WEBVTT 00:00:04.600 --> 00:00:14.060 A relation is a set of ordered pairs. 00:00:21.140 --> 00:00:30.120 Ok, we got a new definition. A function is a relation. 00:00:45.860 --> 00:00:50.740 So it's a relation with a condition that must be fulfilled. 00:00:50.740 --> 00:00:58.600 The condition is each x coordinate. 00:01:02.500 --> 00:01:03.800 Sometimes we call that the input 00:01:05.640 --> 00:01:12.660 So, function, relation, set each x coordinate corresponds to only one y coordinate. 00:01:12.660 --> 00:01:13.410 Okay. 00:01:13.400 --> 00:01:16.040 Are A and B both relations? 00:01:18.960 --> 00:01:20.860 Oh, it's not supposed to be a hard question. 00:01:21.320 --> 00:01:22.070 Yeah. 00:01:21.740 --> 00:01:23.020 They are what? 00:01:23.740 --> 00:01:24.490 Because they are sets of ordered pairs. 00:01:24.820 --> 00:01:26.280 They're supposed to just set the work. 00:01:31.760 --> 00:01:33.260 Is A a function? 00:01:33.480 --> 00:01:34.300 Is B a function? 00:01:34.420 --> 00:01:37.160 Are they both functions or neither of them functions? 00:01:37.800 --> 00:01:38.550 don't you discuss 00:01:38.880 --> 00:01:40.440 Is C a function? 00:01:41.060 --> 00:01:41.810 No. 00:01:41.820 --> 00:01:42.660 No, why not? 00:01:43.220 --> 00:01:45.840 Because 2 has 2 y-coordinates. 00:01:46.400 --> 00:01:54.380 Because the x-coordinate of 2 corresponds to 5 and corresponds to 8. 00:01:58.280 --> 00:02:03.720 Here in A, 1 corresponds to 3, 5 corresponds to 3. 00:02:03.940 --> 00:02:08.400 We don't care as long as the same x value only corresponds to one y value. 00:02:09.240 --> 00:02:15.980 So when we're looking at the list, if it's in roster notation like this, all you have 00:02:15.980 --> 00:02:16.730 to do is check. 00:02:16.600 --> 00:02:19.520 As long as it doesn't repeat an x value, it's a function, guaranteed. 00:02:20.220 --> 00:02:24.280 A repeated x-value means the relation is not a function. 00:02:25.160 --> 00:02:31.220 This is the same as relation A, just the graph of the set. 00:02:38.180 --> 00:02:48.100 Relation C contains (2, 5), (2, 8), and (6, 3). 00:02:48.620 --> 00:02:52.240 This is relation C. It contains three ordered pairs. 00:02:52.900 --> 00:02:55.520 And I think you guys learned last year a test to see 00:02:55.520 --> 00:02:56.960 whether a relation is a function. 00:02:57.840 --> 00:02:58.590 What is it? 00:02:58.600 --> 00:02:59.350 line test. 00:02:59.540 --> 00:03:01.340 Called the vertical line test. 00:03:03.820 --> 00:03:05.480 How many remember that from last year? 00:03:07.700 --> 00:03:12.240 If every vertical line only touches it once, it is a function. 00:03:14.800 --> 00:03:19.340 Or if any vertical line touches the relation twice, it's not a function. 00:03:21.160 --> 00:03:23.200 Two ways to say the vertical line test. 00:03:25.180 --> 00:03:28.500 Let's write that down. 00:03:38.340 --> 00:03:44.280 If any vertical line touches the relation twice, it 00:03:50.200 --> 00:03:51.040 Is this a relation? 00:03:52.680 --> 00:03:53.430 Yes. 00:03:53.300 --> 00:03:59.280 Yeah, the set of ordered pairs is some linear relation. 00:04:00.040 --> 00:04:01.620 Is this also a function? 00:04:02.100 --> 00:04:02.850 Yes. 00:04:02.720 --> 00:04:03.560 Yes, and how do we know? 00:04:04.080 --> 00:04:07.200 Because every x-coordinate has only one y-coordinate. 00:04:07.540 --> 00:04:12.940 Either by the definition, every x-coordinate has only one y-coordinate, or corresponds only one y-coordinate, or? 00:04:13.540 --> 00:04:14.540 Passes the vertical line. 00:04:14.680 --> 00:04:18.400 Passes the vertical line test. Every single vertical line touches it most once. 00:04:19.640 --> 00:04:20.390 Okay. 00:04:26.440 --> 00:04:30.600 We're going to learn in a couple months about circles. 