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 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 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 00:00:29,867 --> 00:00:34,060 HERBERT GROSS: Hi, our lecture today is entitled inverse 9 00:00:34,060 --> 00:00:37,510 functions, and it's almost what you could call a natural 10 00:00:37,510 --> 00:00:41,280 follow-up to our lecture of last time when we talked 11 00:00:41,280 --> 00:00:44,220 briefly about 1:1 and onto functions. 12 00:00:44,220 --> 00:00:47,100 Inverse functions have a tremendous application as we 13 00:00:47,100 --> 00:00:52,430 progress through calculus, but of even more exciting impact 14 00:00:52,430 --> 00:00:55,320 is the fact that inverse functions are valuable in 15 00:00:55,320 --> 00:00:56,340 their own right. 16 00:00:56,340 --> 00:00:58,760 They are a pre-calculus topic. 17 00:00:58,760 --> 00:01:02,110 In fact, they appear as early in the curriculum as 18 00:01:02,110 --> 00:01:04,560 approximately the first grade. 19 00:01:04,560 --> 00:01:08,580 See roughly speaking, inverse functions in plain English, 20 00:01:08,580 --> 00:01:12,680 mean that all we've done is made a switch in emphasis. 21 00:01:12,680 --> 00:01:14,600 Let's take a look at that. 22 00:01:14,600 --> 00:01:17,980 Let's go back roughly to our first grade curriculum when 23 00:01:17,980 --> 00:01:21,760 one learns that 2 plus 3 equals 5, or that 5 24 00:01:21,760 --> 00:01:23,990 minus 3 equals 2. 25 00:01:23,990 --> 00:01:27,820 Both of these statements say the same thing, but with a 26 00:01:27,820 --> 00:01:29,340 change in emphasis. 27 00:01:29,340 --> 00:01:35,020 It's as if 2 is being emphasized here while 5 is 28 00:01:35,020 --> 00:01:36,770 being emphasized here. 29 00:01:36,770 --> 00:01:38,220 This is rather interesting you see. 30 00:01:38,220 --> 00:01:41,740 For example, in the new mathematics, one talks about 31 00:01:41,740 --> 00:01:44,530 subtraction being the inverse of addition. 32 00:01:44,530 --> 00:01:47,470 And this is the same inverse that we want to talk about 33 00:01:47,470 --> 00:01:50,540 today as it applies to mathematics in general. 34 00:01:50,540 --> 00:01:53,410 What do we that subtraction is the inverse of addition? 35 00:01:53,410 --> 00:01:57,790 It may sound fancy, but all it means is that if you know how 36 00:01:57,790 --> 00:02:01,650 to add, if you define subtraction properly, you 37 00:02:01,650 --> 00:02:03,920 automatically know how to subtract. 38 00:02:03,920 --> 00:02:06,910 And this, of course, is what's prevalent in the old 39 00:02:06,910 --> 00:02:09,940 change-making technique of going into a store, making a 40 00:02:09,940 --> 00:02:13,100 purchase, paying for the purchase, and when you receive 41 00:02:13,100 --> 00:02:17,420 your change, the clerk rarely, if ever, performs subtraction. 42 00:02:17,420 --> 00:02:20,280 You may recall that what he does is he adds onto the 43 00:02:20,280 --> 00:02:23,590 amount of the purchase the amount necessary to make up 44 00:02:23,590 --> 00:02:26,450 the denomination of the bill with which you paid him. 45 00:02:26,450 --> 00:02:30,120 In other words, what we're saying here is, for example, 46 00:02:30,120 --> 00:02:33,910 that one may think of 5 minus 3 as being what? 47 00:02:33,910 --> 00:02:40,070 That number, which must be added onto 3 to give 5. 48 00:02:40,070 --> 00:02:42,420 You see, in this sense, subtraction is 49 00:02:42,420 --> 00:02:44,130 the inverse of addition. 50 00:02:44,130 --> 00:02:45,770 Once we know how to add, we 51 00:02:45,770 --> 00:02:47,860 automatically know how to subtract. 52 00:02:47,860 --> 00:02:50,110 Now you see, this idea goes with us. 53 00:02:50,110 --> 00:02:53,450 We learned that multiplication and division are inverses of 54 00:02:53,450 --> 00:02:54,160 one another. 55 00:02:54,160 --> 00:02:57,200 And as we go on through higher mathematics, even on the 56 00:02:57,200 --> 00:03:00,570 pre-calculus level, we find additional examples of this. 57 00:03:00,570 --> 00:03:05,340 For example, when one knows how to use exponents, one 58 00:03:05,340 --> 00:03:08,230 automatically knows how to study logarithms. 59 00:03:08,230 --> 00:03:12,380 Namely, if y equals the log of x to the base b, this is a 60 00:03:12,380 --> 00:03:16,330 synonym for saying that b to the y equals x. 61 00:03:16,330 --> 00:03:18,360 And what is the basic difference between these two 62 00:03:18,360 --> 00:03:19,270 statements? 63 00:03:19,270 --> 00:03:22,170 They are paraphrases of one another. 64 00:03:22,170 --> 00:03:26,160 In one case it seems that the number y is being emphasized 65 00:03:26,160 --> 00:03:28,630 and in the other case it's the number x that's being 66 00:03:28,630 --> 00:03:30,610 emphasized. 67 00:03:30,610 --> 00:03:34,460 And as in the case of most examples of paraphrasing, 68 00:03:34,460 --> 00:03:38,970 which of the two forms we use depends on what problem it is 69 00:03:38,970 --> 00:03:40,570 that we're trying to solve. 70 00:03:40,570 --> 00:03:43,960 In other words, if we know one of these two, then we 71 00:03:43,960 --> 00:03:47,260 automatically can study the other in terms of the one with 72 00:03:47,260 --> 00:03:48,900 which we feel more familiar. 73 00:03:48,900 --> 00:03:52,420 This, of course, continues when one gets to trigonometry 74 00:03:52,420 --> 00:03:54,330 and studies the so-called inverse 75 00:03:54,330 --> 00:03:55,830 trigonometric functions. 76 00:03:55,830 --> 00:03:59,280 If y equals the inverse sine of x that's the same thing as 77 00:03:59,280 --> 00:04:01,840 saying that x equals sine y. 78 00:04:01,840 --> 00:04:03,930 Again, what is the basic difference? 79 00:04:03,930 --> 00:04:07,580 In one case, it seems that y is being emphasized. 80 00:04:07,580 --> 00:04:10,930 In the other case, it's x that's being emphasized. 81 00:04:10,930 --> 00:04:15,680 In terms of the usual calculus jargon of independent variable 82 00:04:15,680 --> 00:04:18,370 versus dependent variable it appears that what? 83 00:04:18,370 --> 00:04:21,839 In one case, x is the independent variable, y the 84 00:04:21,839 --> 00:04:23,010 dependent variable. 85 00:04:23,010 --> 00:04:26,890 In the other case, y is the independent variable, x the 86 00:04:26,890 --> 00:04:28,350 dependent variable. 87 00:04:28,350 --> 00:04:32,850 To generalize this result what we're saying is simply this. 88 00:04:32,850 --> 00:04:37,740 If y equals f of x, you see where y is being emphasized, 89 00:04:37,740 --> 00:04:39,170 the dependent variable. 90 00:04:39,170 --> 00:04:43,780 If we wish to switch the emphasis, then we write x 91 00:04:43,780 --> 00:04:47,290 equals f inverse of y. 92 00:04:47,290 --> 00:04:50,850 This is read f inverse of y and it's the function which in 93 00:04:50,850 --> 00:04:53,750 a sense, inverts the roles of x and y. 94 00:04:53,750 --> 00:04:56,540 Let's see what this means more explicitly in terms of a 95 00:04:56,540 --> 00:04:58,660 particular example. 96 00:04:58,660 --> 00:05:01,220 Let's suppose that we're given the equation y 97 00:05:01,220 --> 00:05:03,040 equals 2x minus 7. 98 00:05:03,040 --> 00:05:07,230 Or to represent this somewhat more abstractly, y equals f of 99 00:05:07,230 --> 00:05:10,800 x where f of x is 2x minus 7. 