1 00:00:00,040 --> 00:00:01,940 ANNOUNCER: The following content is provided under a 2 00:00:01,940 --> 00:00:03,690 Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,980 offer high quality educational resources for free. 5 00:00:09,980 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:42,450 --> 00:00:46,150 HERBERT GROSS: Hi, our lesson today concerns functions. 9 00:00:46,150 --> 00:00:48,870 And in the certain manner of speaking, this is perhaps 10 00:00:48,870 --> 00:00:52,070 where a course in calculus should usually begin. 11 00:00:52,070 --> 00:00:55,370 After all, what will be studying is a relationship 12 00:00:55,370 --> 00:00:57,090 between certain variables. 13 00:00:57,090 --> 00:01:00,180 This is what we mean by a function in general. 14 00:01:00,180 --> 00:01:03,220 Or at least, in particular, with our emphasis on real 15 00:01:03,220 --> 00:01:05,580 variables and graphs that we were talking about. 16 00:01:05,580 --> 00:01:09,180 And what we would like to do now is to motivate the concept 17 00:01:09,180 --> 00:01:13,490 of function from a more general point of view in terms 18 00:01:13,490 --> 00:01:16,690 of the language of sets that we have talked about and read 19 00:01:16,690 --> 00:01:18,890 about in our supplementary notes. 20 00:01:18,890 --> 00:01:23,230 Let's then begin with a basic general definition of what a 21 00:01:23,230 --> 00:01:27,280 function is, after which we will specialize and talk about 22 00:01:27,280 --> 00:01:29,130 functions of a real variable for the 23 00:01:29,130 --> 00:01:32,150 remainder of the lecture. 24 00:01:32,150 --> 00:01:36,720 We begin with a definition written this way. 25 00:01:36,720 --> 00:01:39,360 A function f from A to B-- 26 00:01:39,360 --> 00:01:40,650 see, it's written this way. 27 00:01:40,650 --> 00:01:45,230 It means f is a function from A to B means that f is a rule, 28 00:01:45,230 --> 00:01:52,990 which assigns to each element in set A, an element in set B. 29 00:01:52,990 --> 00:01:54,880 Now of course, this may sound like just an 30 00:01:54,880 --> 00:01:56,570 empty bunch of words. 31 00:01:56,570 --> 00:01:59,180 But if we come here and look a little bit at a picture, we 32 00:01:59,180 --> 00:02:01,560 get the idea as to what's going on. 33 00:02:01,560 --> 00:02:04,260 Namely, we may visualize our sets as circles. 34 00:02:04,260 --> 00:02:06,050 And what does the function do? 35 00:02:06,050 --> 00:02:13,740 The function assigns to each element of A, an element of B. 36 00:02:13,740 --> 00:02:16,280 In fact, notice the geometric wording again. 37 00:02:16,280 --> 00:02:20,300 This element is mapped into this element by f. 38 00:02:20,300 --> 00:02:23,790 This element is mapped into this element by f. 39 00:02:23,790 --> 00:02:26,010 And this element is mapped into this one. 40 00:02:26,010 --> 00:02:28,710 And while we're looking at this, perhaps it would be a 41 00:02:28,710 --> 00:02:33,740 good chance to emphasize not to read more into a definition 42 00:02:33,740 --> 00:02:35,220 than what's already there. 43 00:02:35,220 --> 00:02:39,900 You see when we say that f assigns to each element of A, 44 00:02:39,900 --> 00:02:42,980 what we mean is we will not allow something 45 00:02:42,980 --> 00:02:45,600 like this to happen. 46 00:02:45,600 --> 00:02:49,770 We will not say, for example, let f send this element into 47 00:02:49,770 --> 00:02:51,890 this one and this one. 48 00:02:51,890 --> 00:02:54,220 This becomes non well-defined. 49 00:02:54,220 --> 00:02:55,740 It becomes ambiguous. 50 00:02:55,740 --> 00:02:58,750 In other words, we do not want to have to make the value 51 00:02:58,750 --> 00:03:02,410 judgment as to which of two things we are going to look 52 00:03:02,410 --> 00:03:05,440 back at as being what A maps into. 53 00:03:05,440 --> 00:03:07,390 And why that's the case we'll mention 54 00:03:07,390 --> 00:03:10,260 as our course proceeds. 55 00:03:10,260 --> 00:03:15,200 Also, notice that when we say that each element of A is 56 00:03:15,200 --> 00:03:19,470 assigned to an element of B, two things are implied here. 57 00:03:19,470 --> 00:03:22,890 First of all, notice that we do not insist that all 58 00:03:22,890 --> 00:03:25,460 of B be used up. 59 00:03:25,460 --> 00:03:28,400 You see, in other words, here's some surplus B's over 60 00:03:28,400 --> 00:03:32,760 here, which are not used up by A's with respect to f. 61 00:03:32,760 --> 00:03:37,280 And secondly, whereas we prohibit the same A from 62 00:03:37,280 --> 00:03:42,920 having two different images in B, we do not prohibit one B 63 00:03:42,920 --> 00:03:48,160 from being the image of two different elements in A. In 64 00:03:48,160 --> 00:03:50,850 other words, notice that even though both of these elements 65 00:03:50,850 --> 00:03:55,280 here are assigned to the same element in B with respect to 66 00:03:55,280 --> 00:03:59,070 f, there certainly is no violation of our definition 67 00:03:59,070 --> 00:04:04,970 that each element here was assigned to an element here. 68 00:04:04,970 --> 00:04:08,090 By the way, this leads in the literature to three different 69 00:04:08,090 --> 00:04:11,310 terms that we should define right now. 70 00:04:11,310 --> 00:04:16,560 One of these is the set A itself and that's called the 71 00:04:16,560 --> 00:04:21,100 domain of f. 72 00:04:21,100 --> 00:04:23,470 You see this is what f is defined on. 73 00:04:23,470 --> 00:04:26,250 The domain is the set A over here. 74 00:04:26,250 --> 00:04:27,780 Often abbreviated-- 75 00:04:27,780 --> 00:04:29,570 well, there are many different abbreviations. 76 00:04:29,570 --> 00:04:33,690 Sometimes one just writes D-O-M with an f or D-O-M with 77 00:04:33,690 --> 00:04:34,210 a subscript. 78 00:04:34,210 --> 00:04:36,560 Or a capital D with a subscript f. 79 00:04:36,560 --> 00:04:40,520 But we will adapt to these notations as we need them. 80 00:04:40,520 --> 00:04:44,980 But for our purposes all the domain of f is, it's the set 81 00:04:44,980 --> 00:04:50,350 A. It's the set on which f operates to assign values, ok. 82 00:04:50,350 --> 00:04:55,520 And then the companion to that is the set B. And B, almost 83 00:04:55,520 --> 00:04:59,170 again in graphic terms, is called the range of f. 84 00:05:05,110 --> 00:05:08,460 And by the way, as you look at the range of f, you may get 85 00:05:08,460 --> 00:05:12,020 the feeling that somehow or other, this little element 86 00:05:12,020 --> 00:05:14,380 over here is kind of out of place. 87 00:05:14,380 --> 00:05:18,710 That maybe somehow or other let me just circle this. 88 00:05:18,710 --> 00:05:21,670 Let me call this the set C. Somehow or other, you get the 89 00:05:21,670 --> 00:05:28,000 feeling that C describes much better what f does to A than 90 00:05:28,000 --> 00:05:31,930 B. Because you see, each of the elements in C is used up 91 00:05:31,930 --> 00:05:35,232 as being the image of at least one element in A. Somehow or 92 00:05:35,232 --> 00:05:38,240 other you see, we could have deleted this element B without 93 00:05:38,240 --> 00:05:40,130 too much loss of continuity. 94 00:05:40,130 --> 00:05:44,790 And to get around this we interject still a third 95 00:05:44,790 --> 00:05:53,790 definition, C. It's called the image of f. 96 00:05:53,790 --> 00:05:55,620 You see, image verses range. 