1 00:00:04,600 --> 00:00:14,060 A relation is a set of ordered pairs. 2 00:00:21,140 --> 00:00:30,120 Ok, we got a new definition. A function is a relation. 3 00:00:45,860 --> 00:00:50,740 So it's a relation with a condition that must be fulfilled. 4 00:00:50,740 --> 00:00:58,600 The condition is each x coordinate. 5 00:01:02,500 --> 00:01:03,800 Sometimes we call that the input 6 00:01:05,640 --> 00:01:12,660 So, function, relation, set each x coordinate corresponds to only one y coordinate. 7 00:01:12,660 --> 00:01:13,410 Okay. 8 00:01:13,400 --> 00:01:16,040 Are A and B both relations? 9 00:01:18,960 --> 00:01:20,860 Oh, it's not supposed to be a hard question. 10 00:01:21,320 --> 00:01:22,070 Yeah. 11 00:01:21,740 --> 00:01:23,020 They are what? 12 00:01:23,740 --> 00:01:24,490 Because they are sets of ordered pairs. 13 00:01:24,820 --> 00:01:26,280 They're supposed to just set the work. 14 00:01:31,760 --> 00:01:33,260 Is A a function? 15 00:01:33,480 --> 00:01:34,300 Is B a function? 16 00:01:34,420 --> 00:01:37,160 Are they both functions or neither of them functions? 17 00:01:37,800 --> 00:01:38,550 don't you discuss 18 00:01:38,880 --> 00:01:40,440 Is C a function? 19 00:01:41,060 --> 00:01:41,810 No. 20 00:01:41,820 --> 00:01:42,660 No, why not? 21 00:01:43,220 --> 00:01:45,840 Because 2 has 2 y-coordinates. 22 00:01:46,400 --> 00:01:54,380 Because the x-coordinate of 2 corresponds to 5 and corresponds to 8. 23 00:01:58,280 --> 00:02:03,720 Here in A, 1 corresponds to 3, 5 corresponds to 3. 24 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. 25 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 26 00:02:15,980 --> 00:02:16,730 to do is check. 27 00:02:16,600 --> 00:02:19,520 As long as it doesn't repeat an x value, it's a function, guaranteed. 28 00:02:20,220 --> 00:02:24,280 A repeated x-value means the relation is not a function. 29 00:02:25,160 --> 00:02:31,220 This is the same as relation A, just the graph of the set. 30 00:02:38,180 --> 00:02:48,100 Relation C contains (2, 5), (2, 8), and (6, 3). 31 00:02:48,620 --> 00:02:52,240 This is relation C. It contains three ordered pairs. 32 00:02:52,900 --> 00:02:55,520 And I think you guys learned last year a test to see 33 00:02:55,520 --> 00:02:56,960 whether a relation is a function. 34 00:02:57,840 --> 00:02:58,590 What is it? 35 00:02:58,600 --> 00:02:59,350 line test. 36 00:02:59,540 --> 00:03:01,340 Called the vertical line test. 37 00:03:03,820 --> 00:03:05,480 How many remember that from last year? 38 00:03:07,700 --> 00:03:12,240 If every vertical line only touches it once, it is a function. 39 00:03:14,800 --> 00:03:19,340 Or if any vertical line touches the relation twice, it's not a function. 40 00:03:21,160 --> 00:03:23,200 Two ways to say the vertical line test. 41 00:03:25,180 --> 00:03:28,500 Let's write that down. 42 00:03:38,340 --> 00:03:44,280 If any vertical line touches the relation twice, it 43 00:03:50,200 --> 00:03:51,040 Is this a relation? 44 00:03:52,680 --> 00:03:53,430 Yes. 45 00:03:53,300 --> 00:03:59,280 Yeah, the set of ordered pairs is some linear relation. 46 00:04:00,040 --> 00:04:01,620 Is this also a function? 47 00:04:02,100 --> 00:04:02,850 Yes. 48 00:04:02,720 --> 00:04:03,560 Yes, and how do we know? 49 00:04:04,080 --> 00:04:07,200 Because every x-coordinate has only one y-coordinate. 50 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? 51 00:04:13,540 --> 00:04:14,540 Passes the vertical line. 52 00:04:14,680 --> 00:04:18,400 Passes the vertical line test. Every single vertical line touches it most once. 53 00:04:19,640 --> 00:04:20,390 Okay. 54 00:04:26,440 --> 00:04:30,600 We're going to learn in a couple months about circles. 