100 00:05:10,800 --> 00:05:12,870 Now you see, without mentioning the word inverse 101 00:05:12,870 --> 00:05:15,970 function, it turns out that early in our high school 102 00:05:15,970 --> 00:05:19,640 career we were finding inverse functions as soon as, for 103 00:05:19,640 --> 00:05:23,750 example, someone were to give us this problem and say solve 104 00:05:23,750 --> 00:05:26,160 for x in terms of y. 105 00:05:26,160 --> 00:05:28,640 Solve for x in terms of y. 106 00:05:28,640 --> 00:05:34,820 You see, if we solve for x in terms of y, we now have what? 107 00:05:34,820 --> 00:05:36,490 x is being emphasized. 108 00:05:36,490 --> 00:05:38,940 These two statements tell us the same thing. 109 00:05:38,940 --> 00:05:42,640 But now we write what? x equals f inverse of y. 110 00:05:42,640 --> 00:05:46,920 And f inverse of y is just y plus 7 over 2. 111 00:05:46,920 --> 00:05:50,490 Notice again the connection between f inverse and f. 112 00:05:50,490 --> 00:05:52,460 How one undoes the other. 113 00:05:52,460 --> 00:05:55,240 In terms of our function machine idea, what we're 114 00:05:55,240 --> 00:06:00,370 saying is we may visualize the f machine whereby the input is 115 00:06:00,370 --> 00:06:04,220 x and the output will be twice x minus 7. 116 00:06:04,220 --> 00:06:06,420 In other words, the output will always be twice 117 00:06:06,420 --> 00:06:08,460 the input minus 7. 118 00:06:08,460 --> 00:06:11,990 Now the question that comes up is, suppose we reverse the 119 00:06:11,990 --> 00:06:14,600 roles of the output and the input. 120 00:06:14,600 --> 00:06:20,560 In other words, suppose now we let the input be y, what will 121 00:06:20,560 --> 00:06:22,160 the output be? 122 00:06:22,160 --> 00:06:25,440 If we reverse the terminal so to speak, what we have shown 123 00:06:25,440 --> 00:06:29,050 is that now the f inverse machine would be what? 124 00:06:29,050 --> 00:06:33,400 The input is y, the output is y plus 7 over 2. 125 00:06:33,400 --> 00:06:35,670 And by the way, a question that we shall come back to 126 00:06:35,670 --> 00:06:39,080 very shortly that plays a rather important role here and 127 00:06:39,080 --> 00:06:41,740 which I'll emphasize from another point of view is that 128 00:06:41,740 --> 00:06:46,440 if you get into the idea of always wanting to call the 129 00:06:46,440 --> 00:06:50,110 input x and the output y, which is how we get geared to 130 00:06:50,110 --> 00:06:53,130 do things in terms of calculus. 131 00:06:53,130 --> 00:06:57,430 x is always the horizontal axis, y the vertical axis, and 132 00:06:57,430 --> 00:07:00,120 we always agree to plot the independent 133 00:07:00,120 --> 00:07:01,560 variable along the x-axis. 134 00:07:01,560 --> 00:07:04,240 In other words, the input along the x-axis, the output 135 00:07:04,240 --> 00:07:05,520 along the y-axis. 136 00:07:05,520 --> 00:07:09,360 Then the question is, could we have called this x and called 137 00:07:09,360 --> 00:07:11,810 this x plus 7 over 2? 138 00:07:11,810 --> 00:07:14,480 And we'll go with this in more detail in a little while. 139 00:07:14,480 --> 00:07:18,640 But obviously, what we call the name of the input should 140 00:07:18,640 --> 00:07:21,380 not affect how the machine behaves. 141 00:07:21,380 --> 00:07:24,870 By the way, as a little aside, I thought it might be 142 00:07:24,870 --> 00:07:29,100 interesting to show why we use such notation as f inverse. 143 00:07:29,100 --> 00:07:30,600 f to the minus 1. 144 00:07:30,600 --> 00:07:32,170 It's rather interesting here. 145 00:07:32,170 --> 00:07:37,320 Let's suppose we let a number go into the f machine. 146 00:07:37,320 --> 00:07:38,810 Call that number c. 147 00:07:38,810 --> 00:07:42,120 Notice that any number that goes into the f machine has as 148 00:07:42,120 --> 00:07:46,470 its output twice that number minus 7. 149 00:07:46,470 --> 00:07:51,470 Suppose we now let that number be the input of the f inverse 150 00:07:51,470 --> 00:07:54,950 machine, what does the f inverse machine do? 151 00:07:54,950 --> 00:08:03,030 It adds 7 onto any input and then divides that result by 2. 152 00:08:03,030 --> 00:08:07,420 In other words, if we now run 2c minus 7 through the f 153 00:08:07,420 --> 00:08:13,360 inverse machine, we have 2c minus 7 plus 7 154 00:08:13,360 --> 00:08:16,480 over 2 equals c. 155 00:08:16,480 --> 00:08:20,260 In other words, notice how the f inverse machine 156 00:08:20,260 --> 00:08:22,420 undoes the f machine. 157 00:08:22,420 --> 00:08:26,370 If we wanted to use the language of last time in terms 158 00:08:26,370 --> 00:08:30,250 of composition of functions, what we do is what? 159 00:08:30,250 --> 00:08:36,980 What we're saying is that if you combine f followed by f 160 00:08:36,980 --> 00:08:42,980 inverse, f inverse following f, that that gives you what we 161 00:08:42,980 --> 00:08:47,000 can call the identity function. 162 00:08:47,000 --> 00:08:48,400 The identity function. 163 00:08:48,400 --> 00:08:53,310 Namely, if the input is c, the output will again be c. 164 00:08:53,310 --> 00:09:00,610 In other words, f inverse of f of c is just c back again. 165 00:09:00,610 --> 00:09:04,090 In a similar way, notice that we can reverse these roles. 166 00:09:04,090 --> 00:09:07,290 We saw last time that composition of functions 167 00:09:07,290 --> 00:09:09,620 depends on which order you combine the functions. 168 00:09:09,620 --> 00:09:13,500 But notice that if you run d through the f inverse machine, 169 00:09:13,500 --> 00:09:16,470 the output will be d plus 7 over 2. 170 00:09:16,470 --> 00:09:19,900 If this becomes the input of the f machine, remember what 171 00:09:19,900 --> 00:09:21,290 the f machine does. 172 00:09:21,290 --> 00:09:23,740 It doubles the input and subtracts 7. 173 00:09:26,480 --> 00:09:32,190 In other words, again, f of f inverse of d 174 00:09:32,190 --> 00:09:34,250 gives me d back again. 175 00:09:34,250 --> 00:09:37,600 In other words, in terms of composition of functions, f 176 00:09:37,600 --> 00:09:41,200 followed by f inverse or f inverse followed by f is what 177 00:09:41,200 --> 00:09:43,520 we call the identity function. 178 00:09:43,520 --> 00:09:46,220 That one is truly the inverse of the other from that 179 00:09:46,220 --> 00:09:47,750 particular point of view. 180 00:09:47,750 --> 00:09:51,120 However, let's correlate what we're talking about now with 181 00:09:51,120 --> 00:09:53,960 the circle diagrams that we used in our last lecture. 182 00:09:57,680 --> 00:10:01,200 You see, first of all, let's recall that unless our 183 00:10:01,200 --> 00:10:05,270 function is both 1:1 and onto, we do not 184 00:10:05,270 --> 00:10:07,130 have an inverse function. 185 00:10:07,130 --> 00:10:12,490 Namely, for example, if our function had not been onto, 186 00:10:12,490 --> 00:10:13,800 then when we-- 187 00:10:13,800 --> 00:10:14,900 see, here's the idea again. 188 00:10:14,900 --> 00:10:16,400 Let me make sure this is clear. 189 00:10:16,400 --> 00:10:20,270 To get an inverse function, essentially all we do is this. 190 00:10:20,270 --> 00:10:23,680 If f is a function from A to B, the inverse function is 191 00:10:23,680 --> 00:10:27,530 defined by reversing the input and the output. 192 00:10:27,530 --> 00:10:31,330 Which means in terms of this diagram, we reverse the sense 193 00:10:31,330 --> 00:10:32,340 of our arrows. 