00:04:32.740 --> 00:04:33.700 Is this a relation? 00:04:34.800 --> 00:04:35.550 Yes. 00:04:36.620 --> 00:04:37.900 Is this a set of ordered pairs? 00:04:38.640 --> 00:04:39.390 Yeah. 00:04:40.180 --> 00:04:46.380 Anything on a coordinate plane in black here is just a set of ordered pairs. 00:04:47.160 --> 00:04:48.120 It is a relation. 00:04:48.620 --> 00:04:49.400 Is it a function? 00:04:49.780 --> 00:04:50.530 No. 00:04:50.520 --> 00:04:51.270 No. 00:04:55.440 --> 00:04:58.380 At least one vertical line touches it twice, so it is not a function. 00:04:59.180 --> 00:05:01.680 We can still do lots of math with relations. 00:05:02.880 --> 00:05:03.740 We'll learn about that later. 00:05:04.440 --> 00:05:07.760 But just distinguishing what is a function, what is not a function. 00:05:08.400 --> 00:05:09.360 OK, another definition. 00:05:09.360 --> 00:05:27.280 And the domain of a relation or function is the set 00:05:27.280 --> 00:05:28.380 is the range 00:05:35.480 --> 00:05:45.720 The range of a relation is the set of all y-coordinates. 00:05:48.380 --> 00:05:53.060 Sometimes we say the domain is the set of all inputs and the range is the set of all outputs. 00:05:53.900 --> 00:05:56.600 So what's the domain of A? 00:05:58.320 --> 00:05:59.440 1, 4, 5. 00:05:59.900 --> 00:06:00.820 1, 4, 5. 00:06:01.300 --> 00:06:02.050 Close. 00:06:04.140 --> 00:06:08.500 It's the set containing 1, 4, 5. 00:06:08.500 --> 00:06:11.720 These are sets of numbers. 00:06:14.460 --> 00:06:16.500 And the range? 00:06:20.500 --> 00:06:22.180 of three and six. 00:06:26.740 --> 00:06:30.220 We've got some linear function. 00:06:33.700 --> 00:06:36.080 What is the domain of this? 00:06:39.360 --> 00:06:40.480 And how can we figure that out? 00:06:45.080 --> 00:06:46.500 It is all real numbers. 00:06:47.180 --> 00:06:47.930 Why is that? 00:06:50.960 --> 00:06:53.180 This is going down and to the left. 00:06:53.580 --> 00:06:56.760 So anywhere on the x-coordinate or x-axis. 00:06:56.980 --> 00:06:59.120 If you go down you will eventually have a y-coordinate. 00:07:00.700 --> 00:07:06.300 And let's say this is y equals x plus 5. 00:07:06.400 --> 00:07:07.150 I'm just making up. 00:07:07.760 --> 00:07:08.660 It could be something like that. 00:07:09.760 --> 00:07:13.220 Okay, you can plug in any real number for x and always get a y coordinate. 00:07:15.080 --> 00:07:16.700 So the domain is all real numbers. 00:07:17.280 --> 00:07:18.920 could either write it like this. 00:07:21.960 --> 00:07:25.000 Or in interval notation it would be? 00:07:26.780 --> 00:07:27.900 Negative infinity to infinity. 00:07:33.720 --> 00:07:37.180 And what about the range? 00:07:45.000 --> 00:07:51.000 This goes forever to the right, but it's also going forever up and it's going forever to the left and down. 00:07:51.260 --> 00:07:56.280 So if it takes on all y values down and all y values up, so the range is also 00:08:01.580 --> 00:08:04.040 What's the domain of this? 00:08:06.480 --> 00:08:09.280 Look left and right, it goes forever to the right, forever to the left. 00:08:10.680 --> 00:08:12.820 Every x coordinate has a y coordinate. 00:08:14.480 --> 00:08:29.020 All real numbers. And what is the range? It only has one number in the range. The set containing four. 00:08:33.060 --> 00:08:36.000 We're going to learn something later called quadratic functions. 00:08:36.000 --> 00:08:38.560 This is the example of a quadratic function. 00:08:42.740 --> 00:08:54.100 What is the domain of this? All real numbers. It goes forever. Even it's going 00:08:54.100 --> 00:08:58.140 out very fast, it goes forever to the right. There's always a y coordinate. 