55 00:04:32,740 --> 00:04:33,700 Is this a relation? 56 00:04:34,800 --> 00:04:35,550 Yes. 57 00:04:36,620 --> 00:04:37,900 Is this a set of ordered pairs? 58 00:04:38,640 --> 00:04:39,390 Yeah. 59 00:04:40,180 --> 00:04:46,380 Anything on a coordinate plane in black here is just a set of ordered pairs. 60 00:04:47,160 --> 00:04:48,120 It is a relation. 61 00:04:48,620 --> 00:04:49,400 Is it a function? 62 00:04:49,780 --> 00:04:50,530 No. 63 00:04:50,520 --> 00:04:51,270 No. 64 00:04:55,440 --> 00:04:58,380 At least one vertical line touches it twice, so it is not a function. 65 00:04:59,180 --> 00:05:01,680 We can still do lots of math with relations. 66 00:05:02,880 --> 00:05:03,740 We'll learn about that later. 67 00:05:04,440 --> 00:05:07,760 But just distinguishing what is a function, what is not a function. 68 00:05:08,400 --> 00:05:09,360 OK, another definition. 69 00:05:09,360 --> 00:05:27,280 And the domain of a relation or function is the set 70 00:05:27,280 --> 00:05:28,380 is the range 71 00:05:35,480 --> 00:05:45,720 The range of a relation is the set of all y-coordinates. 72 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. 73 00:05:53,900 --> 00:05:56,600 So what's the domain of A? 74 00:05:58,320 --> 00:05:59,440 1, 4, 5. 75 00:05:59,900 --> 00:06:00,820 1, 4, 5. 76 00:06:01,300 --> 00:06:02,050 Close. 77 00:06:04,140 --> 00:06:08,500 It's the set containing 1, 4, 5. 78 00:06:08,500 --> 00:06:11,720 These are sets of numbers. 79 00:06:14,460 --> 00:06:16,500 And the range? 80 00:06:20,500 --> 00:06:22,180 of three and six. 81 00:06:26,740 --> 00:06:30,220 We've got some linear function. 82 00:06:33,700 --> 00:06:36,080 What is the domain of this? 83 00:06:39,360 --> 00:06:40,480 And how can we figure that out? 84 00:06:45,080 --> 00:06:46,500 It is all real numbers. 85 00:06:47,180 --> 00:06:47,930 Why is that? 86 00:06:50,960 --> 00:06:53,180 This is going down and to the left. 87 00:06:53,580 --> 00:06:56,760 So anywhere on the x-coordinate or x-axis. 88 00:06:56,980 --> 00:06:59,120 If you go down you will eventually have a y-coordinate. 89 00:07:00,700 --> 00:07:06,300 And let's say this is y equals x plus 5. 90 00:07:06,400 --> 00:07:07,150 I'm just making up. 91 00:07:07,760 --> 00:07:08,660 It could be something like that. 92 00:07:09,760 --> 00:07:13,220 Okay, you can plug in any real number for x and always get a y coordinate. 93 00:07:15,080 --> 00:07:16,700 So the domain is all real numbers. 94 00:07:17,280 --> 00:07:18,920 could either write it like this. 95 00:07:21,960 --> 00:07:25,000 Or in interval notation it would be? 96 00:07:26,780 --> 00:07:27,900 Negative infinity to infinity. 97 00:07:33,720 --> 00:07:37,180 And what about the range? 98 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. 99 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 100 00:08:01,580 --> 00:08:04,040 What's the domain of this? 101 00:08:06,480 --> 00:08:09,280 Look left and right, it goes forever to the right, forever to the left. 102 00:08:10,680 --> 00:08:12,820 Every x coordinate has a y coordinate. 103 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. 104 00:08:33,060 --> 00:08:36,000 We're going to learn something later called quadratic functions. 105 00:08:36,000 --> 00:08:38,560 This is the example of a quadratic function. 106 00:08:42,740 --> 00:08:54,100 What is the domain of this? All real numbers. It goes forever. Even it's going 107 00:08:54,100 --> 00:08:58,140 out very fast, it goes forever to the right. There's always a y coordinate. 