194 00:10:32,340 --> 00:10:35,430 We reverse which end the arrowhead goes on. 195 00:10:35,430 --> 00:10:38,490 And what we're saying is, if we had a function from A to B, 196 00:10:38,490 --> 00:10:41,860 which was not onto, then you see when we reverse the 197 00:10:41,860 --> 00:10:45,870 arrowheads, f is not defined on all of b. 198 00:10:45,870 --> 00:10:49,860 In other words, the domain of f, the domain of the inverse 199 00:10:49,860 --> 00:10:52,880 function, would not exist because it would not be 200 00:10:52,880 --> 00:10:54,490 defined on all of B. 201 00:10:54,490 --> 00:10:59,380 Secondly, if two different elements of A went into the 202 00:10:59,380 --> 00:11:02,840 same element of B when we reversed the arrowheads, the 203 00:11:02,840 --> 00:11:05,570 resulting function would not be single-valued. 204 00:11:05,570 --> 00:11:08,890 And hence, in terms of modern mathematics, it would not be a 205 00:11:08,890 --> 00:11:11,680 well defined function. 206 00:11:11,680 --> 00:11:15,290 So in other words, for the inverse to exist it must be 207 00:11:15,290 --> 00:11:19,590 that the original function is both 1:1 and onto. 208 00:11:19,590 --> 00:11:22,790 And as an example of that, this is what this diagram here 209 00:11:22,790 --> 00:11:23,780 represents. 210 00:11:23,780 --> 00:11:26,140 And to make sure that we can read this all right, I have 211 00:11:26,140 --> 00:11:29,440 singled out a typical element of capital A, a typical 212 00:11:29,440 --> 00:11:32,740 element of capital B. Remember what our notation is. 213 00:11:32,740 --> 00:11:36,390 The notation is that f of a equals b. 214 00:11:36,390 --> 00:11:39,110 That the image of a under f is b. 215 00:11:39,110 --> 00:11:42,700 And to use that in terms of the inverse language, if I 216 00:11:42,700 --> 00:11:45,340 called g the function that I get when I reversed the 217 00:11:45,340 --> 00:11:48,140 arrowheads, g of b equals a. 218 00:11:48,140 --> 00:11:50,870 And g is what I'm calling f inverse. 219 00:11:50,870 --> 00:11:55,200 By way of further review, the domain of f is equal to the 220 00:11:55,200 --> 00:11:56,780 image of f inverse. 221 00:11:56,780 --> 00:11:59,750 And that's A. The image of f is the same as 222 00:11:59,750 --> 00:12:01,820 the domain of f inverse. 223 00:12:01,820 --> 00:12:03,180 And that's B. 224 00:12:03,180 --> 00:12:05,300 And now, what the question is, is this. 225 00:12:05,300 --> 00:12:08,850 Notice that as long as you want to use the same diagram, 226 00:12:08,850 --> 00:12:13,180 all we have to do to express f inverse in terms of f is sort 227 00:12:13,180 --> 00:12:15,350 of to reverse the arrowheads. 228 00:12:15,350 --> 00:12:19,150 The question that comes up is, suppose you insist that the 229 00:12:19,150 --> 00:12:21,550 domain be listed first. 230 00:12:21,550 --> 00:12:25,030 In other words, when we're going to talk about g or f 231 00:12:25,030 --> 00:12:29,890 inverse in this case, that's a function from B to A. So why 232 00:12:29,890 --> 00:12:31,720 don't we list B first 233 00:12:31,720 --> 00:12:33,640 And you see again, we can do this. 234 00:12:33,640 --> 00:12:38,050 Here are the elements of B, here are the elements of A. 235 00:12:38,050 --> 00:12:42,250 And all we have to do is see what happens over here. 236 00:12:42,250 --> 00:12:45,510 For example, if we come back to here notice that the first 237 00:12:45,510 --> 00:12:50,830 element listed in B comes from the third element listed in A. 238 00:12:50,830 --> 00:12:55,120 So when I make up the inverse function, I just capitalize on 239 00:12:55,120 --> 00:12:57,740 this by writing the same thing. 240 00:12:57,740 --> 00:12:59,290 The only problem is-- 241 00:12:59,290 --> 00:13:01,430 and this is going to become a crucial one-- 242 00:13:01,430 --> 00:13:04,880 is the fact that if somehow or other you couldn't see these 243 00:13:04,880 --> 00:13:08,640 labels, if you couldn't see these labels and all you knew 244 00:13:08,640 --> 00:13:11,170 was that the first set was called the domain and the 245 00:13:11,170 --> 00:13:13,090 second set was called the image. 246 00:13:13,090 --> 00:13:17,760 If you now looked at these two functions, you see they 247 00:13:17,760 --> 00:13:20,470 wouldn't look anything at all alike. 248 00:13:20,470 --> 00:13:23,910 In other words, f and f inverse, while not independent 249 00:13:23,910 --> 00:13:26,330 of one another, do look quite different. 250 00:13:26,330 --> 00:13:30,370 For example, notice that f inverse causes the first two 251 00:13:30,370 --> 00:13:33,320 elements in here to sort of crisscross as they have 252 00:13:33,320 --> 00:13:37,070 images, and the second two elements of here crisscross. 253 00:13:37,070 --> 00:13:40,870 Notice though in terms of f, it's the second and third that 254 00:13:40,870 --> 00:13:44,570 crisscross and the first and fourth that don't intersect at 255 00:13:44,570 --> 00:13:45,060 all this way. 256 00:13:45,060 --> 00:13:48,570 In other words, if you look at this curve or this diagram and 257 00:13:48,570 --> 00:13:51,870 compare it with this diagram, notice that there is a 258 00:13:51,870 --> 00:13:54,800 difference in what seems to be going on. 259 00:13:54,800 --> 00:13:57,340 Well again, this is quite abstract. 260 00:13:57,340 --> 00:14:01,020 Let's try to relate this as much as possible to the 261 00:14:01,020 --> 00:14:04,780 language of calculus and our coordinate geometry graphing 262 00:14:04,780 --> 00:14:05,790 techniques. 263 00:14:05,790 --> 00:14:10,080 To begin with, let's suppose that we have a function f 264 00:14:10,080 --> 00:14:13,690 whose domain is the closed interval from a to b and whose 265 00:14:13,690 --> 00:14:17,390 range is the closed interval from to c to d. 266 00:14:17,390 --> 00:14:20,300 And the question that we'd like to raise is, under what 267 00:14:20,300 --> 00:14:23,580 conditions will f possess an inverse function? 268 00:14:23,580 --> 00:14:24,930 What does onto mean? 269 00:14:24,930 --> 00:14:27,860 What does 1:1 mean and things of this type? 270 00:14:27,860 --> 00:14:30,720 Well, the first thing I'd like to point out is that if the 271 00:14:30,720 --> 00:14:34,780 graph y equals f of x looks something like this. 272 00:14:34,780 --> 00:14:37,690 See, notice that the domain is from a to b. 273 00:14:37,690 --> 00:14:40,410 The image is from c to d. 274 00:14:40,410 --> 00:14:43,600 Notice the fact that we have a break in the curve over here, 275 00:14:43,600 --> 00:14:46,470 tells us that our function is not onto. 276 00:14:46,470 --> 00:14:51,730 Namely, given this number p, which is in our image between 277 00:14:51,730 --> 00:14:55,000 c and d, there is no element of the domain 278 00:14:55,000 --> 00:14:56,720 that maps into p. 279 00:14:56,720 --> 00:15:00,090 So, in other words, if there is a break in the curve, the 280 00:15:00,090 --> 00:15:01,950 function is not onto and hence, it 281 00:15:01,950 --> 00:15:04,800 will not have an inverse. 282 00:15:04,800 --> 00:15:07,530 Now suppose there is no break in the curve. 283 00:15:07,530 --> 00:15:10,870 Let's suppose now that the curve doubles back. 284 00:15:10,870 --> 00:15:13,230 It comes up and doubles back. 285 00:15:13,230 --> 00:15:17,030 Now what my claim is is that the function will not be 1:1. 286 00:15:17,030 --> 00:15:18,770 Well, how can we see that? 287 00:15:18,770 --> 00:15:22,100 Pick any part where the curve doubles back, pick a point 288 00:15:22,100 --> 00:15:25,100 like this in that range. 