WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:02.200 Welcome to Educator.com. 00:00:02.200 --> 00:00:07.200 In today's session, we are going to talk about several special functions. 00:00:07.200 --> 00:00:13.500 The first one we are going to discuss is the step function; and the step function makes a unique-looking graph. 00:00:13.500 --> 00:00:17.800 It is a function that is constant for different intervals of real numbers. 00:00:17.800 --> 00:00:27.800 And the result is a graph that is a series of horizontal line segments, so they look like steps; and that is where the name "step function" comes from. 00:00:27.800 --> 00:00:30.200 The best way to understand this is through an example. 00:00:30.200 --> 00:00:53.200 So, for example, if apples were sold at a price of a dollar per pound, and the price is such that 00:00:53.200 --> 00:01:10.000 you are charged \$1 for each pound, or any part of a pound--in other words, they round up in order to determine the price; 00:01:10.000 --> 00:01:18.800 if you had a pound and a half of apples, they are going to charge you \$1 for the pound, and then another \$1 for the half pound. 00:01:18.800 --> 00:01:28.500 So, they charged you a dollar for a pound, and then any part of a pound is considered a full pound in terms of pricing. 00:01:28.500 --> 00:01:44.400 So, if x is the number of pounds, and y is the cost, then let's see what kind of values we get and what the graph looks like. 00:01:44.400 --> 00:01:53.200 OK, if I have .8 pounds, I am going to get charged \$1: they are going to round up. 00:01:53.200 --> 00:02:04.000 If I have a full pound, I am going to get charged \$1; if I have a little bit over a pound--I have 1.2 pounds-- 00:02:04.000 --> 00:02:15.500 I am going to get charged \$1 for the first pound, and then that .2 is going to be another dollar, so \$2. 00:02:15.500 --> 00:02:24.900 1.4 pounds--again, \$2, and on up...2 pounds--a dollar for the first pound and a dollar for the second pound. 00:02:24.900 --> 00:02:35.600 2.5--\$2, and then the .5 is another \$1, so that bumps me up to \$3; until I hit 3 pounds, it is \$3, also. 00:02:35.600 --> 00:02:46.100 3.8: \$1 for the first pound, \$1 for the second pound, \$1 for the third pound, and another \$1 for that .8, so \$4. 00:02:46.100 --> 00:02:50.300 So, you can see that the function is constant for different intervals. 00:02:50.300 --> 00:02:59.700 So, for this first interval, from a little bit above 0, all the way to 1, the y-value is constant; it is \$1. 00:02:59.700 --> 00:03:07.700 For the next interval, which is just above 1, all the way to 2, including 2, it is going to be \$2. 00:03:07.700 --> 00:03:13.000 Once I get above 2, up to and including 3, it is \$3; and so on. 00:03:13.000 --> 00:03:17.800 So, this is constant for different intervals of real numbers. 00:03:17.800 --> 00:03:27.400 Looking at what the graph looks like: any part of a pound up to and including a dollar for that first pound is a dollar. 00:03:27.400 --> 00:03:33.500 Now, 0 pounds of apples are going to be \$0; so I am not going to include 0. 00:03:33.500 --> 00:03:43.900 But even .1...the slightest part of a pound is going to be \$1, so I put an open circle to indicate that 0 is not included; 00:03:43.900 --> 00:03:52.700 but just above that is, all the way up to and including 1; since 1 is included (1 is also a dollar), I am going to put that as a closed circle. 00:03:52.700 --> 00:04:03.400 So here, I have pounds; and here, I have the cost on the y-axis in dollars. 00:04:03.400 --> 00:04:14.200 Now, once I get just above 1 (say 1.1 pounds), they are going to charge me \$2--not including 1, but just above it--open circle. 00:04:14.200 --> 00:04:20.600 All the way up to 2...at 2 pounds, I will also be charged \$2. 00:04:20.600 --> 00:04:30.600 Once I hit just above 2, I am going to be charged \$3, all the way to 3 and including 3. 00:04:30.600 --> 00:04:46.900 And on...so, you can see how this looks like a series of steps, and how this is a result of the fact that the function is constant for different intervals of x. 00:04:46.900 --> 00:04:52.000 This function is the same for this entire interval; then it is the same for the second interval; and on. 00:04:52.000 --> 00:04:55.400 So, this is a step function. 