1 00:00:01,110 --> 00:00:03,310 What is a function? 2 00:00:03,970 --> 00:00:08,470 Now, one definition of a function is that a function is a 3 00:00:08,470 --> 00:00:12,970 rule that Maps 1 unique number to another unique number. So In 4 00:00:12,970 --> 00:00:17,845 other words, If I start off with a number and I apply my 5 00:00:17,845 --> 00:00:21,595 function, I finish up with another unique number. So for 6 00:00:21,595 --> 00:00:25,345 example, let's suppose that my function added three to any 7 00:00:25,345 --> 00:00:30,595 number I could start off with a #2. I apply my function and I 8 00:00:30,595 --> 00:00:32,845 finish up with the number 5. 9 00:00:33,440 --> 00:00:37,318 Start off with the number 8 and I apply my function and I finish 10 00:00:37,318 --> 00:00:38,703 up with the number 11. 11 00:00:39,370 --> 00:00:45,730 And if I start off with a number X and I apply my function, I 12 00:00:45,730 --> 00:00:50,818 finish up with the number X +3 and we can show this 13 00:00:50,818 --> 00:00:56,330 mathematically by writing F of X equals X plus three, where X is 14 00:00:56,330 --> 00:01:00,570 our inputs, which we often called the arguments of the 15 00:01:00,570 --> 00:01:06,506 function and X +3 is our output. Now suppose I had an argument of 16 00:01:06,506 --> 00:01:09,898 two, I could write down F of two 17 00:01:09,898 --> 00:01:14,220 equals. 2 + 3, which gives us an output of five. 18 00:01:15,080 --> 00:01:21,112 Suppose I had an argument of eight. I could write down F of 19 00:01:21,112 --> 00:01:26,680 eight equals 8 + 3, which gives me an output of 11. 20 00:01:27,330 --> 00:01:33,322 And suppose I had an argument of minus six. I could write F of 21 00:01:33,322 --> 00:01:39,804 minus 6. Equals minus 6 + 3, which would give me minus three 22 00:01:39,804 --> 00:01:41,136 for my output. 23 00:01:42,030 --> 00:01:48,810 And if I had an argument of zed, I could write down F of said 24 00:01:48,810 --> 00:01:50,618 equals zed plus 3. 25 00:01:51,600 --> 00:01:57,264 And likewise, if I had an argument of X squared, I could 26 00:01:57,264 --> 00:02:01,984 write down F of X squared equals X squared +3. 27 00:02:02,720 --> 00:02:07,062 Now it is with me pointing out here that's a lower first sight. 28 00:02:07,062 --> 00:02:10,736 It might appear that we can choose any value for argument. 29 00:02:10,736 --> 00:02:14,744 That's not always the case, as we shall see later, but because 30 00:02:14,744 --> 00:02:19,086 we do have some choice on what number we can pick the argument 31 00:02:19,086 --> 00:02:22,092 to be, this is sometimes called the independent variable. 32 00:02:22,710 --> 00:02:26,910 And because output depends on our choice of arguments, the 33 00:02:26,910 --> 00:02:29,010 output is sometimes called the 34 00:02:29,010 --> 00:02:35,650 dependent variable. Now let's have a look at an example 35 00:02:35,650 --> 00:02:39,430 F of X equals 3 X 36 00:02:39,430 --> 00:02:42,510 squared. Minus 4. 37 00:02:43,080 --> 00:02:48,274 Now, as we said before, X is our input which we call the argument 38 00:02:48,274 --> 00:02:50,129 and that is the independent 39 00:02:50,129 --> 00:02:54,791 variable. And our output which is 3 X squared minus four is 40 00:02:54,791 --> 00:02:55,814 our dependent variable. 41 00:02:56,860 --> 00:03:01,720 Now we can choose different values for arguments, which is 42 00:03:01,720 --> 00:03:08,524 often a good place to start when we get function like this. So F 43 00:03:08,524 --> 00:03:15,814 of zero will give us 3 * 0 squared takeaway 4, which is 0 - 44 00:03:15,814 --> 00:03:18,730 4, which is just minus 4. 