 Welcome back to our lecture series Math 1050, college algebra for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. This section, 2.1, is the first section in chapter two. We saw in chapter one all these many different topics related to functions. We talked about graphs and the transformations, composition, inverses, so many other things, right? And it really might feel like a baptism by fire for us into college algebra. And this was done intentionally, right? Our focus for this series will always be on functions and using functions and understanding functions. And so we were exposed to basically the important concepts related to functions. But for many of us, as we were going through that, there was a lot of stuff that we didn't know, right? We didn't understand, or maybe it's like we learned it once, but I'm not really strong on it right now. That was okay. If that's the experience you felt, by all means know that you were okay in that regard. What we're going to be doing in subsequent chapters, like in chapter two, is we're going to be focusing on specific function families and revisiting many of those topics we learned about previously, like how do you graph this function? What's the domain or range of this function? How do you solve equations or inequalities involving this function? And many things like onto that, looking at applications of these functions. And so chapter two is going to focus on the simplest of all types of functions, that idea of linear functions. So what is a linear function after all? A linear function as defined here on the screen. A linear function is a function that can be written in the form f of x equals mx plus b. So that is, there's a single variable x, which that variable will have no exponents attached to it, it'll just be the first power. It will likely have some coefficient in front of it. That coefficient m is often referred to as the slope of the linear function, given the geometric application of slope, which we've talked about before with average rates of change. We'll talk about some more in this section of course as well. And then also there potentially could be a constant value right here. We will see in the not too distant future that this value actually corresponds to the y-intercept. And when I say not too distant future, I mean it happens at this moment, right? When you want to find the y-intercept of any function, you just look at f of zero. Well for a linear function, when you plug in x as zero, everything's going to cancel away except for this constant term right here. So this term b is none other than the y-intercept of the graph. And so you often get that labeling of things, right? b is the y-intercept, m is the slope. And so this equation right here is often referred to as the slope-intercept form of a line. Now why do we call such a thing a linear function? Well it's because the graph of a linear function is a line in the usual geometric sense. So let's explore a few of those things related to linear functions. Like if we have a specific linear function in hand f of x equals 3x minus 5, could we solve for f of x equals 4? What does it mean to determine x given the y-coordinate 4? Well if you want to solve a linear equation, you're going to replace the y-coordinate with the given one 4. This equals 3x minus 5. When one solves a linear equation, you basically follow the following template, right? Especially when it's in this form, mx plus b. You're going to subtract the b from both sides. So in this case, since we have a negative 5, we want to add 5 to both sides to the inverse operation. It'll cancel out on the side where you had the b. And on the other side, you're going to combine the terms 4 and 5 give us a 9. And so then we have 3x equals 9. Then you're going to divide both sides by the slope. So we're going to divide both sides by 3 here. The 3s will cancel out. And you'll notice on the left-hand side, we'll just have x. And then x would equal 9-thirds, which itself is equal to 3. So this would be the solution to the equation. How do we check to see if this is the right answer? We'll plug this number back into the function and see what happens. f of 3 is equal to 3 times 3 minus 5. And if you did everything correctly, this checking it should kind of feel like you solved it backwards, right? 3 times 3 is 9. And I feel like we saw a 9 earlier, didn't we? And then 9 minus 5 gives us back a 4, right? Didn't we do 4 plus 5 to get 9? And so sure enough, f of 3 equals 4, which means 3 is the solution to this equation. So solving 4 and evaluating linear functions is generally a straightforward process. You might have to add or subtract a constant from both sides. You'll probably have to divide by the coefficient, a.k.a. the slope of the function. And that's how one solves linear functions. When it comes to graphing linear functions, I mentioned earlier that the graph of a linear function is a line, which is why we actually call it a linear function. Now, if you want to graph a linear function, the following strategy is basically what you could do the following, right? You want to look for two points on the graph of the line because after all, two points determine a line. We play connect the dots in this situation. If I can give you two points on the graph, then I can connect the dots with a straight edge and that would give you the graph of the line. Well, which two points should I look for? Well, the y intercept is usually a natural candidate. Look at f of 0 here. You're just going to get negative 1, like we mentioned before, right? If your function is f of x equals 2x minus 1, then the y intercept would be negative 1. We can get that very quickly. And that tells us the point 0 comma negative 1 will be on the graph, as illustrated to the right on the screen here. If you want to not find another graph point, you could look for the x intercept. The x intercept would come down to solving the equation 0 equals 2x minus 1. And that's an easy equation to do. Let's take