00:04:55.400 --> 00:05:01.400 A second type of function that you will be working with is an absolute value function. 00:05:01.400 --> 00:05:17.300 And these functions have special properties: looking first at f(x) = |x|, just the simplest case, here f(x) is just going to be the absolute value of x. 00:05:17.300 --> 00:05:29.300 When x is 0, f(x) is also 0; when x is 1, the absolute value is 1; and so on for positive numbers. 00:05:29.300 --> 00:05:43.000 Now, let's look at negative numbers: for -1, the absolute value is 1; -2--the absolute value of x is 2; and on. 00:05:43.000 --> 00:05:57.900 The result is a certain shape of graph: when x is 0, f(x) is 0; x is 1, f(x) is 1; x is 2, f(x) is 2; and on. 00:05:57.900 --> 00:06:12.400 Now, for negative numbers: -1, f(x) is 1; -2, f(x) is 2; -3, f(x) is 3; and it is going to continue on like that. 00:06:12.400 --> 00:06:25.600 So, absolute value graphs are v-shaped; so we are going to end up with a v-shaped graph. 00:06:25.600 --> 00:06:33.300 Depending on the function, the graph can be shifted up, or it can be shifted to the right or to the left; and let's see how that could happen. 00:06:33.300 --> 00:06:44.100 Let's now let f(x) equal (here f(x) equals |x|)...let's say f(x) equals the absolute value of x, plus 1. 00:06:44.100 --> 00:06:51.800 OK, so we are given x, and the absolute value of x is 0; we are adding 1 to them, so this is going to become 1. 00:06:51.800 --> 00:07:05.600 The absolute value of x is 1; 1 + 1 is 2; 3; 4; the absolute value of -1 is 1; 1 + 1 is 2. 00:07:05.600 --> 00:07:12.100 The absolute value of -2 is 2; add one to that--it is 3; and add 1 to 3 to get 4. 00:07:12.100 --> 00:07:19.500 OK, so it is the same as this, except increased by 1: each value of the function has been increased by 1. 00:07:19.500 --> 00:07:23.300 So now, let's see what my graph is going to look like. 00:07:23.300 --> 00:07:31.100 Right here, I have the graph for f(x) = |x|; now, I am going to look at this graph. 00:07:31.100 --> 00:07:45.300 When x is 0, f(x) is 1; when x is 1, f(x) is 2; when x is 2, f(x) is 3; so you can see what is happening. 00:07:45.300 --> 00:08:12.300 And then here, I have x is 3, f(x) is 4; negative values--when x is -1, f(x) is 2; when x is -2, f(x) is 3; when x is -3, f(x) is 4. 00:08:12.300 --> 00:08:29.000 OK, I am drawing the line through this: this is the graph of f(x) = |x| + 1. 00:08:29.000 --> 00:08:33.400 So you see that this graph, the v-shaped graph, is simply shifted up by 1. 00:08:33.400 --> 00:08:38.700 And again, you can also shift this from side to side; and we will see an example of that later on. 00:08:38.700 --> 00:08:46.300 So, in the absolute value function, it is very important to find both negative and positive values of the function. 00:08:46.300 --> 00:08:56.700 So, assign x 0; assign it some positive values; and it is very important to find what f(x) will be when x is negative, 00:08:56.700 --> 00:09:01.500 because if I didn't--if I picked only positive values--I would end up with half of a graph. 00:09:01.500 --> 00:09:07.900 So, to get the entire v-shape, choose negative and positive values for x. 00:09:07.900 --> 00:09:13.200 The third special function that we are going to discuss is called a piecewise function. 00:09:13.200 --> 00:09:20.600 And a piecewise function is a function that is described using two or more different expressions. 00:09:20.600 --> 00:09:24.400 The result is a graph that consists of two or more pieces. 00:09:24.400 --> 00:09:33.400 Just starting out with one that consists of two pieces: the notation is usually like this--one large brace on the left: 00:09:33.400 --> 00:09:50.400 f(x) equals x + 2 for values of x that are less than 3; and f(x) equals 2x - 3 for values of x that are greater than or equal to 3. 00:09:50.400 --> 00:09:57.000 So you see that, for different intervals of the domain, the function is defined differently. 00:09:57.000 --> 00:10:00.600 So, it is a function that is described using two or more different expressions. 00:10:00.600 --> 00:10:04.700 So, for the part of the domain where x is less than 3, this is the function. 