97 00:05:55,620 --> 00:05:58,120 Image is the part that's actually used up 98 00:05:58,120 --> 00:05:59,870 by f with the mapping. 99 00:05:59,870 --> 00:06:02,690 And sometimes this is even abbreviated as follows. 100 00:06:02,690 --> 00:06:06,070 You use the same f as one uses for the function and then a 101 00:06:06,070 --> 00:06:08,910 capital A in parentheses to indicate what? 102 00:06:08,910 --> 00:06:14,410 This is a set of all things of the form f of x where x is in 103 00:06:14,410 --> 00:06:17,420 A. In other words, again, the standard notation if we want 104 00:06:17,420 --> 00:06:18,470 to look at it this way. 105 00:06:18,470 --> 00:06:24,340 If we call this element here a and this element here b, to 106 00:06:24,340 --> 00:06:31,510 indicate that a was mapped into b by the function f or 107 00:06:31,510 --> 00:06:38,640 the rule f, we often write that as f of a equals b. 108 00:06:38,640 --> 00:06:42,730 Now in many places, including our text, there is no 109 00:06:42,730 --> 00:06:46,600 distinction made between the image and the range. 110 00:06:46,600 --> 00:06:50,240 And the reason for this is that in many situations, the 111 00:06:50,240 --> 00:06:54,000 image and the range turn out to be quite the same thing. 112 00:06:54,000 --> 00:06:56,830 In fact, if they do turn out to be the same thing, we call 113 00:06:56,830 --> 00:06:59,220 that a rather special type of function. 114 00:06:59,220 --> 00:07:04,030 Namely, the function from A to B is said to be onto if its 115 00:07:04,030 --> 00:07:06,120 image equals its range. 116 00:07:06,120 --> 00:07:10,125 Now again, in terms of words, that doesn't say very much. 117 00:07:10,125 --> 00:07:12,030 The image equals the range. 118 00:07:12,030 --> 00:07:14,430 What does that mean, image equals the range? 119 00:07:14,430 --> 00:07:16,650 Perhaps the best way to see this is 120 00:07:16,650 --> 00:07:18,440 by means of an example. 121 00:07:18,440 --> 00:07:20,420 The example I have in mind is this. 122 00:07:20,420 --> 00:07:23,180 And this happens many, many times in mathematics. 123 00:07:23,180 --> 00:07:25,090 One begins with a set. 124 00:07:25,090 --> 00:07:27,550 In this case, I picked A to be the set consisting of the 125 00:07:27,550 --> 00:07:29,900 numbers 1, 2, and 3. 126 00:07:29,900 --> 00:07:34,610 Frequently, one defines a function explicitly on A 127 00:07:34,610 --> 00:07:37,350 without any regard to a second set B. 128 00:07:37,350 --> 00:07:40,010 For example, in this illustration I've said, let f 129 00:07:40,010 --> 00:07:45,000 of a be 4a for each a in set capital A. Well, what are the 130 00:07:45,000 --> 00:07:46,700 elements of capital A? 131 00:07:46,700 --> 00:07:48,170 They are 1, 2, and 3. 132 00:07:48,170 --> 00:07:50,730 So according to this recipe, what do we have? 133 00:07:50,730 --> 00:07:54,250 Well, f of 1 is 4. 134 00:07:54,250 --> 00:07:56,840 f of 2, see, it's what? 135 00:07:56,840 --> 00:07:59,500 4 times 2 is 8. 136 00:07:59,500 --> 00:08:02,030 And f of 3. 137 00:08:02,030 --> 00:08:02,660 That's what? 138 00:08:02,660 --> 00:08:04,740 4 times 3 is 12. 139 00:08:04,740 --> 00:08:10,560 Let me now invent a new set B. And the elements of B, as you 140 00:08:10,560 --> 00:08:16,210 could probably guess, are going to be 4, 8, and 12. 141 00:08:16,210 --> 00:08:21,490 And now I look at my function f from A to B. 142 00:08:21,490 --> 00:08:26,750 Notice that in this case the function from A to B uses up 143 00:08:26,750 --> 00:08:29,290 all of B. In fact, in terms of a diagram, you see 144 00:08:29,290 --> 00:08:30,570 here's A. Has what? 145 00:08:30,570 --> 00:08:32,539 1, 2, and 3. 146 00:08:32,539 --> 00:08:37,159 And here's B, which is made up of 4, 8, and 12. 147 00:08:37,159 --> 00:08:38,909 Now, what does have f do? 148 00:08:38,909 --> 00:08:41,490 It maps 1 into 4. 149 00:08:41,490 --> 00:08:43,460 It maps 2 into 8. 150 00:08:43,460 --> 00:08:46,780 And it maps 3 into 12. 151 00:08:46,780 --> 00:08:48,080 What's happened here? 152 00:08:48,080 --> 00:08:51,250 A has not only mapped into B, but all of B 153 00:08:51,250 --> 00:08:52,730 is used up in this. 154 00:08:52,730 --> 00:08:56,100 In other words, notice then that the range of B and the 155 00:08:56,100 --> 00:08:59,970 image of B in this particular case happen to be the same. 156 00:08:59,970 --> 00:09:03,290 This will happen in every single case where we start 157 00:09:03,290 --> 00:09:07,620 with a set A and define a function on A. Namely, we see 158 00:09:07,620 --> 00:09:12,000 what f of x is for each x in A, take the collection of all 159 00:09:12,000 --> 00:09:16,810 those images, call that set B, and then you see by default so 160 00:09:16,810 --> 00:09:22,010 to speak, B will be both the range and the image of f. 161 00:09:22,010 --> 00:09:25,170 And it's in this sense that in most textbook examples that we 162 00:09:25,170 --> 00:09:28,480 deal with, we need not make any distinction between the 163 00:09:28,480 --> 00:09:31,700 range of the function and the image of the function. 164 00:09:31,700 --> 00:09:35,030 But roughly speaking then, just to keep things straight, 165 00:09:35,030 --> 00:09:39,030 a function is called onto if the entire range is used up in 166 00:09:39,030 --> 00:09:40,110 the mapping. 167 00:09:40,110 --> 00:09:42,710 But if there are elements of the range which are not used 168 00:09:42,710 --> 00:09:45,510 up as images, then the function is simply called 169 00:09:45,510 --> 00:09:48,110 into, or not onto. 170 00:09:48,110 --> 00:09:50,370 But at any rate, this is the concept of 171 00:09:50,370 --> 00:09:52,490 what we mean by onto. 172 00:09:52,490 --> 00:09:56,190 Now, a second feature that one talks about with functions 173 00:09:56,190 --> 00:09:59,850 which in no way is connected with onto, this, but which is 174 00:09:59,850 --> 00:10:03,540 a very important independent feature is something which is 175 00:10:03,540 --> 00:10:06,170 called a 1:1 function. 176 00:10:06,170 --> 00:10:08,032 Let's look at that for a moment too. 177 00:10:11,970 --> 00:10:15,910 For example, let's suppose I have a function f defined on 178 00:10:15,910 --> 00:10:23,930 my set A. The question is, f maps a1 into a particular 179 00:10:23,930 --> 00:10:27,850 element of B and it maps a2 into a particular element of 180 00:10:27,850 --> 00:10:28,630 B. 181 00:10:28,630 --> 00:10:31,840 Now, there are two possibilities that can happen. 182 00:10:31,840 --> 00:10:42,030 One is that f of a1 and f of a2 will be 183 00:10:42,030 --> 00:10:44,660 different elements of b. 184 00:10:44,660 --> 00:10:47,000 In other words, what will happen is, is that two 185 00:10:47,000 --> 00:10:51,120 distinct elements of A will have distinct images in B. 186 00:10:51,120 --> 00:10:56,300 On the other hand, it's possible that the two 187 00:10:56,300 --> 00:11:00,910 different elements of A have the same element of the same 188 00:11:00,910 --> 00:11:02,410 image in B. 189 00:11:02,410 --> 00:11:06,030 Now, by and large, whereas nothing is wrong when this 190 00:11:06,030 --> 00:11:10,440 happens, it does cut down our operating speed to some extent 191 00:11:10,440 --> 00:11:11,450 when it does. 192 00:11:11,450 --> 00:11:15,530 Because you see, frequently to study a particular function, 193 00:11:15,530 --> 00:11:20,580 we may want to look at the image rather than the domain. 