment. The single tens are kind of the most important here. If you throw out these empty sets, then it actually looks like we have a partition of the domain. If you look at what are the pre-images, the non-empty pre-images of individual elements, this actually is gonna partition the domain into pieces. And we'll actually prove this in just a second. If you have a partition, that means there's gotta be an equivalence relation in hand here. We'll talk about that relation, that equivalence relation in just a second. Before doing so though, I actually wanted to discuss non-examples. It's not just good enough to talk about things that are functions. We have to also discuss things which are not functions. What can go wrong? Consider the following relation. G, which again, this is a relation between the two, the same two sets. Here's the set A right here, here's the set B. A is one, two, three, four. B is one, two, three, four, five. So the following is not a function relationship here. We have one, one, two, one, three, three, four, three and one, two. So the issue here comes to the first and last pair. The number one is actually used in two different order pairs. With regard to our diagram, we're getting two arrows that come out of one. One gets mapped to one, one gets mapped to two. So we have that f of one equals one and it equals two, which one is it? It can't be one or it can't be both. It's gotta be one or the other. At least to be a function, it has to be a specific evaluation. So this would then tell us that we don't have a function because the number one in the domain shows up more than once as an ordered pair. And so this is not a function relation because it's not a function because one shows up twice. There's two arrows coming out of it. Now be aware, there's no problem with two arrows coming into a number of the code domain. The problem is you can't have two arrows coming out of a number of the domain. That makes it not be a function. Another problem you have is the following. Take this relation. H equals one, one, two, one and three, three. With regard to the diagram, you have an arrow coming out of one, it points to one. You have an arrow coming out of two, which points to one. You have an arrow coming out of three, which goes to three in this situation. The problem is there's nothing coming out of four. So this also is an example of a relation that is not a function. It's a function if there is exactly one assignment given to every element in the domain, one and only one. So there has to be at least one, which this one doesn't have anything coming out of four. So it turns out that F of four, it does not exist in this situation. There's no assignment given to it that fails to be a function. And then of course we also can't have multiple arrows coming out, multiple assignments that we had before. Now again, I don't want you to confuse this with the fact that there's no arrows pointing at five or four or two in this relation. That's not a problem. You can have multiple arrows pointing to a number in the code domain. You can have no arrows pointing to a number in the code domain. That's not a problem. The problem comes to the domain. The domain needs to make sure that everyone has an arrow coming out of it. And you have to make sure you don't have multiple arrows coming out of the same number. If those things happen, it's not a function. Therefore a function is when there's one and only one arrow coming out of each element in the domain. All right, so now let's get back to this idea of equivalence relation that I hinted towards a little bit ago. If you do have a functional relation, so F is a function from A to B, then we can define a relation on the domain A by the following rule. We say that X is related to Y if and only if they have the same image that is F of X equals F of Y. This is then an equivalence relation which then means that if you have an equivalence relation, the equivalence classes are going to form a partition of the domain. Now the equivalence classes are gonna be exactly the pre-images of singletons of the co-domain. Well, the non-empty pre-images in that situation. So therefore that partition we had mentioned earlier was being induced by this equivalence relation. We can very quickly prove that this is an equivalence relation. Let X be inside of the domain. We have to show that this relation is reflexive, symmetric, and transitive. Reflexivity is pretty easy. If you take any element of the domain, F of X equals F of X. F of X exists because it's a function. And clearly F of X equals F of X. This means that X is related to itself and therefore the relation is reflexive. All right, now let's show that it's symmetric. Let's take two elements in the domain X and Y and suppose that X is related to Y. By definition of X is related to Y, that means the image of X under F is equal to F of Y, the image of Y here. But this is the quality of elements here. If F of X equals F of Y, that means F of Y equals F of X. And therefore Y is related to X. This shows that this is a symmetric relation. The last one here is gonna be transitivity. It's a very simple argument here. Suppose we have three elements, X, Y, and Z, such that X is related to Y and Y is related to Z. Well, if X is related to Y, that means that F of X is