1 00:00:01,010 --> 00:00:05,160 In this video, we'll define the inverse, the function F. 2 00:00:05,970 --> 00:00:12,522 Let's suppose we have a function F that sends XY. So I mean F 3 00:00:12,522 --> 00:00:14,394 of X equals Y. 4 00:00:16,150 --> 00:00:21,430 The inverse of F is denoted F to the minus one. 5 00:00:22,090 --> 00:00:27,570 And it's the function that sends why back to X? 6 00:00:28,210 --> 00:00:32,032 For this to be an inverse, it needs to work for every X 7 00:00:32,032 --> 00:00:33,208 that F acts on. 8 00:00:37,580 --> 00:00:44,270 Here's a simple example of how 9 00:00:44,270 --> 00:00:49,845 to workout an inverse function. 10 00:00:50,930 --> 00:00:54,498 Let's have F of 11 00:00:54,498 --> 00:00:56,800 X. Equals 4X. 12 00:00:57,320 --> 00:01:04,860 Now F of X takes number and multiplies it before. 13 00:01:06,220 --> 00:01:11,720 And we want F inverse to take 4X back to X. 14 00:01:12,670 --> 00:01:17,753 Now, this must mean that F inverse of X must divide by 4. 15 00:01:17,753 --> 00:01:19,708 So F inverse of X. 16 00:01:20,720 --> 00:01:24,768 Equals 1/4 of X. 17 00:01:27,240 --> 00:01:34,296 Now we can see from this that since F inverse of X is 1/4 18 00:01:34,296 --> 00:01:39,840 X and one over F of X is one over 4X. 19 00:01:41,140 --> 00:01:45,670 That F inverse and one over F of X and not the same thing, even 20 00:01:45,670 --> 00:01:48,690 though the notation makes it look like they should be. 21 00:01:49,370 --> 00:01:52,970 Because whenever up of X is one over 4X. 22 00:01:54,000 --> 00:01:59,222 But that's not equal to 1/4 X, because here The X is on the 23 00:01:59,222 --> 00:02:02,579 denominator, and there the X is on the numerator. 24 00:02:04,770 --> 00:02:11,610 Let's workout another inverse 25 00:02:11,610 --> 00:02:13,320 function. 26 00:02:14,370 --> 00:02:17,198 This time will have F of X. 27 00:02:17,950 --> 00:02:23,520 Equals. 3X. +2. 28 00:02:24,990 --> 00:02:30,358 Now what does F of X do we start with X? 29 00:02:31,530 --> 00:02:33,310 We multiplied by three. 30 00:02:33,980 --> 00:02:36,209 To get 3X. 31 00:02:36,940 --> 00:02:39,768 Then we add onto. 32 00:02:39,770 --> 00:02:46,898 Now, since we want F inverse to take F of X and 33 00:02:46,898 --> 00:02:49,274 give us back X. 34 00:02:50,180 --> 00:02:53,330 To workout if inverse we need to undo every 35 00:02:53,330 --> 00:02:54,730 operation that ever did. 36 00:02:55,790 --> 00:02:57,770 So if we started with X. 37 00:02:58,780 --> 00:03:03,554 First, we undo the last operation that F did, so we 38 00:03:03,554 --> 00:03:07,460 take away too. That gives us X minus 2. 39 00:03:08,800 --> 00:03:12,256 And before that, we'd multiplied by three. So to undo that, we 40 00:03:12,256 --> 00:03:18,974 divide by three. And that gives us X minus 41 00:03:18,974 --> 00:03:22,370 two, all divided by 42 00:03:22,370 --> 00:03:29,811 three. So F inverse of X in this case is X minus two, all 43 00:03:29,811 --> 00:03:31,332 divided by three. 44 00:03:33,140 --> 00:03:40,173 Here's one more example of how we can undo the operations of F 45 00:03:40,173 --> 00:03:45,042 to workout its inverse. Let's have F of X. 46 00:03:45,760 --> 00:03:49,310 Being 7 minus X cubed. 47 00:03:50,670 --> 00:03:54,894 And I'll rewrite the slightly to make it easier to workout. 