289 00:15:25,100 --> 00:15:26,760 Call that point q. 290 00:15:26,760 --> 00:15:31,290 Noticed that q is in the proper range of f now. 291 00:15:31,290 --> 00:15:33,590 y equals f of x. 292 00:15:33,590 --> 00:15:38,190 Now the question is, given the y value of q, are there any 293 00:15:38,190 --> 00:15:41,410 x-values that map into q under y? 294 00:15:41,410 --> 00:15:43,630 And the answer is yes, there are. 295 00:15:43,630 --> 00:15:46,560 In fact, there are more than one. 296 00:15:46,560 --> 00:15:51,830 Namely, notice that both f of x1 and f of x2 equal q. 297 00:15:51,830 --> 00:15:56,950 In other words, in this case, f of x1 equals f of x2, even 298 00:15:56,950 --> 00:16:01,080 though x1 is unequal to x2. 299 00:16:01,080 --> 00:16:04,090 That means that this function is not 1:1. 300 00:16:04,090 --> 00:16:07,550 And because it's not 1:1, it doesn't have a well defined 301 00:16:07,550 --> 00:16:09,220 inverse function. 302 00:16:09,220 --> 00:16:12,630 Well, putting these two cases together, what it means for 303 00:16:12,630 --> 00:16:15,270 the function to be onto, what it means for the function to 304 00:16:15,270 --> 00:16:20,870 be 1:1, it turns out that if our curve is unbroken, then 305 00:16:20,870 --> 00:16:25,310 the only way our function can have an inverse function is 306 00:16:25,310 --> 00:16:29,350 that the curve must either always be rising or always be 307 00:16:29,350 --> 00:16:32,480 falling, and it can't have a break in it. 308 00:16:32,480 --> 00:16:36,020 And by the way, as an aside, let me point out here the 309 00:16:36,020 --> 00:16:38,750 difference between a continuous variable-- 310 00:16:38,750 --> 00:16:42,320 meaning one that's defined on a whole interval-- 311 00:16:42,320 --> 00:16:43,600 and a discrete variable-- 312 00:16:43,600 --> 00:16:46,030 meaning where you get isolated pieces of data. 313 00:16:46,030 --> 00:16:50,350 Notice, for example, if you plot y versus x the way we do 314 00:16:50,350 --> 00:16:53,480 in a lab experiment where for a particular value of x, you 315 00:16:53,480 --> 00:16:55,170 measure a value of y. 316 00:16:55,170 --> 00:17:00,670 Notice that the data can double back without the 317 00:17:00,670 --> 00:17:03,850 function being multi-valued. 318 00:17:03,850 --> 00:17:06,550 In other words, notice for example, that even though the 319 00:17:06,550 --> 00:17:08,109 curve doubles back here-- 320 00:17:08,109 --> 00:17:09,560 I can't call it a curve. 321 00:17:09,560 --> 00:17:11,060 The data doubles back. 322 00:17:11,060 --> 00:17:14,869 Notice, for example, that no two different pieces of data 323 00:17:14,869 --> 00:17:16,990 have the same y-coordinate. 324 00:17:16,990 --> 00:17:20,760 In other words, given this point here as being q, there 325 00:17:20,760 --> 00:17:23,280 is only one piece of data that has its 326 00:17:23,280 --> 00:17:25,319 y-coordinate equal to q. 327 00:17:25,319 --> 00:17:29,570 However, of course, keep in mind it is possible that 328 00:17:29,570 --> 00:17:32,330 another piece of data will have the same coordinate. 329 00:17:32,330 --> 00:17:35,440 All I'm saying is that the idea of whether the curve 330 00:17:35,440 --> 00:17:39,130 always has to be rising or following certainly depends on 331 00:17:39,130 --> 00:17:41,120 whether you have a continuous curve or not. 332 00:17:41,120 --> 00:17:45,820 Well, again, let's continue on with what inverse functions 333 00:17:45,820 --> 00:17:47,030 are all about. 334 00:17:47,030 --> 00:17:50,970 You see, this comes up with our whole idea of why do we 335 00:17:50,970 --> 00:17:54,670 make fun or why do we minimize single-valued 336 00:17:54,670 --> 00:17:56,770 functions in calculus? 337 00:17:56,770 --> 00:18:00,380 And the answer is that single-valued-- 338 00:18:00,380 --> 00:18:00,850 I'm sorry. 339 00:18:00,850 --> 00:18:04,490 Why do we always stick to single-valued functions and do 340 00:18:04,490 --> 00:18:06,500 away with multi-valued functions? 341 00:18:06,500 --> 00:18:09,410 And the answer is if you have a smooth curve, we can always 342 00:18:09,410 --> 00:18:12,900 break down a multi-valued function into a union of 343 00:18:12,900 --> 00:18:14,370 single-valued functions. 344 00:18:14,370 --> 00:18:18,240 For example, if we take the curve c here as being y equals 345 00:18:18,240 --> 00:18:20,680 f of x, which plots like this. 346 00:18:20,680 --> 00:18:25,640 Notice if I take the points at which I have vertical tangents 347 00:18:25,640 --> 00:18:27,970 and break the curve up at those particular points. 348 00:18:27,970 --> 00:18:29,500 In this case, I'll get what curves? 349 00:18:29,500 --> 00:18:31,910 c1, c2, and c3. 350 00:18:31,910 --> 00:18:37,070 Notice that c is the union of c1, c2, and c3, but that each 351 00:18:37,070 --> 00:18:43,320 of the curves c1, c2, and c3 are either always rising or 352 00:18:43,320 --> 00:18:44,940 always falling. 353 00:18:44,940 --> 00:18:50,720 In a similar way, when we have a function which doubles 354 00:18:50,720 --> 00:18:54,660 back-- and by the way, notice what the connection is between 355 00:18:54,660 --> 00:18:57,870 multi-valued and not 1:1. 356 00:18:57,870 --> 00:19:01,110 You see, notice that in terms of a function versus its 357 00:19:01,110 --> 00:19:07,040 inverse function idea, that if a function is multi-valued the 358 00:19:07,040 --> 00:19:11,140 inverse function cannot be 1:1. 359 00:19:11,140 --> 00:19:13,960 In other words, the idea being that when you interchange the 360 00:19:13,960 --> 00:19:18,540 domain and the range, sort of the curve flips over idea, all 361 00:19:18,540 --> 00:19:20,660 I want you to see here is that what? 362 00:19:20,660 --> 00:19:24,720 If you're given a function which is not single-valued, if 363 00:19:24,720 --> 00:19:27,960 we take the points at which horizontal tangents occur and 364 00:19:27,960 --> 00:19:34,980 break down the curve like this, we can break the curve 365 00:19:34,980 --> 00:19:38,840 down into a union of 1:1 functions. 366 00:19:38,840 --> 00:19:41,940 The hardship being of course, that when you start with a 367 00:19:41,940 --> 00:19:44,900 point like this, analytically speaking, it's rather 368 00:19:44,900 --> 00:19:48,170 difficult unless you invent some scheme to know which of 369 00:19:48,170 --> 00:19:52,340 the points here you want to single out. 370 00:19:52,340 --> 00:19:55,090 In terms of our previous experience, it's sort of like 371 00:19:55,090 --> 00:19:58,320 saying to a person, I am thinking of the angle whose 372 00:19:58,320 --> 00:19:59,820 sine is 1/2. 373 00:19:59,820 --> 00:20:03,010 There are, you see, infinitely many functions whose sine is 374 00:20:03,010 --> 00:20:04,760 equal to 1/2. 375 00:20:04,760 --> 00:20:08,210 Of course if we say to the person, I am thinking of the 376 00:20:08,210 --> 00:20:12,200 angle whose sine is 1/2 and the angle is between minus 90 377 00:20:12,200 --> 00:20:16,690 degrees and plus 90 degrees, then the only possible answer 378 00:20:16,690 --> 00:20:19,050 is the angle must be 30 degrees. 379 00:20:19,050 --> 00:20:21,740 But notice that when you have a function which is not 380 00:20:21,740 --> 00:20:24,420 single-valued, the inverse will be a 381 00:20:24,420 --> 00:20:25,690 multi-valued function. 382 00:20:25,690 --> 00:20:28,180 And we'll talk more about that in a little while. 