194 00:11:20,580 --> 00:11:24,790 And somehow or other, you see if two different elements can 195 00:11:24,790 --> 00:11:29,260 map into the same element in the image, then you see when 196 00:11:29,260 --> 00:11:33,760 we look at the image we have no way of knowing which of 197 00:11:33,760 --> 00:11:35,880 these two elements we're talking about. 198 00:11:35,880 --> 00:11:39,700 So in other words then, if it should turn out that no two 199 00:11:39,700 --> 00:11:45,070 different elements in A can have the same image in B, in 200 00:11:45,070 --> 00:11:48,220 other words, notice what this thing here says. 201 00:11:48,220 --> 00:11:49,590 You see, if what? 202 00:11:49,590 --> 00:11:51,220 This is the image of a1. 203 00:11:51,220 --> 00:11:52,700 This is the image of a2. 204 00:11:52,700 --> 00:11:53,390 It says what? 205 00:11:53,390 --> 00:11:57,530 If a1 and a2 have the same image, then they must be the 206 00:11:57,530 --> 00:11:58,840 same element. 207 00:11:58,840 --> 00:12:01,820 Now if that happens, and again as I show you over here, it 208 00:12:01,820 --> 00:12:02,950 doesn't have to happen. 209 00:12:02,950 --> 00:12:06,570 If that happens, the function is called 1:1. 210 00:12:06,570 --> 00:12:10,060 For example, here is a picture I've drawn in which 211 00:12:10,060 --> 00:12:11,310 a function is 1:1. 212 00:12:14,540 --> 00:12:18,190 By the way, I've drawn this picture so that my function f 213 00:12:18,190 --> 00:12:19,330 is both 1:1. 214 00:12:19,330 --> 00:12:20,040 Meaning what? 215 00:12:20,040 --> 00:12:23,120 That no two elements in A have the same image in B. And 216 00:12:23,120 --> 00:12:28,210 secondly, it also happened to be onto here. 217 00:12:28,210 --> 00:12:32,090 Namely, no element of B was left out by f. 218 00:12:32,090 --> 00:12:33,550 Now that wasn't crucial. 219 00:12:33,550 --> 00:12:38,040 For example, if I do this notice now that the function 220 00:12:38,040 --> 00:12:40,410 from A to B is still 1:1. 221 00:12:40,410 --> 00:12:44,050 No two different elements in A have the same image in B. But 222 00:12:44,050 --> 00:12:47,240 now you see the function is no longer onto because there 223 00:12:47,240 --> 00:12:52,250 happened to be elements in B which are not mapped into 224 00:12:52,250 --> 00:12:55,820 under F by elements of A. 225 00:12:55,820 --> 00:12:58,900 Now, what is nice of course is that if a function happens to 226 00:12:58,900 --> 00:13:03,440 be both 1:1 and onto, notice that we can induce a new 227 00:13:03,440 --> 00:13:08,770 function, which I'll call g from B to A by essentially 228 00:13:08,770 --> 00:13:10,020 reversing the arrowheads here. 229 00:13:12,980 --> 00:13:15,990 You see, if the function is both 1:1 and onto, by 230 00:13:15,990 --> 00:13:19,020 reversing the arrowheads, instead of getting a function 231 00:13:19,020 --> 00:13:22,900 from A to B, I do get a function from B to A. This 232 00:13:22,900 --> 00:13:26,540 function is called the inverse function and will play a very 233 00:13:26,540 --> 00:13:30,150 important role in much of our course which follows. 234 00:13:30,150 --> 00:13:33,350 The important point to notice, however, is that if the 235 00:13:33,350 --> 00:13:38,510 function is not both 1:1 and onto, you cannot reverse the 236 00:13:38,510 --> 00:13:39,880 arrowheads, believe it or not. 237 00:13:39,880 --> 00:13:41,480 Well, you say, I can reverse them, can't I? 238 00:13:41,480 --> 00:13:43,630 Why can't I reverse them over here? 239 00:13:43,630 --> 00:13:44,670 And the answer is well, look it. 240 00:13:44,670 --> 00:13:48,190 If we include these being in here, suppose we reverse the 241 00:13:48,190 --> 00:13:49,690 arrowheads now. 242 00:13:49,690 --> 00:13:51,240 Look at B. 243 00:13:51,240 --> 00:13:52,930 What is the domain of g? 244 00:13:52,930 --> 00:13:57,720 Well, for B to be the domain, every element of B has to be 245 00:13:57,720 --> 00:14:00,190 assigned to something in A by g. 246 00:14:00,190 --> 00:14:03,520 But look at these two elements over here, g doesn't act on 247 00:14:03,520 --> 00:14:04,550 those at all. 248 00:14:04,550 --> 00:14:08,270 In other words, if the original function is not onto, 249 00:14:08,270 --> 00:14:10,800 then when you reverse the arrowheads you haven't defined 250 00:14:10,800 --> 00:14:14,230 the new function on your whole domain here. 251 00:14:14,230 --> 00:14:16,840 In another sense, if the function was not 252 00:14:16,840 --> 00:14:18,430 1:1 when you started. 253 00:14:18,430 --> 00:14:20,940 In other words, suppose this happened. 254 00:14:20,940 --> 00:14:22,800 So f was not 1:1. 255 00:14:22,800 --> 00:14:25,810 Now you see when you try to reverse your arrowheads, 256 00:14:25,810 --> 00:14:30,370 notice that the element here in B is assigned to two 257 00:14:30,370 --> 00:14:33,410 different elements in A. And we agreed that we wouldn't 258 00:14:33,410 --> 00:14:35,730 allow that to happen. 259 00:14:35,730 --> 00:14:37,480 OK, so far so good. 260 00:14:37,480 --> 00:14:40,620 Notice that that particular part of our course has nothing 261 00:14:40,620 --> 00:14:43,060 to do with real variables and the like. 262 00:14:43,060 --> 00:14:46,110 Meaning when we're talking about sets they can be sets of 263 00:14:46,110 --> 00:14:47,500 arbitrary numbers. 264 00:14:47,500 --> 00:14:51,450 Now what I'd like to do is zero in, on our specific 265 00:14:51,450 --> 00:14:54,130 calculus of a single variable course. 266 00:14:54,130 --> 00:14:56,840 And let's go back to our old friend who somehow or other 267 00:14:56,840 --> 00:14:58,960 has made an appearance in every lecture that 268 00:14:58,960 --> 00:15:01,070 we've had so far. 269 00:15:01,070 --> 00:15:04,770 Let's go back to s equals 16t squared. 270 00:15:04,770 --> 00:15:08,140 Only now, we're not going to repeat the same old stuff that 271 00:15:08,140 --> 00:15:09,250 we did before with it. 272 00:15:09,250 --> 00:15:12,580 We're now going to get slightly more sophisticated. 273 00:15:12,580 --> 00:15:16,020 Namely, when we talk about s equals 16t squared, what 274 00:15:16,020 --> 00:15:17,700 problem was being done here? 275 00:15:17,700 --> 00:15:20,470 You are assuming that there is no air resistance. 276 00:15:20,470 --> 00:15:22,940 An object is being held above the ground. 277 00:15:22,940 --> 00:15:26,750 You release the object and the distance s that the object 278 00:15:26,750 --> 00:15:30,080 falls in feet after t seconds is given 279 00:15:30,080 --> 00:15:32,780 by s equal 16t squared. 280 00:15:32,780 --> 00:15:36,570 Now if we think about that for a while, we realize that that 281 00:15:36,570 --> 00:15:38,570 does not tell the whole picture. 282 00:15:38,570 --> 00:15:43,240 Obviously, the s equals 16t squared applies only to the 283 00:15:43,240 --> 00:15:46,380 time in which that object is falling. 284 00:15:46,380 --> 00:15:49,800 Perhaps what we should have said was this, that until you 285 00:15:49,800 --> 00:15:53,960 release the object it doesn't fall any distance at all. 286 00:15:53,960 --> 00:15:58,100 Then from the instance you release it, it starts to fall 287 00:15:58,100 --> 00:16:01,450 a distance s given by 16t squared. 288 00:16:01,450 --> 00:16:04,310 Not forever, but until it hits the ground. 289 00:16:04,310 --> 00:16:07,560 Let's call t sub g the time at which this 290 00:16:07,560 --> 00:16:09,040 thing hits the ground. 