00:10:04.700 --> 00:10:12.700 For the interval of the domain that is greater than or equal to 3, this is the function that I am going to use. 00:10:12.700 --> 00:10:25.500 Let's see what happens: let's first use this part of the function where f(x), or y, equals x + 2. 00:10:25.500 --> 00:10:29.800 And for the domain, remember that x is going to be less than 3. 00:10:29.800 --> 00:10:35.900 So, I will go ahead and start out with 2: when x is 2, f(x) (or y) is 4. 00:10:35.900 --> 00:10:48.300 When x is 1 (I have to remain at x-values less than 3), f(x) is 3; when x is 0, f(x) is 2. 00:10:48.300 --> 00:10:55.800 Just picking a negative number: when x is -4, f(x) is -2. 00:10:55.800 --> 00:11:29.100 I am going to graph that here; OK, when x is 2, f(x), or y, is 4; x is 1, y is 3; x is 0, y is 2; x is -4, y is -2. 00:11:29.100 --> 00:11:38.000 OK, now, this is for values of x that are less than 3; 3 is about here. 00:11:38.000 --> 00:11:43.200 Therefore, anything just below 3, but not including 3, is going to be part of this graph. 00:11:43.200 --> 00:11:56.400 So, we are going to use an open circle here to indicate that 3 is not going to be included as part of this function--the domain of this function. 00:11:56.400 --> 00:12:03.400 So, it is going to begin at just below 3 and continue on indefinitely; that is the first piece of the graph. 00:12:03.400 --> 00:12:15.100 The second piece of the graph is for x such that x is greater than or equal to 3; and here, y is going to be 2x - 3. 00:12:15.100 --> 00:12:18.600 So, it is greater than or equal to 3, so I am first going to let x equal 3. 00:12:18.600 --> 00:12:35.300 3 times 2 is 6, minus 3 is 3; getting larger--when x is 4, it is 2 times 4 is 8, minus 3 is 5; when x is 5, 5 times 2 is 10, minus 3 is 7. 00:12:35.300 --> 00:12:48.200 Now, this does include 3--this section of the graph--this piece; so, when x is 3, y is 3, right here. 00:12:48.200 --> 00:13:21.000 When x is 4, y is 5, right here; when x is 5 (let me shift that over just a bit), y is 7; OK. 00:13:21.000 --> 00:13:27.500 Now, if you look, this actually did end up including all possible values of x (all real numbers), 00:13:27.500 --> 00:13:36.000 because when x is less than 3, I use this function; and then, as soon as x becomes 3 or greater, I shift to this other function. 00:13:36.000 --> 00:13:42.300 So, you can see how there are two pieces to the graph; and you actually can have situations where there are more than 2. 00:13:42.300 --> 00:13:56.500 You could be given, say, f(x) is 4x + 3, 2x + 7, and x - 1, and then given limits on the domain for each of those. 00:13:56.500 --> 00:14:00.200 So, there are at least two pieces; however, there can be more. 00:14:00.200 --> 00:14:04.700 In Example 1, we have a greatest integer function. 00:14:04.700 --> 00:14:08.500 Before we start working in this, let's just review what we mean by the greatest integer function. 00:14:08.500 --> 00:14:15.000 So, when you see this notation with the brackets, let's say that you have a number in here, such as 4.7. 00:14:15.000 --> 00:14:26.300 What this is saying is that this value is equal to the greatest integer that is less than or equal to 4.7; so that is 4. 00:14:26.300 --> 00:14:35.500 It is the greatest integer less than or equal to whatever is in here; if it was 2.8, it would be 2. 00:14:35.500 --> 00:14:42.200 Be careful with negative numbers: let's say I have -3.2--the temptation is to say, "Oh, that is equal to -3"; 00:14:42.200 --> 00:14:53.000 but if you look at it on the number line, -3.2 is right about here; OK, so if I have -3.2, 00:14:53.000 --> 00:15:03.300 and I am trying to find the greatest integer that is less than or equal to 3.2, it is going to have to be something over here--smaller. 00:15:03.300 --> 00:15:07.600 So, it is actually going to be -4; so just be careful when you are working with negative numbers. 00:15:07.600 --> 00:15:20.400 Whatever is inside here--whatever that value is--the function is equal to the greatest integer that is less than or equal to this value in here. 00:15:20.400 --> 00:15:27.100 Understanding that, you can then find the graph; so let's find a bunch of points for this, 00:15:27.100 --> 00:15:34.900 so we make sure we know what is going to happen with various situations. 