48 00:03:54,894 --> 00:03:58,734 Would love F of X written as minus X cubed? 49 00:03:59,760 --> 00:04:04,464 Plus Seven, so now it's a bit easier to see which operations 50 00:04:04,464 --> 00:04:07,208 were doing 2X. We start with X. 51 00:04:08,300 --> 00:04:10,300 We cubit. 52 00:04:11,620 --> 00:04:13,120 That gives us X cubed. 53 00:04:14,250 --> 00:04:19,780 Then we send X cubed minus X Cube, so we times Y minus one. 54 00:04:20,280 --> 00:04:24,444 And then finally we 55 00:04:24,444 --> 00:04:27,567 add on 7. 56 00:04:28,790 --> 00:04:35,622 So again to workout F inverse we start 57 00:04:35,622 --> 00:04:39,465 with X. And we undo all the 58 00:04:39,465 --> 00:04:44,989 operations of ETH. So we start by taking off 7 to undo the plus 59 00:04:44,989 --> 00:04:52,524 7 bit. And then we multiplied by minus one. So now 60 00:04:52,524 --> 00:04:59,850 we divide by minus one. That gives us 7 minus X. 61 00:05:01,050 --> 00:05:03,798 And we started off by cubing. 62 00:05:04,620 --> 00:05:06,195 So this time we take cube roots. 63 00:05:06,790 --> 00:05:14,590 That gives us the cube root of 7 minus X. 64 00:05:15,260 --> 00:05:18,739 So this time F inverse of X. 65 00:05:19,650 --> 00:05:20,850 Is cube roots? 66 00:05:21,530 --> 00:05:23,510 7 minus X. 67 00:05:25,720 --> 00:05:32,216 Now we can also use algebraic manipulation to 68 00:05:32,216 --> 00:05:34,652 workout in versus. 69 00:05:35,270 --> 00:05:38,762 I'll show you how to do this with our second example, which 70 00:05:38,762 --> 00:05:40,520 was. F of X. 71 00:05:41,040 --> 00:05:43,170 Equals 3X. 72 00:05:43,770 --> 00:05:51,078 +2. Now remember that we want to take F of X and send 73 00:05:51,078 --> 00:05:52,594 it back to X. 74 00:05:53,330 --> 00:06:00,519 So if I set F of X equals Y equals 3 X +2. 75 00:06:01,200 --> 00:06:02,830 We want to F inverse. 76 00:06:03,520 --> 00:06:06,264 To take Y and give us back X. 77 00:06:07,020 --> 00:06:11,387 So we need to workout how to get 2X from Y? 78 00:06:12,080 --> 00:06:17,580 So if I write down again, Y equals 3 X +2. 79 00:06:17,590 --> 00:06:18,970 I can rearrange this. 80 00:06:19,760 --> 00:06:22,136 And I get why minus 2? 81 00:06:22,760 --> 00:06:29,518 Equals 3X. So X equals Y minus two, all 82 00:06:29,518 --> 00:06:31,390 divided by three. 83 00:06:33,020 --> 00:06:38,494 So to get from Y and go to X, you need to take why take off 2 84 00:06:38,494 --> 00:06:39,782 and divided by three? 85 00:06:40,870 --> 00:06:44,468 This means that if inverse of Y. 86 00:06:45,220 --> 00:06:47,560 Equals Y minus two, all divided 87 00:06:47,560 --> 00:06:52,818 by three. But that's just the same as saying that F inverse of 88 00:06:52,818 --> 00:06:56,531 X. Is X minus two all divided by 89 00:06:56,531 --> 00:07:00,387 three? So this is how we use algebraic manipulation to 90 00:07:00,387 --> 00:07:01,296 workout in versus. 91 00:07:02,960 --> 00:07:09,750 Now we can use this method to workout some slightly 92 00:07:09,750 --> 00:07:13,145 more difficult in versus 2. 