291 00:16:09,040 --> 00:16:13,230 You see this recipe that we called s equals 16t squared is 292 00:16:13,230 --> 00:16:14,930 not in effect forever. 293 00:16:14,930 --> 00:16:19,510 It's in effect only when t is between 0 and t sub g. 294 00:16:19,510 --> 00:16:23,300 And by the way, hopefully once the object hits the ground it 295 00:16:23,300 --> 00:16:24,840 won't fall any further. 296 00:16:24,840 --> 00:16:30,340 In other words, for any time after t sub g, the distance 297 00:16:30,340 --> 00:16:33,830 that it's fallen is 16t sub g squared. 298 00:16:33,830 --> 00:16:36,570 Meaning this is the distance that its fallen when it hits 299 00:16:36,570 --> 00:16:39,380 the ground and it stays there. 300 00:16:39,380 --> 00:16:43,390 If we wanted to graph this, you see, and notice how we are 301 00:16:43,390 --> 00:16:45,350 refining our previous result. 302 00:16:45,350 --> 00:16:49,000 The graph is not this, you see, the graph is what? 303 00:16:49,000 --> 00:16:52,230 The distance is 0 until t equals 0. 304 00:16:52,230 --> 00:16:55,520 Then the distance that it falls increases up till the 305 00:16:55,520 --> 00:16:57,330 time the object hits the ground. 306 00:16:57,330 --> 00:16:59,460 And then it levels off like this. 307 00:16:59,460 --> 00:17:04,310 And by the way, in terms of making a few asides, notice 308 00:17:04,310 --> 00:17:08,720 that this curve here does represent a 1:1 function. 309 00:17:08,720 --> 00:17:13,950 Namely, if you pick two different times in this strip, 310 00:17:13,950 --> 00:17:15,890 you have two different distances. 311 00:17:15,890 --> 00:17:19,640 Two different times cannot yield the same distance. 312 00:17:19,640 --> 00:17:24,390 As opposed to the fact, let's call this t1 and t2. 313 00:17:24,390 --> 00:17:27,609 As opposed to the fact that once the thing hits the 314 00:17:27,609 --> 00:17:30,830 ground, our function is no longer 1:1. 315 00:17:30,830 --> 00:17:34,100 In fact, the any two values of t once the thing has hit the 316 00:17:34,100 --> 00:17:37,220 ground, we have the same s value. 317 00:17:37,220 --> 00:17:39,670 In other words, what we're saying is what? 318 00:17:39,670 --> 00:17:42,830 That once the object hits the ground, it really makes no 319 00:17:42,830 --> 00:17:45,570 difference what t is, s is still going to 320 00:17:45,570 --> 00:17:48,370 be 16t sub g squared. 321 00:17:48,370 --> 00:17:49,990 Now that was just an aside. 322 00:17:49,990 --> 00:17:53,590 The reason I mentioned this is to motivate a very important 323 00:17:53,590 --> 00:17:56,250 type of domain that takes place when we deal with 324 00:17:56,250 --> 00:17:57,970 functions of real numbers. 325 00:17:57,970 --> 00:18:01,680 In most cases, when we do a physical experiment it's over 326 00:18:01,680 --> 00:18:02,940 some time interval. 327 00:18:02,940 --> 00:18:04,980 We put something into effect and say, let's 328 00:18:04,980 --> 00:18:06,900 measure it for one hour. 329 00:18:06,900 --> 00:18:10,420 Or let's measure if from now until 3 o'clock tomorrow. 330 00:18:10,420 --> 00:18:13,990 In other words, in general, whereas a domain of a function 331 00:18:13,990 --> 00:18:17,140 can be anything we want it to be, in most real life 332 00:18:17,140 --> 00:18:20,790 laboratory situations, our domain happens to be a 333 00:18:20,790 --> 00:18:24,520 connected interval, whatever that means intuitively. 334 00:18:24,520 --> 00:18:28,000 In fact, let's try to talk about that in more detail. 335 00:18:28,000 --> 00:18:32,020 In other words, a very special type of domain that one uses 336 00:18:32,020 --> 00:18:34,670 when one talks about functions of a real variable. 337 00:18:34,670 --> 00:18:36,805 They are called intervals. 338 00:18:36,805 --> 00:18:41,360 Written as sets, if a is less than b, we talk about what? 339 00:18:41,360 --> 00:18:48,650 The set of all x which greater than a and less than b. 340 00:18:48,650 --> 00:18:52,350 By the way, that's called the open interval from a to b. 341 00:18:52,350 --> 00:18:55,980 It's written this way with parentheses. 342 00:18:55,980 --> 00:19:01,280 The set of all elements from less than b and greater than a 343 00:19:01,280 --> 00:19:05,090 inclusively is called the closed set or the closed 344 00:19:05,090 --> 00:19:06,880 interval from a to b. 345 00:19:06,880 --> 00:19:08,390 And it's written this way. 346 00:19:08,390 --> 00:19:11,920 And pictorially, you can't tell these apart. 347 00:19:11,920 --> 00:19:16,840 Namely, if this is a and this is b, both of these 348 00:19:16,840 --> 00:19:18,190 pictorially are what? 349 00:19:18,190 --> 00:19:20,540 An interval as we think of it intuitively. 350 00:19:20,540 --> 00:19:21,840 Namely, it's this stretch. 351 00:19:21,840 --> 00:19:26,670 But in one case, the endpoints a and b are included. 352 00:19:26,670 --> 00:19:30,600 And in the other case, the endpoints are excluded. 353 00:19:30,600 --> 00:19:32,810 They're included in the closed interval. 354 00:19:32,810 --> 00:19:35,270 They're excluded in the open interval. 355 00:19:35,270 --> 00:19:39,070 And again, notice that since a point has no thickness, we 356 00:19:39,070 --> 00:19:42,540 have no way of telling just by looking at the figure which of 357 00:19:42,540 --> 00:19:45,550 these two is meant unless we draw in the appropriate 358 00:19:45,550 --> 00:19:47,150 diagram this way. 359 00:19:47,150 --> 00:19:50,960 By the way, notice also that an interval can be half open 360 00:19:50,960 --> 00:19:52,330 and have closed. 361 00:19:52,330 --> 00:19:56,850 I mean, for example, one could talk about how about including 362 00:19:56,850 --> 00:20:00,970 the left endpoint but excluding the right endpoint. 363 00:20:00,970 --> 00:20:03,500 See, why couldn't we talk about something like this? 364 00:20:03,500 --> 00:20:06,740 In which case we would have written the half open half 365 00:20:06,740 --> 00:20:09,120 closed interval this particular way. 366 00:20:09,120 --> 00:20:11,750 Now again, this is all notation. 367 00:20:11,750 --> 00:20:13,690 It's things that you can memorize. 368 00:20:13,690 --> 00:20:15,990 Things that are emphasized in the text. 369 00:20:15,990 --> 00:20:18,450 But the thing that I wanted to try to have you see from the 370 00:20:18,450 --> 00:20:22,220 lecture is why we concentrate so heavily on the things 371 00:20:22,220 --> 00:20:23,560 called intervals. 372 00:20:23,560 --> 00:20:26,300 It's because in most situations when we deal with 373 00:20:26,300 --> 00:20:30,180 functions of a real variable, our so-called input, is 374 00:20:30,180 --> 00:20:34,100 usually defined on some continuous interval. 375 00:20:34,100 --> 00:20:37,220 All times from such and such to such and such. 376 00:20:37,220 --> 00:20:40,300 Now, by the way again, notice that the picture-- just as 377 00:20:40,300 --> 00:20:41,810 we've been talking about before. 378 00:20:41,810 --> 00:20:43,520 The picture comes in handy. 379 00:20:43,520 --> 00:20:47,760 Namely, 1/2 being in the open interval from 0 to 1 does not 380 00:20:47,760 --> 00:20:49,410 need a picture to interpret it. 381 00:20:49,410 --> 00:20:53,410 Namely since 1/2 is greater than 0 but less than 1, by 382 00:20:53,410 --> 00:20:55,790 definition 1/2 is in this interval. 