add 1 from both sides. We get now 2x equals 1. Divide both sides by 2. You're ending up with x equals 1 half. So this tells you that the x intercept will be 1 half comma 0, for which we could plot that on the graph right here. 1 half comma 0. And then we could connect the dots, right, to make the line. We could connect the two dots we have, not a big deal. The x intercept is generally not the next location I would be going for. Because again, I want to find any two points. It doesn't matter which two. And therefore, I want something that's going to be easy. Something like f of 0, right? So instead, I might be wanting to look at something like f of 1. Just plug in a number which I can predict that arithmetic is going to be easy. If you take 2 times 1 minus 1, you end up with 2 minus 1, which itself is 1. And so then we get the point 1 comma 1, which you see here on the graph. And so with any two points on the graph, you can connect the dots. And that'll then give you the line that you're looking for. Something like that. All right. That's how one graphs the line of the graph of a linear function. What about, say, the domain or range of a linear function? Well, it turns out this is a very simple question for linear functions. Like we talked about before, at this venture, the only things we'd have to worry about with respect to the domain convention is division by 0, which is not going to happen with a linear function. With a linear function, there's really no division. Your slope could be a fraction, which is division, right? But you're not going to be dividing by any variables whatsoever. And so there's no division by 0. There's no square roots or fourth roots or sixth roots or anything like that. So when it comes to the domain of a linear function, the answer is always going to be the simple truth. It's all real numbers. Every linear function ever created, its domain is going to be all real numbers. On the other hand, the range isn't going to be too tricky, but there is one important exception you have to watch out for. When you look at this graph right here, f of x equals 2x minus 1, you'll notice that you have an upward going arrow on the right. This tells us that as x goes to infinity, y goes to infinity as well. And then we also see this downward arrow on the left, which is telling us that as x approaches negative infinity, y will likewise go towards negative infinity. And so this then tells us that the range will likewise be all real numbers. That on this linear function, we get every possible number as an output. And this is typically what happens with a linear function. Its domain will always be all real numbers, and its range is usually all real numbers. There is one important exception to this. So if we were to redraw our x and y axes like so, then an important exception to this range situation is, what if you have a horizontal line? Oops, that's not a horizontal line. Try that again. There's a horizontal line. And so let's say this is the graph of our function f right here. And then it goes to the point y equals 3, excuse me, that's our y intercept. Well, the equation of this function would look something like the following. f of x equals 0x plus 3, or in other words, f of x just equals 3. It's a constant function. Now, constant functions are linear functions. They're situations where the slope is 0. In this situation, you still do get that the domain of the function is all real numbers. That part's never going to change. The situation in the range though is very different here. In this situation, the range would just be the number 3. There's only one number in the range right there. If you wrote this in interval notation, it would look like the interval from 3 to 3, right? We want all numbers which are greater than or equal to 3 and less than or equal to 3. Well, there's not a lot of options there. It's 3. And so although we're not going to be dealing with constant functions that often, I should mention it is a special case. And this would be the only situation where a linear function would not, where its range wouldn't be all real numbers. It's just a fixed number, which would coincide with the y intercept there. When discussing lines, there is one other important case we should mention. What happens if you have a vertical line? So a horizontal line is going to be given by the equation y equals some specific number. A vertical line will be described very similarly. It would be described as x equals like negative 4, or whatever that number turns out to be, right? A vertical line will look like x equals a number. Now, this is a vertical line. It is a line, but this wouldn't be a linear function. Some of the issues here is that a vertical line doesn't pass the vertical line test because it itself is a vertical line. If you draw a vertical line that overlaps this one, it'll intersect in multiple locations. In fact, infinitely many locations. So a vertical line, although it's a line, is not actually a function. And so we do have to distinguish between that a little bit. We're not going to talk about the domain and range of a vertical line because, again, it's not a function. In regards to slope, though, I do want to mention that with a horizontal line, we do say that it has a slope of zero, which kind of makes sense from this expression right here. On the other hand, for a vertical line, we actually say that the slope is undefined. And that's because if you try to make any sense out of the slope, you're going to get a division by zero when you start considering things like rise over run. Like if you take this point right here at negative four comma zero and this point right here at negative four comma two. The rise over run would look like two over zero, which is not a number. And so we often say that the slope for a vertical line is undefined. So these are two important sort of exceptional lines compared to the other oblique lines we'll typically be focusing on, like this one right here, y equals 2x minus one.