108 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 109 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 110 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 111 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 112 00:09:38,420 --> 00:09:48,320 infinity do I include zero yeah it's on the origin and do we include infinity 113 00:09:48,320 --> 00:09:49,300 Never. 114 00:09:50,760 --> 00:09:51,510 Ok. 115 00:09:53,020 --> 00:09:55,180 This has a smallest y value. 116 00:09:55,420 --> 00:10:01,600 smallest y value is this coordinate right here. 117 00:10:02,340 --> 00:10:04,420 Ok and the range has a minimum. 118 00:10:05,260 --> 00:10:06,010 The minimum is zero. 119 00:10:07,600 --> 00:10:08,350 Ok. 120 00:10:08,240 --> 00:10:12,120 If the function was like this, there would be a maximum. 121 00:10:13,040 --> 00:10:19,220 and we take on all of these y values to wherever that point is. 122 00:10:19,960 --> 00:10:26,940 so relations, domain range, vertical line test, some examples. 123 00:10:28,860 --> 00:10:29,960 Alright, new topic. 124 00:10:32,220 --> 00:10:36,300 What is the absolute value of the number 5? 125 00:10:36,660 --> 00:10:37,410 5. 126 00:10:37,320 --> 00:10:38,070 5. 127 00:10:38,900 --> 00:10:42,500 What is the absolute value of the number negative 5? 128 00:10:42,860 --> 00:10:43,610 5. 129 00:10:43,800 --> 00:10:44,550 5. 130 00:10:46,040 --> 00:10:49,300 What is the absolute value of x? 131 00:10:50,780 --> 00:10:55,060 x is it x? 132 00:10:59,720 --> 00:11:02,040 What if x is negative 5? 133 00:11:05,060 --> 00:11:05,860 that true? 134 00:11:06,660 --> 00:11:07,410 No. 135 00:11:07,580 --> 00:11:08,460 So it's not x. 136 00:11:10,320 --> 00:11:11,070 Uh oh. 137 00:11:10,900 --> 00:11:12,820 What is the absolute value of x? 138 00:11:17,020 --> 00:11:17,770 What's that? 139 00:11:19,880 --> 00:11:21,160 So is it both? 140 00:11:27,620 --> 00:11:33,320 Let's try that with this number. Is the absolute value 5 plus or minus 5? 141 00:11:33,960 --> 00:11:38,400 No, it's just the plus. I like the idea though. 142 00:11:41,400 --> 00:11:42,150 Any other ideas? 143 00:11:54,740 --> 00:11:56,500 Does it depend? 144 00:11:57,740 --> 00:12:00,860 it depends on the value of x. 145 00:12:01,880 --> 00:12:03,900 If x is positive, it's x. 146 00:12:05,380 --> 00:12:06,680 What if x is negative? 147 00:12:08,260 --> 00:12:10,440 is negative x. 148 00:12:12,000 --> 00:12:16,160 The absolute value of x is what we call a piecewise-defined function. 149 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? 150 00:12:32,720 --> 00:12:39,680 Yes, it is the distance from the origin. 151 00:12:39,680 --> 00:12:46,180 To describe that distance with a formula, 152 00:12:51,160 --> 00:12:53,180 We need something better 153 00:12:54,480 --> 00:12:58,640 Okay, so because it depends we have to break it up into his two cases 154 00:12:59,540 --> 00:13:03,360 So the way we do that is with a it's called a piecewise defined function 155 00:13:05,100 --> 00:13:06,840 Did you guys learn this last year 156 00:13:12,720 --> 00:13:19,260 So the absolute value of x is x if what's true about x? 157 00:13:19,800 --> 00:13:20,820 If x is positive. 158 00:13:21,380 --> 00:13:24,400 If x is positive or 0. 159 00:13:26,620 --> 00:13:27,680 So you can write it like this. 160 00:13:27,680 --> 00:13:31,180 if x is greater than or equal to 0. 161 00:13:34,060 --> 00:13:41,940 And then if x is less than 0, what is the absolute value? 162 00:13:42,780 --> 00:13:50,140 It's negative x. It's the opposite of the value of x. 163 00:13:52,760 --> 00:14:10,560 This is the, called the piecewise definition of the absolute value of x. 164 00:14:15,960 --> 00:14:20,620 and sometimes we write it like this 165 00:14:22,220 --> 00:14:24,100 its x 166 00:14:24,100 --> 00:14:28,960 if x is contained in 167 00:14:28,960 --> 00:14:33,680 0 to infinity 168 00:14:33,680 --> 00:14:37,500 first negative x if x 169 00:14:37,500 --> 00:14:42,600 is in the set negative infinity to zero. 170 00:14:49,560 --> 00:14:53,740 This symbol, ∈, is read as ‘is an element of.’ 171 00:14:55,400 --> 00:14:59,080 It's the Greek letter, lowercase Greek letter epsilon. 