383 00:20:55,790 --> 00:20:59,190 On the other hand, by use of a picture, I think it becomes 384 00:20:59,190 --> 00:21:02,810 rather easy to visualize what it is that we're saying when 385 00:21:02,810 --> 00:21:05,260 1/2 is in this particular interval. 386 00:21:05,260 --> 00:21:08,220 Again, notice when somebody says does 0 387 00:21:08,220 --> 00:21:09,650 belong to this interval? 388 00:21:09,650 --> 00:21:12,140 Notice that subtlety about open and closed, 389 00:21:12,140 --> 00:21:13,800 point versus dot. 390 00:21:13,800 --> 00:21:19,410 Namely, 0 does not belong to the open interval from 0 to 1, 391 00:21:19,410 --> 00:21:22,120 but it does belong, for example, to the closed 392 00:21:22,120 --> 00:21:24,010 interval from 0 to 1. 393 00:21:24,010 --> 00:21:26,620 Because what is the basic difference between these two? 394 00:21:26,620 --> 00:21:29,220 In this one, the endpoints are not included. 395 00:21:29,220 --> 00:21:32,300 In this one, the endpoints are included. 396 00:21:32,300 --> 00:21:37,060 Now, a companion to interval is a very important building 397 00:21:37,060 --> 00:21:38,420 block of this course. 398 00:21:38,420 --> 00:21:40,910 It's something called a neighborhood. 399 00:21:40,910 --> 00:21:44,000 Now, in terms of a definition, a neighborhood isn't a very 400 00:21:44,000 --> 00:21:45,610 exciting thing. 401 00:21:45,610 --> 00:21:49,790 A neighborhood of a point c, a neighborhood of x equal c is 402 00:21:49,790 --> 00:21:54,490 simply an interval which contains c inside. 403 00:21:54,490 --> 00:21:56,460 You want c to be inside the interval. 404 00:21:56,460 --> 00:21:58,040 Now what does that mean intuitively? 405 00:21:58,040 --> 00:21:59,830 Well, what it means is pick any 406 00:21:59,830 --> 00:22:02,800 interval which has c inside. 407 00:22:02,800 --> 00:22:05,540 Maybe we can go from this point to this point. 408 00:22:05,540 --> 00:22:10,720 This would be called a neighborhood of c. 409 00:22:10,720 --> 00:22:12,870 By the way, you may notice I've drawn 410 00:22:12,870 --> 00:22:16,040 this as an open interval. 411 00:22:16,040 --> 00:22:18,920 The idea is that we really want c to 412 00:22:18,920 --> 00:22:21,020 be inside the interval. 413 00:22:21,020 --> 00:22:24,180 We do not want the situation where c 414 00:22:24,180 --> 00:22:25,730 is one of the endpoints. 415 00:22:25,730 --> 00:22:28,740 And whereas we'll talk about this in more detail later, the 416 00:22:28,740 --> 00:22:31,760 important point is that in many of our investigations in 417 00:22:31,760 --> 00:22:35,470 calculus we will want to study what's happening just before 418 00:22:35,470 --> 00:22:38,100 we get to a certain point and just after 419 00:22:38,100 --> 00:22:39,340 we leave that point. 420 00:22:39,340 --> 00:22:42,330 And somehow or other, if we let that point be at the very 421 00:22:42,330 --> 00:22:45,220 end of our interval, we have no information. 422 00:22:45,220 --> 00:22:48,240 For example, if that point is the left endpoint, we don't 423 00:22:48,240 --> 00:22:51,120 know what's happening before we get to the point. 424 00:22:51,120 --> 00:22:53,790 If it's the right endpoint, we don't know what's happening 425 00:22:53,790 --> 00:22:54,690 afterwards. 426 00:22:54,690 --> 00:22:57,360 And that's why you'll find in the textbook that a 427 00:22:57,360 --> 00:23:03,500 neighborhood is defined to be an open interval, which 428 00:23:03,500 --> 00:23:04,450 contains c. 429 00:23:04,450 --> 00:23:07,410 In other words, we want to make sure that c is in the 430 00:23:07,410 --> 00:23:08,530 interior here. 431 00:23:08,530 --> 00:23:13,040 By the way, in many cases it turns out algebraically to be 432 00:23:13,040 --> 00:23:16,540 easier if this happens to be what we call a symmetric 433 00:23:16,540 --> 00:23:17,290 neighborhood. 434 00:23:17,290 --> 00:23:18,930 In other words, if c is in the middle. 435 00:23:21,740 --> 00:23:24,300 We won't go into that right now, but if c happens to be in 436 00:23:24,300 --> 00:23:28,630 the middle that's called a symmetric neighborhood. 437 00:23:28,630 --> 00:23:31,080 In fact, another way of writing that is to say what? 438 00:23:31,080 --> 00:23:35,660 Pick some definite distance h and what do you write down? 439 00:23:35,660 --> 00:23:41,000 You write down c minus h to c plus h and that puts c right 440 00:23:41,000 --> 00:23:43,680 in the middle of this particular interval. 441 00:23:43,680 --> 00:23:46,150 And you see, the idea here is that when you're looking at 442 00:23:46,150 --> 00:23:47,760 what's happening to a function, 443 00:23:47,760 --> 00:23:50,280 you may lose symmetry. 444 00:23:50,280 --> 00:23:52,480 For example, in this particular graph that I've 445 00:23:52,480 --> 00:23:55,280 drawn, notice that at this particular point I've marked 446 00:23:55,280 --> 00:23:58,880 off equal intervals on both sides of l. 447 00:23:58,880 --> 00:24:02,160 But notice that when I come down here, they do not project 448 00:24:02,160 --> 00:24:05,530 onto equal intervals on either side of c. 449 00:24:05,530 --> 00:24:09,180 In other words, if this had been a straight line. 450 00:24:09,180 --> 00:24:11,140 Frequently what we do in a case like this is we 451 00:24:11,140 --> 00:24:12,180 say well look it. 452 00:24:12,180 --> 00:24:14,690 If we're interested in seeing what happens near c, 453 00:24:14,690 --> 00:24:17,380 why don't we just-- 454 00:24:17,380 --> 00:24:18,490 this is non-symmetric. 455 00:24:18,490 --> 00:24:21,280 Why don't we just take the smaller of these two widths 456 00:24:21,280 --> 00:24:23,680 and see what happens in the symmetric part? 457 00:24:23,680 --> 00:24:26,310 In other words, if the neighborhood is not symmetric, 458 00:24:26,310 --> 00:24:28,420 we can always make it symmetric. 459 00:24:28,420 --> 00:24:31,390 And so there really isn't that much to worry about in that 460 00:24:31,390 --> 00:24:32,490 particular respect. 461 00:24:32,490 --> 00:24:34,190 But why are we interested in neighborhoods 462 00:24:34,190 --> 00:24:35,210 in the first place? 463 00:24:35,210 --> 00:24:38,370 And the answer is that in many cases what we're going to be 464 00:24:38,370 --> 00:24:41,680 doing is studying what's happening near a particular 465 00:24:41,680 --> 00:24:44,600 point c and want to know what's happened just before 466 00:24:44,600 --> 00:24:47,550 and what's happening just after. 467 00:24:47,550 --> 00:24:49,920 The next important concept that's connected with 468 00:24:49,920 --> 00:24:54,200 neighborhoods is the idea of a deleted neighborhood. 469 00:24:54,200 --> 00:24:58,600 And that in turn, is very strongly connected with 0/0. 470 00:24:58,600 --> 00:25:01,940 For example, consider the function f of x which is x 471 00:25:01,940 --> 00:25:04,910 squared minus 9 over x minus 3. 472 00:25:04,910 --> 00:25:08,720 If we let x equal 3, if the input is 3, notice that the 473 00:25:08,720 --> 00:25:14,580 output becomes 9 minus 9 over 3 minus 3, or 0/0. 474 00:25:14,580 --> 00:25:20,290 On the other hand, if x is any number whatsoever except x 475 00:25:20,290 --> 00:25:23,920 equals 3, no harm is done with this as an input. 476 00:25:23,920 --> 00:25:26,970 Consequently, what one is talking about now is the only 477 00:25:26,970 --> 00:25:30,330 time you get that 0/0 form is when x is 3. 478 00:25:30,330 --> 00:25:33,350 What happens if you're in a neighborhood of 3, but not 479 00:25:33,350 --> 00:25:35,190 equal to 3 itself? 