00:08:58.600 --> 00:09:03.320 The equation for this is y equals x squared. You can plug in any x value, you always 00:09:03.320 --> 00:09:10.300 get a y value. However, what is the range? All real numbers greater than or equal to 00:09:10.300 --> 00:09:24.000 zero. All real numbers greater than or equal to zero. That's true. It never has any negative 00:09:24.000 --> 00:09:38.420 values but it's definitely the set of and how do we write that down zero to 00:09:38.420 --> 00:09:48.320 infinity do I include zero yeah it's on the origin and do we include infinity 00:09:48.320 --> 00:09:49.300 Never. 00:09:50.760 --> 00:09:51.510 Ok. 00:09:53.020 --> 00:09:55.180 This has a smallest y value. 00:09:55.420 --> 00:10:01.600 smallest y value is this coordinate right here. 00:10:02.340 --> 00:10:04.420 Ok and the range has a minimum. 00:10:05.260 --> 00:10:06.010 The minimum is zero. 00:10:07.600 --> 00:10:08.350 Ok. 00:10:08.240 --> 00:10:12.120 If the function was like this, there would be a maximum. 00:10:13.040 --> 00:10:19.220 and we take on all of these y values to wherever that point is. 00:10:19.960 --> 00:10:26.940 so relations, domain range, vertical line test, some examples. 00:10:28.860 --> 00:10:29.960 Alright, new topic. 00:10:32.220 --> 00:10:36.300 What is the absolute value of the number 5? 00:10:36.660 --> 00:10:37.410 5. 00:10:37.320 --> 00:10:38.070 5. 00:10:38.900 --> 00:10:42.500 What is the absolute value of the number negative 5? 00:10:42.860 --> 00:10:43.610 5. 00:10:43.800 --> 00:10:44.550 5. 00:10:46.040 --> 00:10:49.300 What is the absolute value of x? 00:10:50.780 --> 00:10:55.060 x is it x? 00:10:59.720 --> 00:11:02.040 What if x is negative 5? 00:11:05.060 --> 00:11:05.860 that true? 00:11:06.660 --> 00:11:07.410 No. 00:11:07.580 --> 00:11:08.460 So it's not x. 00:11:10.320 --> 00:11:11.070 Uh oh. 00:11:10.900 --> 00:11:12.820 What is the absolute value of x? 00:11:17.020 --> 00:11:17.770 What's that? 00:11:19.880 --> 00:11:21.160 So is it both? 00:11:27.620 --> 00:11:33.320 Let's try that with this number. Is the absolute value 5 plus or minus 5? 00:11:33.960 --> 00:11:38.400 No, it's just the plus. I like the idea though. 00:11:41.400 --> 00:11:42.150 Any other ideas? 00:11:54.740 --> 00:11:56.500 Does it depend? 00:11:57.740 --> 00:12:00.860 it depends on the value of x. 00:12:01.880 --> 00:12:03.900 If x is positive, it's x. 00:12:05.380 --> 00:12:06.680 What if x is negative? 00:12:08.260 --> 00:12:10.440 is negative x. 00:12:12.000 --> 00:12:16.160 The absolute value of x is what we call a piecewise-defined function. 00:12:17.020 --> 00:12:32.320 defined function. How did they define in elementary school? Oh, how many spaces is from the origin? 00:12:32.720 --> 00:12:39.680 Yes, it is the distance from the origin. 00:12:39.680 --> 00:12:46.180 To describe that distance with a formula, 00:12:51.160 --> 00:12:53.180 We need something better 00:12:54.480 --> 00:12:58.640 Okay, so because it depends we have to break it up into his two cases 00:12:59.540 --> 00:13:03.360 So the way we do that is with a it's called a piecewise defined function 00:13:05.100 --> 00:13:06.840 Did you guys learn this last year 00:13:12.720 --> 00:13:19.260 So the absolute value of x is x if what's true about x? 00:13:19.800 --> 00:13:20.820 If x is positive. 00:13:21.380 --> 00:13:24.400 If x is positive or 0. 00:13:26.620 --> 00:13:27.680 So you can write it like this. 00:13:27.680 --> 00:13:31.180 if x is greater than or equal to 0. 00:13:34.060 --> 00:13:41.940 And then if x is less than 0, what is the absolute value? 