383 00:20:28,180 --> 00:20:31,340 Again, I just want to keep this shotgun approach going on 384 00:20:31,340 --> 00:20:35,280 just what an inverse function is in relationship to the 385 00:20:35,280 --> 00:20:36,560 function itself. 386 00:20:36,560 --> 00:20:39,780 Again, let's look at this more abstractly. 387 00:20:39,780 --> 00:20:42,710 Here I have drawn a curve which is 388 00:20:42,710 --> 00:20:44,950 continuous and always rising. 389 00:20:44,950 --> 00:20:47,890 So I can talk about the inverse function. 390 00:20:47,890 --> 00:20:50,710 If the equation is y equals f of x, the 391 00:20:50,710 --> 00:20:52,110 inverse is written what? 392 00:20:52,110 --> 00:20:54,610 x equals f inverse y. 393 00:20:54,610 --> 00:20:57,920 And if this seems a little bit too abstract for you, think of 394 00:20:57,920 --> 00:21:00,260 a concrete representation. 395 00:21:00,260 --> 00:21:02,600 Suppose the curve happened to represent y 396 00:21:02,600 --> 00:21:04,570 equals 10 to the x. 397 00:21:04,570 --> 00:21:07,180 Then another way of saying the same thing would be 398 00:21:07,180 --> 00:21:09,300 x equals log y. 399 00:21:09,300 --> 00:21:11,600 The convention of course here being that you don't usually 400 00:21:11,600 --> 00:21:13,000 write base 10. 401 00:21:13,000 --> 00:21:14,850 But we won't worry about that. 402 00:21:14,850 --> 00:21:16,080 You see, this is what? 403 00:21:16,080 --> 00:21:21,500 Two different ways of expressing the same curve. 404 00:21:21,500 --> 00:21:26,210 Whether I write y equals f of x or x equals f inverse y, I 405 00:21:26,210 --> 00:21:29,010 have the same curve this way. 406 00:21:29,010 --> 00:21:32,380 In terms of our arrows, you see what I'm saying is, if I 407 00:21:32,380 --> 00:21:36,760 start with x1, by going this way, my function determines 408 00:21:36,760 --> 00:21:38,690 the output y1. 409 00:21:38,690 --> 00:21:43,440 Inversely, if I start with y1 and reverse the arrows, I wind 410 00:21:43,440 --> 00:21:44,830 up with x1. 411 00:21:44,830 --> 00:21:48,140 Again, the basic difference being as to which of the two 412 00:21:48,140 --> 00:21:50,300 variables is being emphasized. 413 00:21:50,300 --> 00:21:53,910 What the real problem is, is that most people say look it. 414 00:21:53,910 --> 00:21:56,300 I'm not used to studying curves this way. 415 00:21:56,300 --> 00:22:00,000 I'm not used to looking at the input being along the vertical 416 00:22:00,000 --> 00:22:03,820 axis and the output along the horizontal axis according to 417 00:22:03,820 --> 00:22:05,860 the way I've been trained when we're 418 00:22:05,860 --> 00:22:07,750 studying the inverse function. 419 00:22:07,750 --> 00:22:11,640 In other words, when y is the input, aren't we used to 420 00:22:11,640 --> 00:22:16,140 having y over here and then plotting the 421 00:22:16,140 --> 00:22:18,450 output along this axis? 422 00:22:18,450 --> 00:22:21,540 In other words, the question is given this graph, how do 423 00:22:21,540 --> 00:22:24,240 you arrive at this one? 424 00:22:24,240 --> 00:22:27,450 You see, somehow or other, let's observe that if all you 425 00:22:27,450 --> 00:22:30,990 did was switch your orientation and say let me 426 00:22:30,990 --> 00:22:34,890 switch this by 90 degrees, notice that we would be in a 427 00:22:34,890 --> 00:22:36,080 little bit of trouble. 428 00:22:36,080 --> 00:22:39,230 In other words, if we start with this kind of a set up and 429 00:22:39,230 --> 00:22:43,790 we say, let's rotate through a positive 90 degrees. 430 00:22:43,790 --> 00:22:46,930 Notice now what we would wind up with is what? 431 00:22:46,930 --> 00:22:51,170 Our x-axis would be the way we want it, but the y-axis would 432 00:22:51,170 --> 00:22:54,620 now have the opposite sense of what we usually want our input 433 00:22:54,620 --> 00:22:55,900 axis to look like. 434 00:22:55,900 --> 00:22:59,840 So after we rotate through 90 degrees, it would seem that 435 00:22:59,840 --> 00:23:01,560 the next step is to do what? 436 00:23:04,200 --> 00:23:10,190 Flip with respect to the x-axis. 437 00:23:10,190 --> 00:23:12,130 That means fold this thing over. 438 00:23:12,130 --> 00:23:16,660 In other words, a 90 degree rotation followed by a folding 439 00:23:16,660 --> 00:23:22,840 over gives me the orientation if I insist that the input has 440 00:23:22,840 --> 00:23:29,420 to be along the horizontal axis and the output along the 441 00:23:29,420 --> 00:23:30,380 vertical axis. 442 00:23:30,380 --> 00:23:33,180 What I want you to also notice though, is that if we don't 443 00:23:33,180 --> 00:23:37,390 insist on this, there is no reason why we have to use two 444 00:23:37,390 --> 00:23:38,710 separate diagrams. 445 00:23:38,710 --> 00:23:41,230 Again notice, these are two different ways of 446 00:23:41,230 --> 00:23:43,070 giving the same -- 447 00:23:43,070 --> 00:23:46,040 two different equations for giving the same curve. 448 00:23:46,040 --> 00:23:49,810 It's only when we want to switch the role and make sure 449 00:23:49,810 --> 00:23:54,930 that the input is along the horizontal axis that we have 450 00:23:54,930 --> 00:23:58,210 to go through this kind of a process. 451 00:23:58,210 --> 00:24:00,410 Let's look at this a little bit more concretely. 452 00:24:03,500 --> 00:24:06,170 What I call a semi-concrete illustration. 453 00:24:06,170 --> 00:24:08,960 What I'm saying now is let's suppose this is the curve I've 454 00:24:08,960 --> 00:24:11,040 drawn in here, y equals f of x. 455 00:24:11,040 --> 00:24:15,170 Another way of saying that is x equals f inverse of y. 456 00:24:15,170 --> 00:24:18,450 And the question is, suppose I now want to plot this same 457 00:24:18,450 --> 00:24:22,780 curve, same equation, but now with the y-axis as my 458 00:24:22,780 --> 00:24:24,220 horizontal axis. 459 00:24:24,220 --> 00:24:28,330 You see again, in terms of geometry, how I shift my axes 460 00:24:28,330 --> 00:24:30,710 will not change this equation. 461 00:24:30,710 --> 00:24:33,560 But what the picture of this equation looks like will 462 00:24:33,560 --> 00:24:36,770 certainly depend on how I orient my axes. 463 00:24:36,770 --> 00:24:38,170 So the idea is what? 464 00:24:38,170 --> 00:24:40,600 I simply fold this, rotate this 465 00:24:40,600 --> 00:24:42,770 thing, through 90 degrees. 466 00:24:42,770 --> 00:24:44,450 And if I do that, the resulting 467 00:24:44,450 --> 00:24:46,110 picture looks like this. 468 00:24:46,110 --> 00:24:48,170 And once the picture looks like this, the 469 00:24:48,170 --> 00:24:49,460 next step is what? 470 00:24:49,460 --> 00:24:51,650 Flip this with respect to the x-axis. 471 00:24:54,360 --> 00:24:56,510 And now my picture looks like this. 472 00:24:56,510 --> 00:25:01,750 In other words, x equals f inverse of y here and x equals 473 00:25:01,750 --> 00:25:05,340 f inverse of y here are the same equation. 474 00:25:05,340 --> 00:25:08,410 The reason that the picture looks different is because I 475 00:25:08,410 --> 00:25:11,080 didn't allow myself to use this as the 476 00:25:11,080 --> 00:25:12,340 axis of inputs here. 477 00:25:12,340 --> 00:25:18,490 In other words, again, as soon as I wanted to make this axis 478 00:25:18,490 --> 00:25:21,280 orient so it would be the horizontal axis, this is what 479 00:25:21,280 --> 00:25:23,300 I had to go through over here. 480 00:25:23,300 --> 00:25:26,060 Now you see, the next refinement is that a person 481 00:25:26,060 --> 00:25:29,640 says look it, I'm not used to calling this the y-axis. 482 00:25:29,640 --> 00:25:30,640 What's in a name? 483 00:25:30,640 --> 00:25:33,790 Why don't we always agree to call the horizontal axis the 484 00:25:33,790 --> 00:25:37,370 x-axis and the vertical axis the y-axis? 485 00:25:37,370 --> 00:25:40,520 And if I agree to do that, notice what happens just by 486 00:25:40,520 --> 00:25:43,260 changing the names of the variables. 