equal to F of Y. If Y is related to Z, that means they have the same images. Therefore, F of Y equals F of Z. And since F of X and F of Z are both equal to F of Y, that forces that F of X is equal to F of Z, thus showing that X is related to Z. This shows us that we have a transitive relation. And since we have a reflexive, symmetric, and transitive relation, this then proves that we have an equivalence relation just as was suggested. And so the equivalence classes are gonna be the pre-images of individual elements. And so the pre-image does in fact produce a partition on the domain for every function there is. And in fact, this essentially every equivalence relation is this relation right here because if you have a partition of a set, so here's some set X and you have a partition. So you have one cell here, one cell here, one cell here, one cell here. What we can do is we can label each of the cells. So here's the first one, here's the second one, here's the third one, here's the fourth one. And then we can define a function relation to the set over here, one, two, three, four. And then keep on going depending on how many labels you have. For which you then say, okay, everything in the first class, we map to one. Everything in the second class, we map to two. Everything in the third class, we map to three. Everything in the fourth class, we map to four. And then we keep on going if we have more and more cells in our partition. So every partition naturally invokes a function and every partition also invokes an equivalence relation. So these notions of equivalence classes, that is partitions, equivalence relations are deeply connected to this notion of a function that we're introducing right now. Every equivalence relation can be turned into a function and vice versa. So what I wanna do next then is connect this notion of function that we're talking about now with functions that we've seen previously. Like if I was in a calculus setting, if I was in a college algebra setting, my functions will look something like this. I'd have a formula, f of x equals x cubed. Now it turns out that these formulas do give us a function relation. If you pick any number in the domain, you can then evaluate it. Like f of two means you take two cubed, which is then eight. So the relationship here is that two is mapped to eight. And we can do this again, like f of one means you take one cube, which is equal to one. So one is then mapped to one. That's the function of relationship that's given by this formula. Now, as a function, as a set, we need to think of it in terms of domain and co-domain, right? So f is a function from, well, typically in calculus, you're thinking of functions from the real numbers to the real numbers. So you take any real number in and any real number out. And so that's how you wanna think of it here. Same thing with this function right here, g of x. It's given by the formula e to the x. What that really means is g is a function from the real numbers to the real numbers, like so. And so f, as a set, this first one here, is gonna be the collection of ordered pairs of the form x comma x cubed, where x is able to range over all real numbers. Like so, same thing with this one, g down here. G is the function, g as a function is the collection of ordered pairs of the form x comma e to the x, where x is able to range over all real numbers. So these functions we see in calculus are exactly the types of functions we're talking about right now where this formula is just telling you what the second number is gonna look like because the first number can be any real number. The second number will then have a form dependent upon the first one. Now, really what this means here when you look at this right here, a function is its graph. Like if we were to graph this thing, if you were to graph these things, these ordered pairs are coordinates in the plane, right? We can think of this as a subset of R2. And so when we graph something, we take the x-axis, typically that would be the horizontal axis, the input axis, then the vertical axis is typically the y-axis. And so if you graph something like f of x equals x cubed, you're graphing all of the possible ordered pairs that you see here, so like one of these points here is the point, what did we do earlier, two comma eight? And so as you put all of these ordered pairs together, you connect the dots and you get this graph. So essentially what we've done here is we've defined functions to be their graphs, at least in the typical analytic geometry setting here. That takes care of this one and this one here. Now there is one important caveat that we should mention. Actually, before we mentioned the caveat, let's just do some calculations that we did before here. The domain of these functions, these two functions f and g is all real numbers. Their co-domain is gonna be all real numbers. Their actual image might be something else. The image of f in this situation, of course, is gonna be all real numbers. But for the second one here, the image of g is actually not all real numbers. You only end up with zero to infinity because when you take an exponential expression like e to the zero, sorry, e to the x, you can't produce zero for any real numbers and you can't get negatives for any real numbers as well. So