00:13:42.780 --> 00:13:50.140 It's negative x. It's the opposite of the value of x. 00:13:52.760 --> 00:14:10.560 This is the, called the piecewise definition of the absolute value of x. 00:14:15.960 --> 00:14:20.620 and sometimes we write it like this 00:14:22.220 --> 00:14:24.100 its x 00:14:24.100 --> 00:14:28.960 if x is contained in 00:14:28.960 --> 00:14:33.680 0 to infinity 00:14:33.680 --> 00:14:37.500 first negative x if x 00:14:37.500 --> 00:14:42.600 is in the set negative infinity to zero. 00:14:49.560 --> 00:14:53.740 This symbol, ∈, is read as ‘is an element of.’ 00:14:55.400 --> 00:14:59.080 It's the Greek letter, lowercase Greek letter epsilon. 00:15:02.180 --> 00:15:02.930 Okay. 00:15:05.180 --> 00:15:08.220 It simply means in math when we're writing down symbols, 00:15:08.780 --> 00:15:12.340 It means ‘is an element of,’ 00:15:16.180 --> 00:15:18.920 so if x is contained in the set. 00:15:20.380 --> 00:15:22.900 We use a lot in set notation. 00:15:27.360 --> 00:15:29.680 Okay, and again, when we see one girly bracket, 00:15:29.920 --> 00:15:31.000 this doesn't mean a set. 00:15:31.000 --> 00:15:34.980 This is how we write a piecewise defined function. 00:15:36.140 --> 00:15:40.180 You list all the different pieces, all the different intervals. 00:15:41.380 --> 00:15:47.720 You can either write interval notation or you can write inequalities like I did up here. 00:15:49.600 --> 00:15:50.350 And so on. 00:15:51.400 --> 00:15:53.880 Now, what's this thing I bring up here? 00:15:55.200 --> 00:15:56.060 I actually buy it. 00:15:56.180 --> 00:15:57.020 That's the neatest thing. 00:15:58.540 --> 00:16:00.480 Is it an expression or an equation? 00:16:01.980 --> 00:16:06.660 I call this an absolute value equation because it contains an absolute value. 00:16:08.040 --> 00:16:09.640 Is it true, false or open? 00:16:10.000 --> 00:16:10.750 Open. 00:16:10.720 --> 00:16:11.470 Open. 00:16:11.760 --> 00:16:14.280 Are the solutions numbers or are they ordered pairs? 00:16:14.660 --> 00:16:15.410 Numbers. 00:16:15.600 --> 00:16:16.580 Numbers because? 00:16:16.940 --> 00:16:17.690 One variable. 00:16:17.920 --> 00:16:18.670 There is only one variable. 00:16:21.620 --> 00:16:23.180 We're going to solve this. 00:16:23.300 --> 00:16:24.060 What's it mean to solve? 00:16:26.720 --> 00:16:27.470 How 00:16:27.280 --> 00:16:28.030 so 00:16:29.440 --> 00:16:30.840 if x 00:16:30.840 --> 00:16:33.740 is less than or equal to, sorry, less than 0, 00:16:42.900 --> 00:16:45.040 What does the absolute value of x equal? 00:16:49.140 --> 00:16:50.200 Hold on. 00:16:51.040 --> 00:16:57.520 Since x is less than 0, are we here or are we here? 00:16:59.920 --> 00:17:01.720 If x is less than 0, we're here. 00:17:01.720 --> 00:17:02.470 Yeah. 00:17:04.480 --> 00:17:05.230 OK. 00:17:07.580 --> 00:17:13.500 So the absolute value of x is equal to this guy. 00:17:15.100 --> 00:17:18.140 So I replace this with negative x. 00:17:19.940 --> 00:17:23.340 And now I no longer have an absolute value equation. 00:17:23.460 --> 00:17:25.380 It's got a simple linear equation. 00:17:26.740 --> 00:17:29.200 And we multiply it by negative 1. 00:17:29.200 --> 00:17:31.060 and we get x equals negative 7. 00:17:32.960 --> 00:17:36.360 One of our solutions is going to be negative 7. 00:17:37.540 --> 00:17:43.280 And then case two, if x is greater than or equal to 0, 00:17:46.780 --> 00:17:49.020 what does the absolute value of x equal to? 00:17:49.980 --> 00:17:50.730 7. 00:17:51.380 --> 00:17:52.130 No. 00:17:51.400 --> 00:18:02.780 The absolute value equals x in this case. 00:18:02.780 --> 00:18:12.240 this okay and then of course the solution here is seven so the solutions 00:18:12.240 --> 00:18:24.480 The solutions to the absolute-value equation are both values. 