487 00:25:43,260 --> 00:25:48,180 All that happens is, is that now this becomes y equals f 488 00:25:48,180 --> 00:25:49,440 inverse of x. 489 00:25:49,440 --> 00:25:52,770 This is an important thing to notice then. 490 00:25:52,770 --> 00:25:55,420 In other words, if you insist that the horizontal axis in 491 00:25:55,420 --> 00:25:59,220 both cases will be called the x-axis and the vertical axis 492 00:25:59,220 --> 00:26:04,250 the y-axis, then this would be the curve y equals f of x and 493 00:26:04,250 --> 00:26:08,130 this would be the curve y equals f inverse of x. 494 00:26:08,130 --> 00:26:11,110 But again, the whole thing comes about only when you 495 00:26:11,110 --> 00:26:14,750 insist on how you want your axes oriented. 496 00:26:14,750 --> 00:26:18,830 Let's go back to our problem of y equals 2x minus 7 and see 497 00:26:18,830 --> 00:26:21,140 what this thing means in terms of a graph. 498 00:26:21,140 --> 00:26:26,550 As we saw previously, if y equals 2x minus 7, x is equal 499 00:26:26,550 --> 00:26:30,000 to y plus 7 over 2. 500 00:26:30,000 --> 00:26:31,250 And the idea is what? 501 00:26:31,250 --> 00:26:33,160 Let's see what this thing really means. 502 00:26:33,160 --> 00:26:36,950 If I plot the straight line y equals 2x minus 7, this is the 503 00:26:36,950 --> 00:26:38,880 line that I get. 504 00:26:38,880 --> 00:26:41,650 Notice that as long as I'm going to use the same 505 00:26:41,650 --> 00:26:45,270 orientation of axes here, it makes no difference whether I 506 00:26:45,270 --> 00:26:49,560 call this line y equals 2x minus 7 or whether I call it x 507 00:26:49,560 --> 00:26:51,650 equals y plus 7 over 2. 508 00:26:51,650 --> 00:26:54,450 They are two different names for the same line. 509 00:26:54,450 --> 00:26:59,440 The problem occurs when I insist that the independent 510 00:26:59,440 --> 00:27:03,890 variable always be plotted along the x-axis, the 511 00:27:03,890 --> 00:27:07,040 horizontal axis, and the dependent variable along the 512 00:27:07,040 --> 00:27:08,070 vertical axis. 513 00:27:08,070 --> 00:27:11,170 Again, going through what we did before, I first take this 514 00:27:11,170 --> 00:27:15,790 thing and I rotate it through a positive 90 degrees. 515 00:27:15,790 --> 00:27:20,360 That takes this picture and transforms it into this one. 516 00:27:20,360 --> 00:27:24,600 I now take this and I flip it with respect to the x-axis, 517 00:27:24,600 --> 00:27:28,020 and that gives me this picture here. 518 00:27:28,020 --> 00:27:30,380 Now what is this line here? 519 00:27:30,380 --> 00:27:33,440 It's x equals y plus 7 over 2. 520 00:27:33,440 --> 00:27:36,880 Again, this is the same equation as this one. 521 00:27:36,880 --> 00:27:39,850 The reason that the pictures look differently is the fact 522 00:27:39,850 --> 00:27:43,190 that we have changed the orientation of the axis. 523 00:27:43,190 --> 00:27:47,810 Again, if we now say OK, let's rename this the x-axis, let's 524 00:27:47,810 --> 00:27:53,020 rename this the y-axis, then this becomes what? y equals x 525 00:27:53,020 --> 00:27:54,940 plus 7 over 2. 526 00:27:54,940 --> 00:27:58,840 It's in this sense that we call this curve of this 527 00:27:58,840 --> 00:28:02,760 equation and this equation here, that these two equations 528 00:28:02,760 --> 00:28:05,580 are inverses of one another. 529 00:28:05,580 --> 00:28:08,970 Again, in terms of what we said before, if you pick a 530 00:28:08,970 --> 00:28:12,780 particular value of x and compute y this way, then you 531 00:28:12,780 --> 00:28:15,530 apply this recipe to that. 532 00:28:15,530 --> 00:28:21,540 In other words, if you now take twice y, twice the output 533 00:28:21,540 --> 00:28:26,620 here, and subtract 7, your original input returns. 534 00:28:26,620 --> 00:28:28,120 In other words, this works exactly the 535 00:28:28,120 --> 00:28:29,830 same as we did before. 536 00:28:29,830 --> 00:28:33,230 But again, the whole basic difference is what? 537 00:28:33,230 --> 00:28:37,150 How you want to orient your axis. 538 00:28:37,150 --> 00:28:40,680 That the curves look different because your coordinate system 539 00:28:40,680 --> 00:28:42,660 is different. 540 00:28:42,660 --> 00:28:44,880 Of course, the interesting question now is, if we 541 00:28:44,880 --> 00:28:48,610 compared these two curves, since the-- see, granted that 542 00:28:48,610 --> 00:28:51,480 the function and its inverse are different functions, they 543 00:28:51,480 --> 00:28:52,740 are somewhat related. 544 00:28:52,740 --> 00:28:54,490 They're not random. 545 00:28:54,490 --> 00:28:56,370 How are these two graphs related? 546 00:28:56,370 --> 00:28:59,280 That might be the next natural question to ask. 547 00:28:59,280 --> 00:29:02,060 If we do that, the idea is simply this. 548 00:29:02,060 --> 00:29:06,060 Let's suppose we have y equals f of x as one of our curves. 549 00:29:06,060 --> 00:29:09,370 The curve happens to be invertible, meaning that f is 550 00:29:09,370 --> 00:29:12,820 always rising and it's unbroken, et cetera. 551 00:29:12,820 --> 00:29:17,700 The question is, if we now try to plot y equals f inverse of 552 00:29:17,700 --> 00:29:20,320 x in the same diagram. 553 00:29:20,320 --> 00:29:21,570 See notice now what I'm saying. 554 00:29:21,570 --> 00:29:24,070 In other words, I am not saying x equals f 555 00:29:24,070 --> 00:29:25,150 inverse of y here. 556 00:29:25,150 --> 00:29:29,060 I'm saying suppose I have the curve y equals f of x and also 557 00:29:29,060 --> 00:29:31,620 the curve y equals f inverse of x. 558 00:29:31,620 --> 00:29:35,780 How do these two curves look with respect to an x- and 559 00:29:35,780 --> 00:29:38,090 y-coordinate system? 560 00:29:38,090 --> 00:29:40,450 See, let me do that part more slowly again. 561 00:29:40,450 --> 00:29:42,700 Let me come over here for a moment. 562 00:29:42,700 --> 00:29:47,580 Notice that y equals 2x minus 7 was my original curve in 563 00:29:47,580 --> 00:29:49,010 what I dealt with before. 564 00:29:49,010 --> 00:29:52,200 If I want to keep the same orientation of axis, the 565 00:29:52,200 --> 00:29:56,370 inverse function we saw was y equals x plus 7 over 2. 566 00:29:56,370 --> 00:29:58,410 The question that we're asking quite in general, not in this 567 00:29:58,410 --> 00:30:02,040 specific case, is how are these two curves related? 568 00:30:02,040 --> 00:30:04,370 And the solution goes something like this. 569 00:30:04,370 --> 00:30:08,240 Let's suppose that the point x1, y1 belongs 570 00:30:08,240 --> 00:30:10,050 to the curve c1. 571 00:30:10,050 --> 00:30:15,690 By definition of c1, that says that y1 is f of x1. 572 00:30:15,690 --> 00:30:19,500 By definition of inverse function, see if y1 is f of 573 00:30:19,500 --> 00:30:22,510 x1, that means if you interchange the input and the 574 00:30:22,510 --> 00:30:24,620 output, that's another way of saying what? 575 00:30:24,620 --> 00:30:28,920 That x1 is f inverse of y1. 