172 00:15:02,180 --> 00:15:02,930 Okay. 173 00:15:05,180 --> 00:15:08,220 It simply means in math when we're writing down symbols, 174 00:15:08,780 --> 00:15:12,340 It means ‘is an element of,’ 175 00:15:16,180 --> 00:15:18,920 so if x is contained in the set. 176 00:15:20,380 --> 00:15:22,900 We use a lot in set notation. 177 00:15:27,360 --> 00:15:29,680 Okay, and again, when we see one girly bracket, 178 00:15:29,920 --> 00:15:31,000 this doesn't mean a set. 179 00:15:31,000 --> 00:15:34,980 This is how we write a piecewise defined function. 180 00:15:36,140 --> 00:15:40,180 You list all the different pieces, all the different intervals. 181 00:15:41,380 --> 00:15:47,720 You can either write interval notation or you can write inequalities like I did up here. 182 00:15:49,600 --> 00:15:50,350 And so on. 183 00:15:51,400 --> 00:15:53,880 Now, what's this thing I bring up here? 184 00:15:55,200 --> 00:15:56,060 I actually buy it. 185 00:15:56,180 --> 00:15:57,020 That's the neatest thing. 186 00:15:58,540 --> 00:16:00,480 Is it an expression or an equation? 187 00:16:01,980 --> 00:16:06,660 I call this an absolute value equation because it contains an absolute value. 188 00:16:08,040 --> 00:16:09,640 Is it true, false or open? 189 00:16:10,000 --> 00:16:10,750 Open. 190 00:16:10,720 --> 00:16:11,470 Open. 191 00:16:11,760 --> 00:16:14,280 Are the solutions numbers or are they ordered pairs? 192 00:16:14,660 --> 00:16:15,410 Numbers. 193 00:16:15,600 --> 00:16:16,580 Numbers because? 194 00:16:16,940 --> 00:16:17,690 One variable. 195 00:16:17,920 --> 00:16:18,670 There is only one variable. 196 00:16:21,620 --> 00:16:23,180 We're going to solve this. 197 00:16:23,300 --> 00:16:24,060 What's it mean to solve? 198 00:16:26,720 --> 00:16:27,470 How 199 00:16:27,280 --> 00:16:28,030 so 200 00:16:29,440 --> 00:16:30,840 if x 201 00:16:30,840 --> 00:16:33,740 is less than or equal to, sorry, less than 0, 202 00:16:42,900 --> 00:16:45,040 What does the absolute value of x equal? 203 00:16:49,140 --> 00:16:50,200 Hold on. 204 00:16:51,040 --> 00:16:57,520 Since x is less than 0, are we here or are we here? 205 00:16:59,920 --> 00:17:01,720 If x is less than 0, we're here. 206 00:17:01,720 --> 00:17:02,470 Yeah. 207 00:17:04,480 --> 00:17:05,230 OK. 208 00:17:07,580 --> 00:17:13,500 So the absolute value of x is equal to this guy. 209 00:17:15,100 --> 00:17:18,140 So I replace this with negative x. 210 00:17:19,940 --> 00:17:23,340 And now I no longer have an absolute value equation. 211 00:17:23,460 --> 00:17:25,380 It's got a simple linear equation. 212 00:17:26,740 --> 00:17:29,200 And we multiply it by negative 1. 213 00:17:29,200 --> 00:17:31,060 and we get x equals negative 7. 214 00:17:32,960 --> 00:17:36,360 One of our solutions is going to be negative 7. 215 00:17:37,540 --> 00:17:43,280 And then case two, if x is greater than or equal to 0, 216 00:17:46,780 --> 00:17:49,020 what does the absolute value of x equal to? 217 00:17:49,980 --> 00:17:50,730 7. 218 00:17:51,380 --> 00:17:52,130 No. 219 00:17:51,400 --> 00:18:02,780 The absolute value equals x in this case. 220 00:18:02,780 --> 00:18:12,240 this okay and then of course the solution here is seven so the solutions 221 00:18:12,240 --> 00:18:24,480 The solutions to the absolute-value equation are both values. 