00:15:34.900 --> 00:15:44.800 When x is 0, this inside here is going to be 2; the greatest integer less than or equal to 2 is 2. 00:15:44.800 --> 00:15:53.600 When x is .6, then you are going to get 2.6 in here; the greatest integer less than or equal to 2.6 is 2. 00:15:53.600 --> 00:15:58.300 .8--I get 2.8; again, I round down to 2. 00:15:58.300 --> 00:16:06.700 All right, so when I hit 1, 1 plus 2 is 3, and the greatest integer less than or equal to 3 is 3. 00:16:06.700 --> 00:16:16.200 Slightly above 1: that is going to give me 1.2 + 2 is 3.2; the greatest integer less than or equal to 3.2 is also 3. 00:16:16.200 --> 00:16:20.100 OK, so you can get the idea of what this is going to look like. 00:16:20.100 --> 00:16:28.400 And that continues on; and then, when we hit 2, 2 + 2 is 4; the greatest integer is going to be 4. 00:16:28.400 --> 00:16:44.700 For negative numbers: let's take -.5: -.5 and 2 is 1.5; the greatest integer less than or equal to 1.5 is going to be 1. 00:16:44.700 --> 00:16:53.700 Now, notice: I have a negative number for x, but this did not come out to be a negative number; so that is different from the case I was discussing there. 00:16:53.700 --> 00:17:04.500 Let's go a little bit bigger--let's say -3: -3 and 2 is -1, and that is going to be -1. 00:17:04.500 --> 00:17:17.600 Let's say I take -3.5: -3.5 and 2 is going to equal -1.5: again, just thinking about that to make sure you have it straight, 00:17:17.600 --> 00:17:32.700 I have -1.5; so I have 0; I have -1; I have -1.5; I have -2; the greatest integer less than or equal to this is actually -2. 00:17:32.700 --> 00:17:55.200 OK, now plotting this out: when x is 0, f(x) is 2; when x is slightly above 0 (it's .6), f(x) is 2; .8--it is 2, all the way up until I hit 1. 00:17:55.200 --> 00:18:02.200 At 1, f(x) becomes 3; therefore, 1 is not included in this interval. 00:18:02.200 --> 00:18:08.700 So, you can already see that this is going to be a step function, because we have intervals. 00:18:08.700 --> 00:18:15.000 For different intervals of the domain, we have that same value for the range. 00:18:15.000 --> 00:18:31.900 All right, for values between 1 and 2, f(x) will be 3; once we hit 2, I have to do an open circle, because at 2, the value for f(x) jumps up to 4. 00:18:31.900 --> 00:18:36.300 OK, so you can see what this is going to look like; and that pattern is just going to continue. 00:18:36.300 --> 00:18:57.100 Let's look over here at negative numbers: when x is slightly less than 0, then you are going to end up with an f(x) that is 1. 00:18:57.100 --> 00:19:05.000 So, for values slightly less than 0, but not including 0, this is what you are going to end up with. 00:19:05.000 --> 00:19:21.000 OK, looking, say, when x is -3: when x is -3, f(x) will be -1. 00:19:21.000 --> 00:19:35.200 But when we go slightly more negative than that, when x is -3.5, f(x) is going to be -2; it is going to be down here. 00:19:35.200 --> 00:19:43.700 So, the steps on this side are going to have the open circle on the right. 00:19:43.700 --> 00:19:53.200 And I am going to jump down, and it is not going to include -2, because -2 and 2 is 0; 00:19:53.200 --> 00:19:59.000 so -2 is going to be right here for the x-value, and the f(x) will be 0. 00:19:59.000 --> 00:20:07.300 But as soon as I get to a little bit bigger than -2, the greatest integer is going to be down here. 00:20:07.300 --> 00:20:20.500 OK, and so, we continue on like that with the steps; and you can see how this is a step function. 00:20:20.500 --> 00:20:25.100 You just have to be very careful and pick multiple points until you can see the pattern 00:20:25.100 --> 00:20:34.100 where for a certain interval of the domain, the range is a particular value. 00:20:34.100 --> 00:20:40.200 OK, so that was a step function, and it involved the greatest integer function. 00:20:40.200 --> 00:20:43.800 Example 2: now we are working with absolute value. 00:20:43.800 --> 00:20:47.800 g(x) equals the absolute value of x, minus 3. 