480 00:25:35,190 --> 00:25:38,420 You see what I'm driving at here is pick any number x in 481 00:25:38,420 --> 00:25:41,050 this interval other than 3 itself. 482 00:25:41,050 --> 00:25:44,860 And notice that f of x can be written this way. 483 00:25:44,860 --> 00:25:48,870 As long as x is not equal to 3, we can cancel x minus 3 484 00:25:48,870 --> 00:25:50,650 from numerator and denominator. 485 00:25:50,650 --> 00:25:52,540 Remember, we can't divide by 0. 486 00:25:52,540 --> 00:25:53,950 And now we see what? 487 00:25:53,950 --> 00:25:57,260 That as long as x is not equal to 3, f of x is 488 00:25:57,260 --> 00:25:58,780 perfectly well defined. 489 00:25:58,780 --> 00:26:02,230 And consequently, this is what motivates the concept of a 490 00:26:02,230 --> 00:26:03,520 deleted neighborhood. 491 00:26:03,520 --> 00:26:07,560 Namely, everything is fine in this whole neighborhood except 492 00:26:07,560 --> 00:26:08,750 for 3 itself. 493 00:26:08,750 --> 00:26:12,980 So to avoid that unpleasantry, let's just delete that point. 494 00:26:12,980 --> 00:26:15,570 And that's called a deleted neighborhood. 495 00:26:15,570 --> 00:26:17,770 And you see what we do when the neighborhood is deleted, 496 00:26:17,770 --> 00:26:19,450 we're still going to talk about what? 497 00:26:19,450 --> 00:26:21,830 How close you are to that point. 498 00:26:21,830 --> 00:26:24,900 And by the way, this brings us to another very fascinating 499 00:26:24,900 --> 00:26:29,020 aspect of what's going on between our geometry and our 500 00:26:29,020 --> 00:26:29,990 arithmetic. 501 00:26:29,990 --> 00:26:33,460 Do you really talk about the distance between numbers? 502 00:26:33,460 --> 00:26:35,660 I mean, is 7 near 3? 503 00:26:35,660 --> 00:26:38,116 And the guys says, well, what do you mean, is 7 near 3? 504 00:26:38,116 --> 00:26:42,810 Well, I would say here that 7 is very near to 3. 505 00:26:42,810 --> 00:26:46,120 When you say that 7 is near 3, you certainly don't mean close 506 00:26:46,120 --> 00:26:48,090 to in the geometric sense. 507 00:26:48,090 --> 00:26:50,600 You mean the difference between them is small. 508 00:26:50,600 --> 00:26:53,520 In other words, the next thing that we have to talk about is 509 00:26:53,520 --> 00:26:56,600 how when we talk about being close to a point which is a 510 00:26:56,600 --> 00:27:00,250 geometric term, how do we talk about that algebraically? 511 00:27:00,250 --> 00:27:03,210 You see geometrically, how do you talk about the distance 512 00:27:03,210 --> 00:27:04,890 between x1 and x2? 513 00:27:04,890 --> 00:27:08,340 Well, if you're going this way, it's just what? 514 00:27:08,340 --> 00:27:10,920 x2 minus x1. 515 00:27:10,920 --> 00:27:14,760 If you're going the other way, the direct distance this way, 516 00:27:14,760 --> 00:27:17,580 it's x1 minus x2. 517 00:27:17,580 --> 00:27:21,070 In any event, the distance between these two points is 518 00:27:21,070 --> 00:27:23,080 just the magnitude of the difference 519 00:27:23,080 --> 00:27:24,720 of these two numbers. 520 00:27:24,720 --> 00:27:27,760 And that leads, you see, to the concept that's hit quite 521 00:27:27,760 --> 00:27:29,630 heavily in our text, and that is the 522 00:27:29,630 --> 00:27:31,500 concept of absolute value. 523 00:27:31,500 --> 00:27:34,600 Perhaps one of the most critical analytical geometric 524 00:27:34,600 --> 00:27:38,650 topics that we tackle in our early part of our course. 525 00:27:38,650 --> 00:27:42,250 Analytically, we define the absolute value written with 526 00:27:42,250 --> 00:27:44,240 vertical bars here, x1 minus x2. 527 00:27:44,240 --> 00:27:48,390 The absolute value of x1 minus x2 to be the positive square 528 00:27:48,390 --> 00:27:51,140 root of x1 minus x2 squared. 529 00:27:51,140 --> 00:27:53,800 And in plain English all this says is what? 530 00:27:53,800 --> 00:27:57,510 See when you square and then take the positive square root, 531 00:27:57,510 --> 00:27:59,930 you haven't undone what you've done before. 532 00:27:59,930 --> 00:28:01,010 All you've done is what? 533 00:28:01,010 --> 00:28:03,790 If it's positive here you haven't changed anything. 534 00:28:03,790 --> 00:28:07,350 But if x1 minus x2 are negative, when you square it 535 00:28:07,350 --> 00:28:09,960 and extract the positive square root, all you've done 536 00:28:09,960 --> 00:28:12,690 is changed the sign just like you're supposed to. 537 00:28:12,690 --> 00:28:14,060 Let me give you an example. 538 00:28:14,060 --> 00:28:17,800 Suppose you're faced with the absolute value of x minus 3 is 539 00:28:17,800 --> 00:28:18,710 less than 2. 540 00:28:18,710 --> 00:28:21,490 What does this say geometrically? 541 00:28:21,490 --> 00:28:24,930 Geometrically what it says is that x is within 542 00:28:24,930 --> 00:28:26,910 two units of 3. 543 00:28:26,910 --> 00:28:31,260 in terms of a picture, all you have to do now is draw in 3, 544 00:28:31,260 --> 00:28:35,590 mark off two units on either side, and for x to be within 545 00:28:35,590 --> 00:28:38,200 two units of 3, all you know is is that x 546 00:28:38,200 --> 00:28:39,215 has to be in here. 547 00:28:39,215 --> 00:28:42,230 In other words, look at how easily you can solve this 548 00:28:42,230 --> 00:28:44,380 particular problem. 549 00:28:44,380 --> 00:28:46,680 On the other hand, you can always go back to the basic 550 00:28:46,680 --> 00:28:48,710 definition and say, wait a second. 551 00:28:48,710 --> 00:28:54,720 This means the positive square root of x minus 3 squared is 552 00:28:54,720 --> 00:28:55,880 less than 2. 553 00:28:55,880 --> 00:28:57,900 So I will square both sides. 554 00:28:57,900 --> 00:29:02,620 If you do that you get x minus 3 squared is less than 4. 555 00:29:02,620 --> 00:29:06,860 If you now collect terms and expand, you get x squared 556 00:29:06,860 --> 00:29:10,630 minus 6x plus 9 minus 4. 557 00:29:10,630 --> 00:29:11,320 That's what? 558 00:29:11,320 --> 00:29:15,550 Plus 5 is less than 0. 559 00:29:15,550 --> 00:29:20,920 This factors into x minus 1 times x minus 5 560 00:29:20,920 --> 00:29:22,640 is less than 0. 561 00:29:22,640 --> 00:29:25,700 The only way the product of two numbers can be negative is 562 00:29:25,700 --> 00:29:27,970 if the factors have different signs. 563 00:29:27,970 --> 00:29:32,790 Since this is x minus 5 is less than x minus 1, this must 564 00:29:32,790 --> 00:29:34,670 be the smaller of the two. 565 00:29:34,670 --> 00:29:37,560 This must be the larger of the two. 566 00:29:37,560 --> 00:29:40,920 To say that x minus 1 is greater than 0 is the same as 567 00:29:40,920 --> 00:29:42,990 saying that x is greater than 1. 568 00:29:42,990 --> 00:29:46,570 To say that x minus 5 is less than 0 is the same as saying 569 00:29:46,570 --> 00:29:48,800 that x is less than 5. 570 00:29:48,800 --> 00:29:51,810 You put that all together and notice that even though it 571 00:29:51,810 --> 00:29:55,510 wasn't quite as comfortable, we can obtain the same answer 572 00:29:55,510 --> 00:29:59,040 algebraically as we can obtain geometrically. 573 00:29:59,040 --> 00:30:01,405 In other words, our relationship between algebra 574 00:30:01,405 --> 00:30:04,870 and geometry remains the same. 575 00:30:04,870 --> 00:30:07,040 Again, when you can draw the picture, it's 576 00:30:07,040 --> 00:30:08,520 worth a thousand words. 