the image in the co-domain are actually different. That does happen for some of these functions here. All right, now let's get to that caveat. Also, like in college algebra calculus, you have functions that might be defined not for all real numbers, but still defined by a formula. Like when you look at something like this, one over x, you can't just plug in every real number because if you plug in x equals zero, this thing will be undefined. Or for this one here, if you plug in a negative number, that doesn't give you a real number. So in calculus, you have this so-called domain convention, the domain convention here, which tells us that we choose the domain to be the maximal subset of the real numbers so that the formula is well-defined. Okay, so for the first one, what you would say is this is a function, f1. It goes from the non-zero numbers, so negative infinity to zero union zero to infinity. That's the domain. You take everything with zero and so this maps into the real numbers here. For the second one, f2 here, you then take as your domain non-negative numbers, zero to infinity, and that maps into the real numbers as well. So you shrink the domain to be the largest collection of real numbers such that it's well-defined. Because again, if you stick in negative numbers there, you're gonna get some, you're gonna get some imaginary numbers in that situation. And what we always, we can always stick the co-domain to be the real numbers. But again, the image might be much smaller. Like the image of f1 here is gonna equal actually just non-zero numbers as well. So we get everything with zero. And for the second one, the image is likewise. This is going to be zero to infinity. You can't get any negatives coming out of a square root here. And so we don't usually talk about co-domains in calculus one, because it's usually just always the real numbers. You usually focus on the range, which we're calling the image. The domain, you choose to be big as possible. So when we can, we take the all real numbers, but when we can't, it's understood by the formulas you throw out those numbers for which it's undefined. So in a calculus setting or a college algebra setting, you're able to get around this idea of domain and co-domain because it's implicit inside of the formula. And calculus students are then learn, they learn to compute the domains of these functions. One other thing I should mention that from a calculus perspective, from a college algebra perspective, you have this so-called vertical line test that if the graph passes the vertical line test, then it's the graph of a function, meaning that a vertical line intersects the graph at most at one point. This was the rule that we had introduced earlier that for a functional relation that every number in the domain has one and only one assignment in the co-domain. If you had multiple assignments, then the graph would hit a vertical line in more than one spot. So we don't allow for things where you have two different intersections here because then it's like, well, f of x, which one does it equal? Does it equal this one? Or does it equal this one? We don't know. So to be a function, you need to pass the vertical line test from an analytic geometry point of view. So people often refer to it even when you don't have the geometry as the vertical line test. All right, the previous example, I did use this word that I wanna define right now, this idea of being well-defined. Maybe I didn't use it in the previous example. I used it somewhere in this video here. What does it mean for a function to be well-defined? Well, like we'd see in calculus, it's okay to use a formula to define a function, but we should be explicit on what the domain and the co-domain is because in calculus we have conventions, but students don't really understand the convention, so they're never using it really. The math professor knows what's going on there, and so it's really just to make the math professor feel good, I guess, about these functions, but there is always a co-domain and a domain when one talks about a function. But if you define functions using formulas, you have to be very careful that the relationship is well-defined, okay? So consider the following example. Be aware the relation we're gonna talk about right now is not a functional relation. Suppose we're gonna define a function from the rational numbers to the integers in the following way. F of P over Q, which a typical element of Q would look like P over Q. You have some integer numerator, some integer denominator. Our function is gonna report back the numerator of that fraction. So if you give me something like F of one-half, it gives me back one. If you give me F of 17 over five, it gives you back 17. Seems like a pretty simple function relationship. Let's just report the numerator here. The problem is the following. If you take two different fractions that are actually equal to each other as rational numbers, like take one-half and two-fourths, these are different fractions. I mean, because one has a numerator of one, one has a numerator of two, but they're the same rational number. That's the same ratio here. And so if you look at F of one-half, you're tempted to say this should equal one, but one-half is also two-fourths. If you take F of