00:20:47.800 --> 00:20:52.100 And we already know that the shape of this graph is going to be in a v. 00:20:52.100 --> 00:20:58.400 But we don't know exactly where that v is going to land, so let's plot it out. 00:20:58.400 --> 00:21:03.700 When x is 0, the absolute value of x is 0; minus 3--that gives me -3. 00:21:03.700 --> 00:21:09.400 When x is 1, the absolute value is 1; minus 3...g(x) is -2. 00:21:09.400 --> 00:21:17.300 When x is 2, the absolute value is 2; minus 3 is going to give me -1. 00:21:17.300 --> 00:21:22.700 Now, let's pick some negative numbers for x, because that is really important to do with an absolute value graph. 00:21:22.700 --> 00:21:31.500 When x is -1, the absolute value is 1, minus 3 gives me -2; you can already see that my v shape is going to occur. 00:21:31.500 --> 00:21:38.900 When x is -2, the absolute value is 2; minus 3 is -1. 00:21:38.900 --> 00:21:45.000 The absolute value of -3 is 3; minus 3 is 0; so this is enough to go ahead and plot. 00:21:45.000 --> 00:21:55.900 x is 0; g(x) is -3; x is 1, g(x) is -2; x is 2, g(x) is -1; over here with the negative values, 00:21:55.900 --> 00:22:05.100 when x is -1, g(x) is -2; when x is -2, g(x) is -3; when x is -3, g(x) is 0. 00:22:05.100 --> 00:22:17.800 So, you can see that I have a v-shaped graph, and compared with my graph that would look like this, 00:22:17.800 --> 00:22:28.400 that would have the v starting right here, it has actually shifted down by 3; that is an absolute value function. 00:22:28.400 --> 00:22:37.000 Here you can see that you are given a piecewise function, because there are two different pieces. 00:22:37.000 --> 00:22:40.200 And this could also be written in this notation. 00:22:40.200 --> 00:22:51.600 There are two different sections to the graph; and we see that the function is defined differently for different intervals of the domain. 00:22:51.600 --> 00:23:06.400 Starting with if x is greater than 2 (this is going to be for x-values where x is greater than 2): f(x) is going to be x + 1. 00:23:06.400 --> 00:23:24.400 When x is 3, f(x) is 4; when x is 4, f(x) is 5; when x is 5, f(x) is 6; OK. 00:23:24.400 --> 00:23:36.000 When x is 3, f(x) is 4; when x is 4, f(x) is 5; and it is going to go on up. 00:23:36.000 --> 00:23:42.500 And that is going to go all the way, until just greater than 2. 00:23:42.500 --> 00:23:50.400 2 is not going to be included on this graph, because it is a strict inequality; so I am going to use an open circle, and this is going to continue on. 00:23:50.400 --> 00:24:03.400 Now, for x less than or equal to 2, I have a different situation: I am looking at f(x) is -2x. 00:24:03.400 --> 00:24:23.700 OK, so when x is 2, 2 time -2 is -4; when x is 1, 1 times -2 is -2; when x is 0, f(x) is 0; when x is -2, that is -2 times -2, which is positive 4. 00:24:23.700 --> 00:24:36.600 So, starting with x is 2: when x is 2, f(x) is -4; and that is including the 2. 00:24:36.600 --> 00:24:54.300 When x is 1 (these are values less than or equal to 2, so I am getting smaller), f(x) is -2. 00:24:54.300 --> 00:25:09.500 x is 0; f(x) is also 0; when x is -2, f(x) is up here at 4; OK, so I have a steep line going right through here. 00:25:09.500 --> 00:25:15.500 So, you can see: this is a piecewise function consisting of two pieces; and here, one picks up where the other leaves off. 00:25:15.500 --> 00:25:25.500 For values greater than 2, this is my graph; for values of x less than or equal to 2, this is my graph; so this is a piecewise function. 00:25:25.500 --> 00:25:31.900 OK, this time, in Example 4, we have both greatest integer and absolute value in this function. 00:25:31.900 --> 00:25:39.900 Recall that, for the greatest integer function, what that is saying is that whatever is inside this bracket--let's say it's 1.2-- 00:25:39.900 --> 00:25:45.600 it is asking for the greatest integer less than or equal to 1.2; in that case, this would be 1. 00:25:45.600 --> 00:25:51.000 Or if I had 4.8, it would be 4. 00:25:51.000 --> 00:26:00.100 For negative numbers, like -3.2, the greatest integer less than or equal to -3.2 is -4. 00:26:00.100 --> 00:26:06.900 OK, now, since this is a bit complicated, it is helpful just to take it in stages. 