577 00:30:08,520 --> 00:30:11,470 If you can't draw the picture or you're suspicious about the 578 00:30:11,470 --> 00:30:15,170 picture, especially when it involves point versus dot, 579 00:30:15,170 --> 00:30:19,000 then what you do is resort to the analytic definition. 580 00:30:19,000 --> 00:30:21,000 These do not replace one another, they 581 00:30:21,000 --> 00:30:22,750 work hand in hand. 582 00:30:22,750 --> 00:30:25,630 Finally, what we must talk about now is the 583 00:30:25,630 --> 00:30:26,950 arithmetic of functions. 584 00:30:26,950 --> 00:30:29,820 Can we combine functions to form functions? 585 00:30:29,820 --> 00:30:31,960 And the answer is yes. 586 00:30:31,960 --> 00:30:35,190 First of all, in talking about the arithmetic of functions, 587 00:30:35,190 --> 00:30:36,360 what must we do? 588 00:30:36,360 --> 00:30:38,860 We must first, at least, define what it means for two 589 00:30:38,860 --> 00:30:40,980 functions to be equal. 590 00:30:40,980 --> 00:30:44,670 Well, for two functions to be equal, all we insist on is 591 00:30:44,670 --> 00:30:48,510 that first of all, they're defined on the same domain. 592 00:30:48,510 --> 00:30:52,670 And secondly, that for each input in the domain, each 593 00:30:52,670 --> 00:30:55,660 function gives you the same output. 594 00:30:55,660 --> 00:30:58,770 For example, suppose a is a set whose elements 595 00:30:58,770 --> 00:31:00,410 consists of 0 and 1. 596 00:31:00,410 --> 00:31:03,020 And suppose b is also the set whose elements 597 00:31:03,020 --> 00:31:04,430 consist of 0 and 1. 598 00:31:04,430 --> 00:31:09,580 One such function would be f. 599 00:31:09,580 --> 00:31:11,190 It maps 0 into 0. 600 00:31:11,190 --> 00:31:13,480 It maps 1 into 1. 601 00:31:13,480 --> 00:31:15,760 Another function, which I'll call g-- 602 00:31:18,850 --> 00:31:20,100 see, f does what? 603 00:31:20,100 --> 00:31:24,340 It maps 0 into 0 and 1 into 1. 604 00:31:24,340 --> 00:31:25,970 What does g do? 605 00:31:25,970 --> 00:31:30,350 g maps 0 into 1 and 1 into 0. 606 00:31:30,350 --> 00:31:32,430 Notice that f and g are different. 607 00:31:32,430 --> 00:31:34,590 They both have the same domain. 608 00:31:34,590 --> 00:31:37,530 They both have the same image. 609 00:31:37,530 --> 00:31:40,700 But notice that element for element, they're not the same. 610 00:31:40,700 --> 00:31:43,960 Namely, f and g do different things to 0. 611 00:31:43,960 --> 00:31:45,780 f sends 0 into 0. 612 00:31:45,780 --> 00:31:47,450 g sends 0 into 1. 613 00:31:47,450 --> 00:31:49,860 So I can tell f and g apart. 614 00:31:49,860 --> 00:31:52,860 And because I can tell them apart, they're not equal. 615 00:31:52,860 --> 00:31:56,010 All right, so equality means I can't tell f from g. 616 00:31:56,010 --> 00:31:59,440 Now the next kind of question is, how do you do arithmetic 617 00:31:59,440 --> 00:32:00,090 with f and g? 618 00:32:00,090 --> 00:32:01,660 Can I add two functions? 619 00:32:01,660 --> 00:32:03,430 Can I multiply two functions? 620 00:32:03,430 --> 00:32:05,250 Can I subtract two functions? 621 00:32:05,250 --> 00:32:07,680 And the answer again, turns out to be yes. 622 00:32:07,680 --> 00:32:10,900 And not only yes, but yes in a rather simple way. 623 00:32:10,900 --> 00:32:14,460 Let's again do this by means of examples. 624 00:32:14,460 --> 00:32:19,120 Suppose we defined f of x to be 2x for all x in A. Namely, 625 00:32:19,120 --> 00:32:23,630 if A is 1, 2, 3, f of 1 will be 2, f of 2 will be 4, f 626 00:32:23,630 --> 00:32:25,230 of 3 will be 6. 627 00:32:25,230 --> 00:32:28,840 Let's define another function on A, let's call it g. 628 00:32:28,840 --> 00:32:32,960 g of x will be x plus 1 for each x in A. In other words, g 629 00:32:32,960 --> 00:32:39,000 of 1 will be 2, g of 2 will be 3, g of 3 will be 4. 630 00:32:39,000 --> 00:32:42,340 Now the point is, can I add f of x and g of x? 631 00:32:42,340 --> 00:32:44,650 Well, sure. f of x is 2x. 632 00:32:44,650 --> 00:32:46,490 g of x is x plus 1. 633 00:32:46,490 --> 00:32:51,170 So if I add these I get h of x is 3x plus 1. 634 00:32:51,170 --> 00:32:54,110 In a similar way, could I have multiplied these two? 635 00:32:54,110 --> 00:32:54,590 Well, sure. 636 00:32:54,590 --> 00:32:58,210 Again, f of x is 2x, g of x is x plus 1. 637 00:32:58,210 --> 00:33:00,550 If I multiply these together, I get what? 638 00:33:00,550 --> 00:33:03,930 2x times x plus 1, which is the same as 2x 639 00:33:03,930 --> 00:33:06,040 squared plus 2x. 640 00:33:06,040 --> 00:33:09,540 Now of course this probably doesn't look too smooth 641 00:33:09,540 --> 00:33:11,180 because there's no pictures here. 642 00:33:11,180 --> 00:33:12,750 All we're saying is this. 643 00:33:12,750 --> 00:33:17,760 Here's A and all you're saying is that if you add f and g, 644 00:33:17,760 --> 00:33:18,840 what do you get? 645 00:33:18,840 --> 00:33:25,840 If A is 1, 3x plus 1 is 4. 646 00:33:25,840 --> 00:33:28,760 2 times 3 plus 1 is 7. 647 00:33:28,760 --> 00:33:31,880 3 times 3 plus 1 is 10. 648 00:33:31,880 --> 00:33:34,706 In other words, in this case, our image, if I want to call 649 00:33:34,706 --> 00:33:36,560 it b, would look like this. 650 00:33:36,560 --> 00:33:40,510 This would be the sum of the two functions f and g. 651 00:33:40,510 --> 00:33:42,790 And similarly, for the product I could do the 652 00:33:42,790 --> 00:33:44,050 same kind of a thing. 653 00:33:44,050 --> 00:33:47,690 In other words, I can just arithmetically, since both the 654 00:33:47,690 --> 00:33:51,880 output of f and the g machines are real numbers, and the sum 655 00:33:51,880 --> 00:33:55,350 of two real numbers is a real number, I can add and multiply 656 00:33:55,350 --> 00:33:57,980 functions to form functions. 657 00:33:57,980 --> 00:34:00,950 But there's one other important way of combining 658 00:34:00,950 --> 00:34:02,430 functions in calculus. 659 00:34:02,430 --> 00:34:05,550 A way which is very, very important and one which we may 660 00:34:05,550 --> 00:34:07,700 not have seen too much of before. 661 00:34:07,700 --> 00:34:10,219 And so let me close our lecture for today with an 662 00:34:10,219 --> 00:34:13,949 emphasis on that particular topic. 663 00:34:13,949 --> 00:34:16,530 It's called composition of functions. 664 00:34:16,530 --> 00:34:19,670 And to see what composition of functions means think of a 665 00:34:19,670 --> 00:34:25,659 particular example where maybe the f machine f of x is 2x. 666 00:34:25,659 --> 00:34:26,929 In other words, think of it this way. 667 00:34:26,929 --> 00:34:29,440 We run x through the f machine, the 668 00:34:29,440 --> 00:34:31,580 output will be 2x. 669 00:34:31,580 --> 00:34:36,360 Now we run the output of the f machine into the g machine. 670 00:34:36,360 --> 00:34:38,420 Now what does the g machine do? 671 00:34:38,420 --> 00:34:42,370 If x is the input, x plus 1 means 1 more than the input. 672 00:34:42,370 --> 00:34:46,080 The g machine always adds on 1 to the input to 673 00:34:46,080 --> 00:34:47,110 give you the output. 674 00:34:47,110 --> 00:34:52,860 Well, if the input is 2x, the output will be 2x plus 1. 675 00:34:52,860 --> 00:34:56,699 Notice that these two together can be thought of as being one 676 00:34:56,699 --> 00:35:00,230 function machine, which I'll call the q machine. 