576 00:30:28,920 --> 00:30:36,810 In other words again, if f maps x1 into y1, f inverse 577 00:30:36,810 --> 00:30:40,230 maps y1 into x1, by definition. 578 00:30:40,230 --> 00:30:45,970 Now you see, compare this with the curve c2. 579 00:30:45,970 --> 00:30:48,530 See, this says what? 580 00:30:48,530 --> 00:30:54,690 That the input y1 maps into the output x1. 581 00:30:54,690 --> 00:30:59,380 In other words, notice that if you look at the f inverse 582 00:30:59,380 --> 00:31:04,740 situation here, when the input is y1, the output is x1. 583 00:31:04,740 --> 00:31:09,330 That's another way of saying that y1 comma x1 belongs to 584 00:31:09,330 --> 00:31:10,870 the curve c2. 585 00:31:10,870 --> 00:31:17,070 In other words, if x1, y1 belongs to y1 equals f of x1, 586 00:31:17,070 --> 00:31:22,430 then y1, x1 belongs to y equals f inverse of x. 587 00:31:22,430 --> 00:31:27,170 Now, what is the relationship between the point x1 comma y1 588 00:31:27,170 --> 00:31:30,080 and the point y1 comma x1? 589 00:31:30,080 --> 00:31:33,110 If we draw this little diagram, we observe that we 590 00:31:33,110 --> 00:31:36,010 have a couple of congruent triangles here. 591 00:31:36,010 --> 00:31:38,280 This length equals this length. 592 00:31:38,280 --> 00:31:41,650 This angle equals this angle. 593 00:31:41,650 --> 00:31:43,180 And this gives me a hint. 594 00:31:43,180 --> 00:31:48,020 This makes triangle OPQ isosceles. 595 00:31:48,020 --> 00:31:53,260 I draw the angle bisector of angle O. The angle bisector of 596 00:31:53,260 --> 00:31:56,200 an isosceles triangle is the perpendicular 597 00:31:56,200 --> 00:31:59,080 bisector of the base. 598 00:31:59,080 --> 00:32:01,350 And angle bisector of the vertex angle is the 599 00:32:01,350 --> 00:32:03,350 perpendicular bisector of the base. 600 00:32:03,350 --> 00:32:06,810 Well, you see that makes this angle equal to this angle. 601 00:32:06,810 --> 00:32:09,720 That makes this a 45 degree angle. 602 00:32:09,720 --> 00:32:11,860 In other words, the line that I've drawn is 603 00:32:11,860 --> 00:32:13,790 indeed, y equals x. 604 00:32:13,790 --> 00:32:18,100 And notice that P and Q are symmetrically located with 605 00:32:18,100 --> 00:32:20,780 respect to the line y equals x. 606 00:32:20,780 --> 00:32:25,070 In other words, going back to our original problem here, the 607 00:32:25,070 --> 00:32:31,140 curve c1 and c2 are related by the fact that they are mirror 608 00:32:31,140 --> 00:32:33,600 images of one another with respect to the 609 00:32:33,600 --> 00:32:35,560 line y equals x. 610 00:32:35,560 --> 00:32:38,120 That's exactly what I've drawn over here. 611 00:32:38,120 --> 00:32:40,760 In other words, going back to the problem of how are the 612 00:32:40,760 --> 00:32:45,540 curves y equal 2x minus 7 and y equal x plus 7 over 2 613 00:32:45,540 --> 00:32:49,250 related, the answer is they are mirror images of one 614 00:32:49,250 --> 00:32:52,090 another with respect to the line y equals x. 615 00:32:52,090 --> 00:32:55,210 They are symmetric with respect to that line. 616 00:32:55,210 --> 00:32:58,990 Now you see, let's talk about this from another point of 617 00:32:58,990 --> 00:33:01,440 view also, and show what the tough thing is. 618 00:33:01,440 --> 00:33:04,190 You see, so far my whole discussion seems to have 619 00:33:04,190 --> 00:33:07,810 hinged on the fact that we have a function, which is 620 00:33:07,810 --> 00:33:08,700 invertible. 621 00:33:08,700 --> 00:33:12,090 What if you have a function which is non-invertible? 622 00:33:12,090 --> 00:33:16,540 Going back to something more familiar, why do we, talk 623 00:33:16,540 --> 00:33:18,700 about -- when y equals the square root of x, why do we 624 00:33:18,700 --> 00:33:21,930 have this convention that we take the positive square root? 625 00:33:21,930 --> 00:33:25,520 After all, doesn't the square root of x and minus the square 626 00:33:25,520 --> 00:33:28,150 root of x have the property that when you square them you 627 00:33:28,150 --> 00:33:29,290 get the same result? 628 00:33:29,290 --> 00:33:32,250 Plus or minus squared is always plus. 629 00:33:32,250 --> 00:33:35,860 And the answer is that if you square both sides here and 630 00:33:35,860 --> 00:33:39,210 think of this as being the curve y squared equals x, what 631 00:33:39,210 --> 00:33:43,990 happens is you get a multi-valued function. 632 00:33:43,990 --> 00:33:47,730 One value of x yields two values of y. 633 00:33:47,730 --> 00:33:50,650 And the way we get around that is we break this curve down 634 00:33:50,650 --> 00:33:56,050 into two pieces, c1 and c2, where c1 is always rising. 635 00:33:56,050 --> 00:33:57,970 c2 is always falling here. 636 00:33:57,970 --> 00:34:00,560 In other words, we broke this thing off at the point of 637 00:34:00,560 --> 00:34:01,850 vertical tangency. 638 00:34:01,850 --> 00:34:04,990 And we can now think of this curve as being the union of 639 00:34:04,990 --> 00:34:05,850 two curves. 640 00:34:05,850 --> 00:34:09,080 One of which is y equals the positive square root of x and 641 00:34:09,080 --> 00:34:12,480 the other is y equals the negative square root of x. 642 00:34:12,480 --> 00:34:14,880 Now the question is, what happens when you have a 643 00:34:14,880 --> 00:34:16,820 function which is not single-valued. 644 00:34:16,820 --> 00:34:19,090 In other words, let's just invert this one. 645 00:34:19,090 --> 00:34:23,800 Let's suppose we started with the curve y equals x squared. 646 00:34:23,800 --> 00:34:28,050 You see, now for a given value of y, I'm in trouble. 647 00:34:28,050 --> 00:34:32,520 Because if y1 is positive, there are two different values 648 00:34:32,520 --> 00:34:36,300 of x which yield this particular result. 649 00:34:36,300 --> 00:34:40,139 In other words, both of these have the property that when 650 00:34:40,139 --> 00:34:42,460 you square them you get y1. 651 00:34:42,460 --> 00:34:46,440 And all we're saying is that in a problem such as this, we 652 00:34:46,440 --> 00:34:50,250 can study this curve as two separate pieces. 653 00:34:50,250 --> 00:34:52,440 Call one of these curves k1. 654 00:34:52,440 --> 00:34:55,560 That will be the curve y equals x squared, where x is 655 00:34:55,560 --> 00:34:56,659 non-negative. 656 00:34:56,659 --> 00:34:58,730 So this will be the curve k1. 657 00:34:58,730 --> 00:35:02,890 And call the other one k2, where k2 will be what? 658 00:35:02,890 --> 00:35:06,390 The same curve y equals x squared, but its domain is the 659 00:35:06,390 --> 00:35:07,900 negative values of x. 660 00:35:07,900 --> 00:35:10,680 In other words, k2 will be this one over here. 661 00:35:10,680 --> 00:35:13,620 And now the point is, if we deal with either of these two 662 00:35:13,620 --> 00:35:18,870 pieces separately, we can talk about inverse functions. 663 00:35:18,870 --> 00:35:22,690 Now the point is, which of these two halves do we use? 664 00:35:22,690 --> 00:35:25,150 And this is where the word principal values comes in. 665 00:35:25,150 --> 00:35:28,790 And you see what I'd like you to keep in mind is this, a 666 00:35:28,790 --> 00:35:30,770 little cliche I've written down here. 667 00:35:30,770 --> 00:35:35,220 It's called misinterpretation versus non-comprehension. 