two-fourths, this relation wants to give you back two. So what is the evaluation? Is F of point five equal to one-half? Sorry, is F of point five, point five here, is it equal to one or is it equal to two? It could also equal three, because one-half is three-six. It could also equal four, because point five is four-eighths. So this relation depends entirely on the representation of the ratio and not on the ratio itself. So this is an example of a relation that is not well-defined. And so this is not a function. It's not a function, because basically what we've done here is it doesn't pass the vertical line test that the same number in the domain has different evaluations. If we were to write this relation using ordered pairs, we would have that point five has, it can be paired with one, it can be paired with two, it can be paired with three, et cetera, et cetera, et cetera. So the fraction one-half has multiple relations and that is not a function, that's not a function relation. This is just talking about one-half. Every rational number has this problem. This is not well-defined. So with that example that presented, the definition here is a relation is well-defined if each element in the domain is assigned to a unique element of the range, right? This is just, that's what it means to be a function, so we have to make sure our formulas are in fact well-defined. This thing is not a function because the same fraction will have multiple pairings and thus violates the vertical line test. One last comment I wanna mention about functions is how do you actually prove that two functions are equal to each other? We talk a lot about proofs in this lecture series and on occasion we have to prove that two functions are equal to each other. Now the thing to remember here, if you have two functions from A to B, they could be equal. Now, I should mention, if you have two functions and their domains and co-domains differ, so if you have a function from A to B and you have a function from C to D and you have something like, oh, A doesn't equal C or B doesn't equal D, if their domains are mismatched, they're not the same function. If their co-domains are mismatched, they're not the same function. So the only times we have to show that two functions are the same is when the domains and co-domains are the same. Okay, so that's now it's a possibility. What does it mean for two functions to be the same? Well, functions are themselves sets because they're relations. Relations are sets of ordered pairs. So to show that two sets are equal to each other, you show that they're subsets of each other. So okay, G equaling F means that F is a subset of G and G is a subset of F. That might feel weird when you're talking about relations, might be feeling you're talking about functions, but functions are relations and relations are sets and sets can be subsets of each other. Now let's consider the first one. If I was gonna prove that F is a subset of G, then I'd have to say that every element of F is an element of G, but a typical element of F is gonna be an ordered pair of the form X comma F of X where X is just an element of the domain, right? So I wanna show that the ordered pair X comma F of X is inside of G. Now let's say that I've successfully done that. Now if X comma F of X is inside of G, since X is inside the domain, then there's gonna be some ordered pair X comma G of X that's inside of there, right? Because by definition of a function, every element of the domain has some assignment and that assignment is its image G of X. So I know because G is a function, X comma G of X is inside of the set G, but if I also have shown that the element X comma F of X is inside of there by the vertical line test, the only way that you can have these two pairs because they both involve X is they actually have to be equal to each other. You have to have the X comma F of X is equal to X comma G of X, like so. And if you have two ordered pairs are equal to each other, their first coordinates have to equal, that's obvious, but it has to also be that their second coordinates are equal to each other. So we then get that F of X is equal to G of X, like so. So if F is a subset of G, that could only happen if the image is F of X is equal to G of X for all X inside the domain. And the reverse direction will give you the same thing. So in summary, if you want to show that two sets or two functions are equal to each other, what you actually argue is you argue that the images F of X equals G of X for every X inside of the domain. So if the images of two functions are always the same as you allow the element of the domain to vary, that actually implies that the two functions are equal to each other. And that's the proof template you would use in that setting. But with that said, that's where we're gonna end lecture 30. The only topic we talked about was functions, but that's the end of our video here. Thanks for watching. If you learned anything about functions in this video, please like this video, subscribe to the channel to see more videos like this in the future. Share these videos with friends and colleagues if you think they'd be interested in them too. And as always, if you have any questions, please post them in the comments below and I'll be glad to answer them as soon as I can.