677 00:35:00,230 --> 00:35:02,920 In other words, what happens for the q machine is what? 678 00:35:02,920 --> 00:35:06,740 x runs into the f machine, the output of the f machine 679 00:35:06,740 --> 00:35:10,060 becomes the input of the g machine, and the output of the 680 00:35:10,060 --> 00:35:13,410 g machine is then the output of the q machine. 681 00:35:13,410 --> 00:35:16,520 The q machine is sort of build with component parts here. 682 00:35:16,520 --> 00:35:19,310 And the reason that this is very, very important is that 683 00:35:19,310 --> 00:35:22,390 this comes up in calculus all the time, where the first 684 00:35:22,390 --> 00:35:25,610 variable is related to the second variable, the second 685 00:35:25,610 --> 00:35:27,790 variable has a definite relationship to the third 686 00:35:27,790 --> 00:35:31,100 variable, and we now want to relate the first variable to 687 00:35:31,100 --> 00:35:32,410 the third variable. 688 00:35:32,410 --> 00:35:34,650 And the way we write that-- and I guess 689 00:35:34,650 --> 00:35:35,660 this is hard to see. 690 00:35:35,660 --> 00:35:38,265 This is not an O over here, it's a little circle, like a 691 00:35:38,265 --> 00:35:41,360 dot, and it's called the composition of g and f. 692 00:35:41,360 --> 00:35:42,380 It's not gof. 693 00:35:42,380 --> 00:35:44,480 Its g circle f. 694 00:35:44,480 --> 00:35:46,140 And the q machine is what? 695 00:35:46,140 --> 00:35:48,780 You write it this way and maybe if you look at the 696 00:35:48,780 --> 00:35:52,550 picture you can see exactly what's happening here. 697 00:35:52,550 --> 00:35:56,510 You apply f to x and then apply g to the result. 698 00:35:56,510 --> 00:35:58,930 In other words, just looking at this picture it becomes 699 00:35:58,930 --> 00:36:03,570 rather apparent that q of x is just 2x plus 1. 700 00:36:03,570 --> 00:36:07,320 Notice you see, that the domain of the q machine is the 701 00:36:07,320 --> 00:36:08,530 same as the domain of f. 702 00:36:08,530 --> 00:36:12,360 The input of the q machine is what goes into the f machine. 703 00:36:12,360 --> 00:36:15,560 The output, the image of the q machine, is the 704 00:36:15,560 --> 00:36:17,540 image of the g machine. 705 00:36:17,540 --> 00:36:20,270 In other words, just this particular thing. 706 00:36:20,270 --> 00:36:23,210 Now this type of function combination called 707 00:36:23,210 --> 00:36:25,870 composition, is a very intricate thing. 708 00:36:25,870 --> 00:36:29,050 It depends on the order in which you do these things. 709 00:36:29,050 --> 00:36:30,660 This is a rather interesting point. 710 00:36:30,660 --> 00:36:33,970 For example, when you add two numbers, a and b, it makes no 711 00:36:33,970 --> 00:36:36,270 difference in which order you add them. 712 00:36:36,270 --> 00:36:38,840 On the other hand, when you divide two numbers, a and b, 713 00:36:38,840 --> 00:36:41,380 the quotient does depend on the order in 714 00:36:41,380 --> 00:36:42,580 which you divided them. 715 00:36:42,580 --> 00:36:44,520 Well, the same thing is true here. 716 00:36:44,520 --> 00:36:48,160 Let's call p the function which starts with the g 717 00:36:48,160 --> 00:36:51,010 machine followed by the f machine. 718 00:36:51,010 --> 00:36:54,090 And as this lecture wears on, I think maybe there's a reason 719 00:36:54,090 --> 00:36:56,800 for making this circle look like an O. Maybe this is 720 00:36:56,800 --> 00:36:59,430 starting to look a little bit like fog at this time. 721 00:36:59,430 --> 00:37:00,400 It's not. 722 00:37:00,400 --> 00:37:04,174 All I want you to see is that what we do now is we start-- 723 00:37:04,174 --> 00:37:06,420 see, what are we going to do here now? 724 00:37:06,420 --> 00:37:09,450 We're going to start with the g machine 725 00:37:09,450 --> 00:37:12,240 first, then the f machine. 726 00:37:12,240 --> 00:37:14,000 In other words, what happens now? 727 00:37:14,000 --> 00:37:18,660 If the input is x, the output of the g machine is one more 728 00:37:18,660 --> 00:37:19,690 than the input. 729 00:37:19,690 --> 00:37:23,780 That would make the input x plus 1 to the f machine. 730 00:37:23,780 --> 00:37:24,990 What does f do? 731 00:37:24,990 --> 00:37:28,450 Remember, f doubles. 732 00:37:28,450 --> 00:37:31,280 f doubles the input. 733 00:37:31,280 --> 00:37:32,810 So the output here would be what? 734 00:37:32,810 --> 00:37:35,230 Twice x plus 1. 735 00:37:35,230 --> 00:37:38,450 In other words, what would p of x look like? 736 00:37:38,450 --> 00:37:44,590 If x goes into the p machine, what comes out is twice x plus 737 00:37:44,590 --> 00:37:48,130 1, or 2x plus 2. 738 00:37:48,130 --> 00:37:51,320 On the other hand, when we put the f and the g machine in the 739 00:37:51,320 --> 00:37:56,070 other order and formed q of x, what was the output? 740 00:37:56,070 --> 00:37:57,680 Let's go back here and look. 741 00:37:57,680 --> 00:38:00,610 The output was 2x plus 1. 742 00:38:00,610 --> 00:38:04,570 In other words, do you build a different function machine by 743 00:38:04,570 --> 00:38:06,760 interchanging the f and the g? 744 00:38:06,760 --> 00:38:08,370 And the answer is yes. 745 00:38:08,370 --> 00:38:10,830 In other words, what I'd like you to see for concluding this 746 00:38:10,830 --> 00:38:13,110 part is that whereas everything was pretty 747 00:38:13,110 --> 00:38:16,030 straightforward up until now, the most important new 748 00:38:16,030 --> 00:38:19,440 concept, one which was not so intuitive is the one that's 749 00:38:19,440 --> 00:38:21,330 called the composition of functions. 750 00:38:21,330 --> 00:38:24,070 It's the one that occurs all the time in 751 00:38:24,070 --> 00:38:26,140 related rates problems. 752 00:38:26,140 --> 00:38:29,620 We'll be using it over and over again in this course. 753 00:38:29,620 --> 00:38:33,790 And all I want you to see is that first of all when you use 754 00:38:33,790 --> 00:38:36,610 the composition of functions, what you get depends on the 755 00:38:36,610 --> 00:38:38,020 order in which you combine them. 756 00:38:38,020 --> 00:38:42,120 And that secondly, and most importantly, that neither of 757 00:38:42,120 --> 00:38:44,800 these is the same as this. 758 00:38:44,800 --> 00:38:48,820 That combining two functions is not the same as multiplying 759 00:38:48,820 --> 00:38:51,170 the outputs of two different functions. 760 00:38:51,170 --> 00:38:54,710 Well, I think that's sort of enough of a 761 00:38:54,710 --> 00:38:57,680 mouthful for one sitting. 762 00:38:57,680 --> 00:39:01,250 Our main aim today was to introduce functions, the 763 00:39:01,250 --> 00:39:03,270 language that we're going to be using the rest of the way. 764 00:39:03,270 --> 00:39:07,920 Because after all if we don't have the vocabulary 765 00:39:07,920 --> 00:39:10,550 established it will not be second nature to talk about 766 00:39:10,550 --> 00:39:11,450 the concepts. 767 00:39:11,450 --> 00:39:14,750 Starting next time and beyond we will deal more with 768 00:39:14,750 --> 00:39:17,200 specific calculus contexts. 769 00:39:17,200 --> 00:39:18,910 But until next time, goodbye. 770 00:39:21,990 --> 00:39:24,530 ANNOUNCER: Funding for the publication of this video was 771 00:39:24,530 --> 00:39:29,240 provided by the Gabriella and Paul Rosenbaum Foundation. 772 00:39:29,240 --> 00:39:33,420 Help OCW continue to provide free and open access to MIT 773 00:39:33,420 --> 00:39:37,610 courses by making a donation at ocw.mit.edu/donate.