668 00:35:35,220 --> 00:35:38,060 If you don't understand what something means, there's no 669 00:35:38,060 --> 00:35:40,390 danger you're going to misinterpret it. 670 00:35:40,390 --> 00:35:42,500 The danger is when you think that you know what something 671 00:35:42,500 --> 00:35:45,240 means and you have the thing twisted around. 672 00:35:45,240 --> 00:35:46,370 You see, the idea is this. 673 00:35:46,370 --> 00:35:49,450 Let's go back to our old friend y equals sine x. 674 00:35:49,450 --> 00:35:53,290 Let's pick the value of y equal to 1/2 say. 675 00:35:53,290 --> 00:35:55,650 And now we say to the person the same problem as we asked 676 00:35:55,650 --> 00:35:59,470 before, find the angle whose sine is 1/2. 677 00:35:59,470 --> 00:36:02,200 Well, the point is to find that angle. 678 00:36:02,200 --> 00:36:05,350 If I draw this particular line, I can find all sorts of 679 00:36:05,350 --> 00:36:07,240 candidates. 680 00:36:07,240 --> 00:36:10,370 The point is that what we tried to do instead is to say, 681 00:36:10,370 --> 00:36:12,700 OK, well now restrict the function. 682 00:36:12,700 --> 00:36:17,280 We'll break this down to be a union of several curves. 683 00:36:17,280 --> 00:36:20,900 In other words, it'll be this curve union this one. 684 00:36:20,900 --> 00:36:22,090 Union this one. 685 00:36:22,090 --> 00:36:23,310 Union this one, et cetera. 686 00:36:23,310 --> 00:36:26,210 What do all of these separate pieces have in common? 687 00:36:26,210 --> 00:36:29,600 What they have in common is that what? 688 00:36:29,600 --> 00:36:35,740 They are onto the range from minus 1 to 1 and on that range 689 00:36:35,740 --> 00:36:38,610 they are also 1:1. 690 00:36:38,610 --> 00:36:39,490 1:1 and onto. 691 00:36:39,490 --> 00:36:44,280 Every value of y between minus 1 and 1 is taken on along each 692 00:36:44,280 --> 00:36:45,590 of these pieces. 693 00:36:45,590 --> 00:36:50,930 And no value occurs more than once on any of these pieces. 694 00:36:50,930 --> 00:36:54,360 The point is it's not so crucial whether you take this 695 00:36:54,360 --> 00:36:59,520 particular one or whether you take this particular piece. 696 00:36:59,520 --> 00:37:01,580 That's something that's sort of arbitrary. 697 00:37:01,580 --> 00:37:05,110 What we must do to avoid misinterpretation is unless 698 00:37:05,110 --> 00:37:08,280 otherwise specified we say look it, unless you hear from 699 00:37:08,280 --> 00:37:11,700 me to the contrary, let's always agree that this is the 700 00:37:11,700 --> 00:37:14,310 little piece of the curve that we're talking about, or this 701 00:37:14,310 --> 00:37:17,030 is the piece that we're talking about. 702 00:37:17,030 --> 00:37:18,530 But the idea being what? 703 00:37:18,530 --> 00:37:21,380 Unless you make such a restriction, we cannot talk 704 00:37:21,380 --> 00:37:22,700 about inverse functions. 705 00:37:22,700 --> 00:37:26,260 The idea being that for an inverse function to exist, we 706 00:37:26,260 --> 00:37:28,070 must be able to back map. 707 00:37:28,070 --> 00:37:32,090 We must be able to go from the value in the image to the 708 00:37:32,090 --> 00:37:35,540 value in the domain without any danger of 709 00:37:35,540 --> 00:37:37,890 misinterpretation. 710 00:37:37,890 --> 00:37:42,760 We can conclude our example with returning to our y equals 711 00:37:42,760 --> 00:37:44,420 x squared idea again. 712 00:37:44,420 --> 00:37:48,270 You see the idea is given the curve y equals x squared, we 713 00:37:48,270 --> 00:37:50,590 can think of it in terms of our pieces 714 00:37:50,590 --> 00:37:53,040 k2 and k1 as before. 715 00:37:53,040 --> 00:37:55,530 The accented piece being k2, the 716 00:37:55,530 --> 00:37:58,260 non-accented piece being k1. 717 00:37:58,260 --> 00:38:03,290 And what we're saying is the inverse of k1 is this curve 718 00:38:03,290 --> 00:38:06,480 here, which I'll call k1 inverse. 719 00:38:06,480 --> 00:38:11,240 The inverse of k2 is this curve here, which 720 00:38:11,240 --> 00:38:13,530 I'll call k2 inverse. 721 00:38:13,530 --> 00:38:15,980 In other words, the important thing is I can find the 722 00:38:15,980 --> 00:38:21,240 inverse of either this curve or this curve. 723 00:38:21,240 --> 00:38:22,800 And in fact, how do I do that? 724 00:38:22,800 --> 00:38:26,040 Again, with respect to the 45 degree line, the line y equals 725 00:38:26,040 --> 00:38:31,630 x, notice that k1 and k1 inverse are symmetric with 726 00:38:31,630 --> 00:38:33,880 respect to this 45 degree line. 727 00:38:33,880 --> 00:38:38,870 And similarly, so are k2 and k2 inverse. 728 00:38:38,870 --> 00:38:41,840 The thing I must be very careful about and this is 729 00:38:41,840 --> 00:38:45,320 where problems occur, is I must not confuse-- 730 00:38:45,320 --> 00:38:52,040 for example, what I can't do is take, for example, k2 and 731 00:38:52,040 --> 00:38:54,380 k1 inverse. 732 00:38:54,380 --> 00:38:56,960 Notice the built-in idea here. 733 00:38:56,960 --> 00:39:00,260 These two curves together are not symmetric with respect to 734 00:39:00,260 --> 00:39:01,990 the 45 degree line. 735 00:39:01,990 --> 00:39:04,590 You see what we're saying here is, is what? 736 00:39:04,590 --> 00:39:06,660 That for this particular curve, x 737 00:39:06,660 --> 00:39:08,790 and y are both positive. 738 00:39:08,790 --> 00:39:11,950 So obviously, anything that matches it must have x and y 739 00:39:11,950 --> 00:39:13,070 both positive. 740 00:39:13,070 --> 00:39:14,970 And that doesn't happen over here. 741 00:39:14,970 --> 00:39:16,860 What we're saying is you can't do these things 742 00:39:16,860 --> 00:39:18,200 completely at random. 743 00:39:18,200 --> 00:39:22,150 However, what you can do is either take k1 and match that 744 00:39:22,150 --> 00:39:26,520 with k1 inverse, k2 and match that with k2 inverse. 745 00:39:26,520 --> 00:39:30,470 It's not important which of the two ways you do this, as 746 00:39:30,470 --> 00:39:34,010 long as you understand that there is a danger of getting 747 00:39:34,010 --> 00:39:37,680 mixed up once the curve itself is not 1:1. 748 00:39:37,680 --> 00:39:40,430 In other words, when we break the curve down into 1:1 749 00:39:40,430 --> 00:39:42,930 pieces, we have to make sure that we match 750 00:39:42,930 --> 00:39:44,540 these things up properly. 751 00:39:44,540 --> 00:39:47,630 Now, this is all we're going to say about inverse functions 752 00:39:47,630 --> 00:39:49,290 for the time being. 753 00:39:49,290 --> 00:39:52,690 The rest will be taken care of in the exercises in this unit. 754 00:39:52,690 --> 00:39:56,540 However, we will return to this point very, very strongly 755 00:39:56,540 --> 00:39:59,500 later in our discussion of calculus. 756 00:39:59,500 --> 00:40:02,250 The important point to remember is that a function 757 00:40:02,250 --> 00:40:05,820 and its inverse function give us two different ways of 758 00:40:05,820 --> 00:40:08,260 expressing the same information. 759 00:40:08,260 --> 00:40:10,840 And that we can use whichever one happens 760 00:40:10,840 --> 00:40:13,060 to be to our advantage. 761 00:40:13,060 --> 00:40:15,200 Well, until next time, goodbye. 762 00:40:18,020 --> 00:40:20,550 ANNOUNCER: Funding for the publication of this video was 763 00:40:20,550 --> 00:40:25,260 provided by the Gabriella and Paul Rosenbaum Foundation. 764 00:40:25,260 --> 00:40:29,440 Help OCW continue to provide free and open access to MIT 765 00:40:29,440 --> 00:40:33,640 courses by making a donation at ocw.mit.edu/donate.