222 00:18:30,660 --> 00:18:41,760 sometimes we write this as plus or minus seven these mean the same thing it 223 00:18:41,760 --> 00:18:44,580 really means both positive 7 and negative 7. 224 00:18:47,580 --> 00:18:49,280 OK, we solved the first absolute 225 00:18:49,980 --> 00:18:57,160 Okay, before we get to the definition, what are we going to do first? 226 00:18:58,420 --> 00:18:59,600 Add 4. 227 00:19:01,920 --> 00:19:05,140 So, yeah, get the absolute value all by itself. 228 00:19:09,540 --> 00:19:11,500 Got to get the absolute value all by itself. 229 00:19:12,760 --> 00:19:15,400 Add 4, we got 10. 230 00:19:16,960 --> 00:19:18,140 Divide by 2. 231 00:19:19,740 --> 00:19:21,140 Okay. 232 00:19:21,140 --> 00:19:28,140 And then we split it up into two cases. 233 00:19:36,680 --> 00:19:37,620 Oops. 234 00:19:47,820 --> 00:19:55,400 So, brain-run equals zero, absolute value of x is x, solution's five. 235 00:20:08,400 --> 00:20:13,120 Once we get to here, is it always going to be plus or minus that number? 236 00:20:20,120 --> 00:20:21,000 It is. 237 00:20:21,940 --> 00:20:22,820 Okay. 238 00:20:28,660 --> 00:20:31,000 So if any time you have a 239 00:20:36,740 --> 00:20:41,700 If absolute value of x equals some number a 240 00:20:41,700 --> 00:20:46,440 and a is greater than or equal to zero 241 00:20:47,740 --> 00:20:51,760 Then x will equal plus or minus a. 242 00:20:56,080 --> 00:20:57,600 x will equal 243 00:20:59,740 --> 00:21:03,440 yeah let's just do that plus or minus a. 244 00:21:08,500 --> 00:21:09,680 Okay and then we get 245 00:21:09,680 --> 00:21:14,880 which is two equations x equals a and x equals negative a. 246 00:21:14,880 --> 00:21:17,940 we get plus or minus A. 247 00:21:18,880 --> 00:21:21,000 Now we have something inside the absolute value. 248 00:21:26,140 --> 00:21:27,820 Sorry—inside the absolute value. 249 00:21:29,440 --> 00:21:30,740 What are we going to do with this? 250 00:21:32,220 --> 00:21:33,880 We'll just make this a... 251 00:21:41,240 --> 00:21:43,800 I think of this as my x now. 252 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 5 253 00:21:59,080 --> 00:22:00,500 equals negative 7. 254 00:22:02,180 --> 00:22:08,000 Same thing as x minus 5 equals plus or minus 7, which is two separate equations. 255 00:22:09,140 --> 00:22:14,580 Once we get the two separate equations we're going to solve them separately. 256 00:22:28,520 --> 00:22:29,940 We got two solutions. 257 00:22:30,860 --> 00:22:34,280 We typically write them in numerical order but it really doesn't matter. 258 00:22:34,280 --> 00:22:37,280 the order of a set does not matter 259 00:22:37,520 --> 00:22:41,180 Let me change this. 260 00:23:06,700 --> 00:23:08,860 First step. 261 00:23:25,880 --> 00:23:26,630 And 262 00:23:26,440 --> 00:23:32,980 And once we get the absolute value by itself, we have to write two separate equations. 263 00:23:33,760 --> 00:23:40,960 5x minus 10 equals 1, or 5x minus 10 is negative 1. 264 00:23:51,400 --> 00:23:53,760 Now we just got two simple linear equations. 265 00:23:55,080 --> 00:23:56,160 Add ten plus ten. 266 00:24:29,920 --> 00:24:35,740 I have two solutions. 267 00:24:37,820 --> 00:24:39,580 Okay, so for all these, 268 00:24:39,840 --> 00:24:42,340 once you get the absolute value by itself, 269 00:24:43,200 --> 00:24:44,800 just write two separate equations. 270 00:24:46,200 --> 00:24:48,040 It'll either equal positive that number 271 00:24:48,040 --> 00:24:48,980 or negative that number. 272 00:24:51,920 --> 00:24:52,740 Sound good? 273 00:24:53,340 --> 00:24:54,090 Yeah. 274 00:24:54,140 --> 00:24:54,890 Okay. 275 00:24:59,860 --> 00:25:01,840 I think you'll be able to do the homework. 276 00:25:01,900 --> 00:25:03,280 So we're gonna stop here.