93 00:07:14,670 --> 00:07:21,430 This time will have F of X being X over X minus one. 94 00:07:22,630 --> 00:07:26,026 And we have to set X is greater than one because otherwise 95 00:07:26,026 --> 00:07:29,422 you'll get a zero denominators. So we only look at this function 96 00:07:29,422 --> 00:07:30,837 for X greater than one. 97 00:07:32,270 --> 00:07:38,809 That work at the inverse again will have Y equals X over X 98 00:07:38,809 --> 00:07:44,444 minus one. Remember again we need to get to X from Y, so we 99 00:07:44,444 --> 00:07:48,248 need to write X in terms of Y, so will rearrange this. 100 00:07:49,250 --> 00:07:54,892 Or multiply both sides by X minus one and that gives us why 101 00:07:54,892 --> 00:07:56,628 times X minus one. 102 00:07:56,790 --> 00:08:03,038 Equals X. And we want and X equals out of this with all 103 00:08:03,038 --> 00:08:07,454 the access on one side, so I'll multiply it. These brackets you 104 00:08:07,454 --> 00:08:09,662 get YX minus Y equals X. 105 00:08:10,720 --> 00:08:13,842 Then I'll take all the ex is over to one side, so that gives 106 00:08:13,842 --> 00:08:20,198 us. Wax. Minus X equals Y. 107 00:08:21,450 --> 00:08:27,586 Factor out the X you get X times Y minus one equals Y 108 00:08:27,586 --> 00:08:32,778 and that gives you X equals Y over Y minus one. 109 00:08:34,660 --> 00:08:39,145 So here we have X in terms of Y again so F inverse. 110 00:08:39,800 --> 00:08:45,186 Of Why? Is Y over Y minus one? 111 00:08:46,210 --> 00:08:51,111 And that's just the same as saying F inverse of X equals X 112 00:08:51,111 --> 00:08:56,389 over X minus one. So in this case the inverse of F turns out 113 00:08:56,389 --> 00:08:59,028 to be exactly the same as F. 114 00:08:59,810 --> 00:09:02,386 This example shows how useful algebraic manipulation is 115 00:09:02,386 --> 00:09:05,606 because it would have been really, really difficult to try 116 00:09:05,606 --> 00:09:09,148 and get this inverse by just reversing the operations of F. 117 00:09:09,920 --> 00:09:15,824 Not all functions have 118 00:09:15,824 --> 00:09:23,010 straightforward inverses. Let's look at F of X. 119 00:09:23,510 --> 00:09:28,346 Equals X squared and I'll just quickly sketch you a graph if I 120 00:09:28,346 --> 00:09:33,584 can. Now when 121 00:09:33,584 --> 00:09:37,688 we look 122 00:09:37,688 --> 00:09:41,792 for an 123 00:09:41,792 --> 00:09:43,844 inverse 124 00:09:43,844 --> 00:09:50,100 function. We want to take the value of F of X. 125 00:09:51,170 --> 00:09:52,688 And send it back to X. 126 00:09:53,330 --> 00:09:55,750 But in this case. 127 00:09:58,460 --> 00:10:01,437 There are two X is we could send F of X back to. 128 00:10:04,470 --> 00:10:10,784 That's because F of X is X squared, but F of minus X is 129 00:10:10,784 --> 00:10:12,137 also X squared. 130 00:10:13,090 --> 00:10:14,728 Now we can't define an inverse 131 00:10:14,728 --> 00:10:18,444 for a function. If there are two things we could 132 00:10:18,444 --> 00:10:20,004 define each value to be. 133 00:10:21,490 --> 00:10:25,175 To get around this problem, we restrict how much of the 134 00:10:25,175 --> 00:10:26,515 function we look at. 135 00:10:27,520 --> 00:10:32,616 So instead of defining F of X like this for every X, what we 136 00:10:32,616 --> 00:10:35,164 do is we cut down the graph. 137 00:10:36,350 --> 00:10:40,062 We say F of 138 00:10:40,062 --> 00:10:42,808 X. Equals X squared. 139 00:10:43,770 --> 00:10:47,356 But only look at X greater than or equal to 0. 140 00:10:48,200 --> 00:10:49,984 And that gives us a graph like this. 141 00:10:57,120 --> 00:10:59,325 Now, since we've cut out all the 142 00:10:59,325 --> 00:11:02,798 values this side. Each value of FX. 143 00:11:04,150 --> 00:11:08,630 Only comes from One X. 144 00:11:09,930 --> 00:11:12,810 So in this case we can define F inverse. 