00:18:30.660 --> 00:18:41.760 sometimes we write this as plus or minus seven these mean the same thing it 00:18:41.760 --> 00:18:44.580 really means both positive 7 and negative 7. 00:18:47.580 --> 00:18:49.280 OK, we solved the first absolute 00:18:49.980 --> 00:18:57.160 Okay, before we get to the definition, what are we going to do first? 00:18:58.420 --> 00:18:59.600 Add 4. 00:19:01.920 --> 00:19:05.140 So, yeah, get the absolute value all by itself. 00:19:09.540 --> 00:19:11.500 Got to get the absolute value all by itself. 00:19:12.760 --> 00:19:15.400 Add 4, we got 10. 00:19:16.960 --> 00:19:18.140 Divide by 2. 00:19:19.740 --> 00:19:21.140 Okay. 00:19:21.140 --> 00:19:28.140 And then we split it up into two cases. 00:19:36.680 --> 00:19:37.620 Oops. 00:19:47.820 --> 00:19:55.400 So, brain-run equals zero, absolute value of x is x, solution's five. 00:20:08.400 --> 00:20:13.120 Once we get to here, is it always going to be plus or minus that number? 00:20:20.120 --> 00:20:21.000 It is. 00:20:21.940 --> 00:20:22.820 Okay. 00:20:28.660 --> 00:20:31.000 So if any time you have a 00:20:36.740 --> 00:20:41.700 If absolute value of x equals some number a 00:20:41.700 --> 00:20:46.440 and a is greater than or equal to zero 00:20:47.740 --> 00:20:51.760 Then x will equal plus or minus a. 00:20:56.080 --> 00:20:57.600 x will equal 00:20:59.740 --> 00:21:03.440 yeah let's just do that plus or minus a. 00:21:08.500 --> 00:21:09.680 Okay and then we get 00:21:09.680 --> 00:21:14.880 which is two equations x equals a and x equals negative a. 00:21:14.880 --> 00:21:17.940 we get plus or minus A. 00:21:18.880 --> 00:21:21.000 Now we have something inside the absolute value. 00:21:26.140 --> 00:21:27.820 Sorry—inside the absolute value. 00:21:29.440 --> 00:21:30.740 What are we going to do with this? 00:21:32.220 --> 00:21:33.880 We'll just make this a... 00:21:41.240 --> 00:21:43.800 I think of this as my x now. 00:21:47.400 --> 00:21:59.080 Once we get the absolute value all by itself, just write x minus 5 equals 7 or x minus 5 00:21:59.080 --> 00:22:00.500 equals negative 7. 00:22:02.180 --> 00:22:08.000 Same thing as x minus 5 equals plus or minus 7, which is two separate equations. 00:22:09.140 --> 00:22:14.580 Once we get the two separate equations we're going to solve them separately. 00:22:28.520 --> 00:22:29.940 We got two solutions. 00:22:30.860 --> 00:22:34.280 We typically write them in numerical order but it really doesn't matter. 00:22:34.280 --> 00:22:37.280 the order of a set does not matter 00:22:37.520 --> 00:22:41.180 Let me change this. 00:23:06.700 --> 00:23:08.860 First step. 00:23:25.880 --> 00:23:26.630 And 00:23:26.440 --> 00:23:32.980 And once we get the absolute value by itself, we have to write two separate equations. 00:23:33.760 --> 00:23:40.960 5x minus 10 equals 1, or 5x minus 10 is negative 1. 00:23:51.400 --> 00:23:53.760 Now we just got two simple linear equations. 00:23:55.080 --> 00:23:56.160 Add ten plus ten. 00:24:29.920 --> 00:24:35.740 I have two solutions. 00:24:37.820 --> 00:24:39.580 Okay, so for all these, 00:24:39.840 --> 00:24:42.340 once you get the absolute value by itself, 00:24:43.200 --> 00:24:44.800 just write two separate equations. 00:24:46.200 --> 00:24:48.040 It'll either equal positive that number 00:24:48.040 --> 00:24:48.980 or negative that number. 00:24:51.920 --> 00:24:52.740 Sound good? 00:24:53.340 --> 00:24:54.090 Yeah. 00:24:54.140 --> 00:24:54.890 Okay. 00:24:59.860 --> 00:25:01.840 I think you'll be able to do the homework. 00:25:01.900 --> 00:25:03.280 So we're gonna stop here.