00:26:06.900 --> 00:26:18.100 So, I am going to look first at what the greatest integer of x is; and then, I am going to look for the absolute value of what that is. 00:26:18.100 --> 00:26:30.400 If x is .2, the greatest integer less than or equal to .2 is 0; the absolute value of 0 is 0; so this is the function that we are looking for. 00:26:30.400 --> 00:26:35.700 And the same would hold true of .5: round down to 0; the absolute value is 0. 00:26:35.700 --> 00:26:48.100 When we hit 1, the greatest integer less than or equal to 1 is 1, and the absolute value of that is 1. 00:26:48.100 --> 00:27:00.800 1.2: again, we are going to go down to the greatest integer that is less than or equal to 1.2, which is 1; and the absolute value is 1. 00:27:00.800 --> 00:27:09.400 The same for 1.8, and all the way up until 2; once we hit 2, the greatest integer less than or equal to 2 is 2; the absolute value is 2. 00:27:09.400 --> 00:27:16.100 So, that is working with positive numbers, greatest integer, and the absolute value; it is the same; OK. 00:27:16.100 --> 00:27:30.600 So, let's go to negative: for -.4, the greatest integer that is less than or equal to -.4...I am looking, and I have 0, and 1, 00:27:30.600 --> 00:27:38.900 and -1, and -.4 is about here; so I am going to go down to -1; the absolute value of that is 1. 00:27:38.900 --> 00:27:43.700 Here you can see that the greatest integer is not the same as the absolute value. 00:27:43.700 --> 00:27:54.600 Or for -1, the greatest integer less than or equal to -1 is -1; the absolute value is 1. 00:27:54.600 --> 00:28:06.200 -1.8: the greatest integer that is less than or equal to -1.8...I am going to go down to -2; and the absolute value is 2. 00:28:06.200 --> 00:28:14.200 For -2, the greatest integer less than or equal to -2 is -2; the absolute value is 2. 00:28:14.200 --> 00:28:24.200 So, you see that there are intervals here--intervals of the domain end up with the same value for the function. 00:28:24.200 --> 00:28:28.900 So, I am going to have a step function. 00:28:28.900 --> 00:28:35.000 But remember that absolute value graphs are v-shaped, so I am going to end up with a v-shaped step function. 00:28:35.000 --> 00:28:46.300 Let's plot these out: for 0, the greatest integer of 0 would be 0, and then the absolute value would be 0. 00:28:46.300 --> 00:28:56.100 So, with 0, we are going to include it; and for all values up to but not including 1, the function is going to equal 0. 00:28:56.100 --> 00:28:59.700 Once we get to 1, I have an open circle, because it is not included. 00:28:59.700 --> 00:29:06.100 When x is 1, f(x) is 1; so I am going to jump up here. 00:29:06.100 --> 00:29:14.500 All the values between 1 and 2, but not including 2, will have an f(x), or a y-value, that is 1. 00:29:14.500 --> 00:29:31.000 As soon as I hit 2, open circle: I am going to jump up, and once I hit 2, f(x) is 2, all the way up to, but not including, 3. 00:29:31.000 --> 00:29:37.800 And it is going to go on that way: and you see now, we have the step function, and it is v-shaped like absolute value. 00:29:37.800 --> 00:29:41.900 Let's look over here on the negative side of things. 00:29:41.900 --> 00:29:59.200 For -.4, somewhere in here, it is going to equal 1; -1 is also equal to 1; so here, on the left side, I have a closed circle, and an open circle on the right. 00:29:59.200 --> 00:30:02.300 It is the opposite of what I had over here. 00:30:02.300 --> 00:30:15.500 When I get to less than -1, my value for f(x) is going to jump up to 2; this is a closed circle; 00:30:15.500 --> 00:30:21.700 I get slightly less than, but not including, -1; it is going to jump up to 2. 00:30:21.700 --> 00:30:41.400 -2: my value is also 2, and everything in between; and then, when I get to just slightly more negative than -2, like -2.1, it is going to jump up to 3. 00:30:41.400 --> 00:30:47.100 You can see how this is v-shaped, and it is a step function. 00:30:47.100 --> 00:30:52.700 The step function comes from it being the greatest integer function; the v shape comes from that absolute value. 00:30:52.700 --> 00:31:05.000 And you also just had to be careful how you are doing the open and the closed circles; OK.