145 00:11:14,160 --> 00:11:16,280 And F inverse of X. 146 00:11:17,210 --> 00:11:20,198 Is plus square root of X. 147 00:11:21,320 --> 00:11:28,096 Now we didn't have to define F for X greater than or equal to 148 00:11:28,096 --> 00:11:34,100 0. We could have cut out the other half of the graph instead, 149 00:11:34,100 --> 00:11:38,280 so I could have defined F of X equals X squared. 150 00:11:38,800 --> 00:11:44,350 For X less than or equal to 0 and that would have given us a 151 00:11:44,350 --> 00:11:45,460 graph like this. 152 00:11:50,840 --> 00:11:54,778 Now again, if a pack of value of F of X. 153 00:11:55,570 --> 00:11:57,698 There's only one value. 154 00:11:58,430 --> 00:12:05,366 Of ex. That gives us the F of X, so in this case F inverse 155 00:12:05,366 --> 00:12:06,572 of X defines. 156 00:12:07,230 --> 00:12:10,485 And it's equal to minus root X. 157 00:12:11,420 --> 00:12:16,612 Here's another function where we need to restrict the domain to 158 00:12:16,612 --> 00:12:19,444 be able to define an inverse. 159 00:12:20,810 --> 00:12:22,380 Would have F of X. 160 00:12:22,940 --> 00:12:24,368 Equals sign X. 161 00:12:25,990 --> 00:12:28,978 And this graph. 162 00:12:29,680 --> 00:12:31,822 Looks a bit like this. I'll try 163 00:12:31,822 --> 00:12:38,710 and sketch it. So for X greater than zero, does this kind of 164 00:12:38,710 --> 00:12:41,390 thing and carries on forever. 165 00:12:43,920 --> 00:12:46,728 Then repeats itself down this way as well forever. 166 00:12:47,400 --> 00:12:51,668 Now, if a pick a value of F of X here. 167 00:12:55,060 --> 00:12:59,038 You can see there's certainly more than One X giving this F of 168 00:12:59,038 --> 00:13:02,264 X. In fact, there will be an infinite number. 169 00:13:05,440 --> 00:13:12,076 So we'll certainly need to cut down the domain of this function 170 00:13:12,076 --> 00:13:14,288 to define an inverse. 171 00:13:15,600 --> 00:13:19,464 What we do in this case is we look at X for X greater 172 00:13:19,464 --> 00:13:22,500 than or equal to minus 90 degrees and less than or 173 00:13:22,500 --> 00:13:23,880 equal to plus 90 degrees. 174 00:13:27,360 --> 00:13:30,088 And if I block out the rest of 175 00:13:30,088 --> 00:13:32,530 the function. Hopefully. 176 00:13:33,080 --> 00:13:35,117 You'll be able to see that in 177 00:13:35,117 --> 00:13:39,658 this case. For every F of X there's only one ex giving that 178 00:13:39,658 --> 00:13:42,938 F of X. So for this. 179 00:13:43,470 --> 00:13:45,018 We restrict the domain. 180 00:13:47,790 --> 00:13:51,675 2X is greater than or equal to 181 00:13:51,675 --> 00:13:56,816 minus 90. Less than or equal 2 + 90. 182 00:13:58,590 --> 00:14:00,339 And the inverse. 183 00:14:01,050 --> 00:14:06,030 Of Cynex Is called arc sine X. 184 00:14:07,210 --> 00:14:13,909 Now this notation can be particularly confusing 185 00:14:13,909 --> 00:14:19,629 because. F inverse of X. Like I said before, is not equal to one 186 00:14:19,629 --> 00:14:23,262 over sine X. But you'll often see 187 00:14:23,262 --> 00:14:26,490 things like sine squared of X. 188 00:14:27,580 --> 00:14:31,490 Which means cynex all squared. 189 00:14:32,020 --> 00:14:34,100 Just remember that sign. 190 00:14:34,900 --> 00:14:38,600 Minus one of X is 191 00:14:38,600 --> 00:14:41,260 not. One over sine X. 192 00:14:42,380 --> 00:14:48,555 But the inverse function sign X, which is also called AC sine X. 193 00:14:49,390 --> 00:14:55,121 The functions, calls, and turn also need the domains to be 194 00:14:55,121 --> 00:14:57,726 restricted for us to define 195 00:14:57,726 --> 00:15:01,094 inverses. But we cover these more fully the trig 196 00:15:01,094 --> 00:15:01,680 functions video. 197 00:15:03,130 --> 00:15:10,290 There are some functions that counts have inverses even if 198 00:15:10,290 --> 00:15:13,870 we do restrict the domains. 199 00:15:14,460 --> 00:15:19,044 A good example of this is a constant function and that is 200 00:15:19,044 --> 00:15:21,718 something like say F of X equals 201 00:15:21,718 --> 00:15:26,340 4. The growth of this looks like this. 202 00:15:31,340 --> 00:15:36,350 Now you can see here that the only way we could get One X for 203 00:15:36,350 --> 00:15:41,360 each F of X here is to cut down the domain to a single point, 204 00:15:41,360 --> 00:15:44,366 and this isn't a very useful thing to do. 205 00:15:45,090 --> 00:15:48,282 So in this case, we say that this function has no inverse. 206 00:15:49,890 --> 00:15:57,606 Now there is a noise easy way of getting the graph of 207 00:15:57,606 --> 00:16:03,393 an inverse function for the graph of a function. 208 00:16:04,190 --> 00:16:07,682 If I just get you the graph of some random function that 209 00:16:07,682 --> 00:16:08,846 I can think of. 210 00:16:10,740 --> 00:16:12,430 So we have X here. 211 00:16:13,440 --> 00:16:14,480 That's Why. 212 00:16:15,530 --> 00:16:18,362 And I'll just draw some function that's appropriate, 213 00:16:18,362 --> 00:16:19,778 so something like that. 214 00:16:21,320 --> 00:16:23,840 Notice that. 215 00:16:25,010 --> 00:16:27,120 A point on this graph. 216 00:16:27,950 --> 00:16:29,999 Has coordinates X. 217 00:16:31,450 --> 00:16:32,410 F of X. 218 00:16:38,140 --> 00:16:43,728 Now since. F inverse sends F of X 2X. 219 00:16:44,790 --> 00:16:48,882 We want the coordinates on the graph of F inverse to 220 00:16:48,882 --> 00:16:52,602 be the coordinates F of XX. So these two interchanged. 221 00:16:53,670 --> 00:16:57,790 Now for turnover this transparency so that my old X 222 00:16:57,790 --> 00:17:03,558 axis was, well, my Y axis was and vice versa. Do that for you. 223 00:17:04,490 --> 00:17:09,712 You can see by doing this I've got precisely that I've got F of 224 00:17:09,712 --> 00:17:11,849 X here. And X here. 225 00:17:12,690 --> 00:17:13,908 So these points. 226 00:17:14,620 --> 00:17:17,210 Must form the graph of F inverse. 227 00:17:18,710 --> 00:17:24,261 So notice that in swapping these axes all I did was I reflected 228 00:17:24,261 --> 00:17:25,542 down that line. 229 00:17:27,970 --> 00:17:30,517 The sort of bottom left, top right diagonal reflecting 230 00:17:30,517 --> 00:17:33,347 down here to get from one graph to the other. 231 00:17:36,510 --> 00:17:40,450 And this line is precisely the line Y equals X. 232 00:17:41,270 --> 00:17:42,370 So that must mean. 233 00:17:42,880 --> 00:17:44,758 But the graph of F inverse. 234 00:17:45,730 --> 00:17:50,290 Is the graph of F reflected in the line Y equals X?