45 00:03:19,540 --> 00:03:21,927 If we start off with an argument 46 00:03:21,927 --> 00:03:28,820 of one. We get F of one equals 3 * 1 squared takeaway 4, 47 00:03:28,820 --> 00:03:34,045 which is just three takeaway four which gives us minus one. 48 00:03:34,310 --> 00:03:41,240 If we have an argument of two F of two equals 3 * 2 squared 49 00:03:41,240 --> 00:03:46,784 takeaway, four switches 3 * 4, which is 12 takeaway. Four gives 50 00:03:46,784 --> 00:03:52,264 us 8. And as I said before, there's no reason why we can't 51 00:03:52,264 --> 00:03:56,736 include and negative arguments. So if I put F of minus one in. 52 00:03:56,750 --> 00:04:03,122 The three times minus one squared takeaway 4, which is 3 * 53 00:04:03,122 --> 00:04:08,432 1 gives us three takeaway. Four gives us minus one. 54 00:04:08,450 --> 00:04:11,950 And F of minus 2. 55 00:04:12,010 --> 00:04:17,639 Would give us three times minus 2 squared takeaway 4 which is 3 56 00:04:17,639 --> 00:04:23,268 * 4 gives us 12 and take away four will give us 8. 57 00:04:24,120 --> 00:04:28,436 So what are we going to do with these results? Well, one thing 58 00:04:28,436 --> 00:04:33,416 we can do is put them into a table to help us plot the graph 59 00:04:33,416 --> 00:04:37,732 of the function. So if we do a table of X&F of X. 60 00:04:38,560 --> 00:04:43,455 The values were chosen for RX column, which is. Our arguments 61 00:04:43,455 --> 00:04:47,015 are minus 1 - 2 zero one and 62 00:04:47,015 --> 00:04:54,697 two. So minus 2 - 1 zero 12. 63 00:04:55,430 --> 00:05:00,374 And the corresponding outputs are 8 - 1. 64 00:05:01,500 --> 00:05:05,180 Minus 4 - 1 and 65 00:05:05,180 --> 00:05:09,624 8. And we can use this as I said to help us plot the 66 00:05:09,624 --> 00:05:10,576 graph of the function. 67 00:05:11,680 --> 00:05:18,316 So just to copy the table down again. So we've got handy. 68 00:05:18,990 --> 00:05:25,750 We have minus 2 - 1 zero 12. 69 00:05:26,420 --> 00:05:32,468 8 - 1 - 4 - 1 and 8. 70 00:05:33,140 --> 00:05:35,360 So we plot our graph. 71 00:05:36,240 --> 00:05:42,540 Of F of X on the vertical axis, so output and arguments. 72 00:05:43,690 --> 00:05:47,058 On the horizontal axis. 73 00:05:47,060 --> 00:05:50,108 So we've got 74 00:05:50,108 --> 00:05:54,240 one too. Minus 1 - 2. 75 00:05:55,250 --> 00:05:57,749 And we've got. 76 00:05:57,750 --> 00:06:00,830 Minus 1 - 2 - 3 - 4. 77 00:06:01,820 --> 00:06:04,298 I'm going off. 78 00:06:05,560 --> 00:06:12,640 Up to 81234567 and eight. So our first point is minus 2 79 00:06:12,640 --> 00:06:16,180 eight so we can put the 80 00:06:16,180 --> 00:06:22,318 appear. Our second point, minus 1 - 1 should appear down here. 81 00:06:23,480 --> 00:06:26,459 0 - 4. 82 00:06:26,460 --> 00:06:29,220 Here one and minus one. 83 00:06:30,480 --> 00:06:34,752 Yeah, but up on two and eight which will appear over here 84 00:06:34,752 --> 00:06:39,024 and we can see we can draw a smooth curve through these 85 00:06:39,024 --> 00:06:42,228 points, which will be the graph of the function. 86 00:06:49,090 --> 00:06:53,398 OK, now why are we drawing a graph of a function? Because 87 00:06:53,398 --> 00:06:58,065 this is quite useful to us because we can now read off the 88 00:06:58,065 --> 00:07:02,014 output of a function for any given arguments straight off the 89 00:07:02,014 --> 00:07:04,527 graph without the need to do any 90 00:07:04,527 --> 00:07:08,025 calculations. So for example, if we look at two and arguments of 91 00:07:08,025 --> 00:07:11,175 two, we know that's going to give us 8 before I do work that 92 00:07:11,175 --> 00:07:15,570 out. But if we looked and we wanted to figure out. 93 00:07:16,360 --> 00:07:21,148 But the output would be when the argument was one point 5. If we 94 00:07:21,148 --> 00:07:25,594 follow our lineup and across you can see that that gives us a 95 00:07:25,594 --> 00:07:29,698 value between 2:00 and 3:00 for the output, and if we substitute 96 00:07:29,698 --> 00:07:33,460 in 1.5 back into our original expression for the function, you 97 00:07:33,460 --> 00:07:36,880 can see actually gives us an exact value of 2.75. 98 00:07:37,820 --> 00:07:42,008 Now earlier on when I discussed uniqueness, I said that a unique 99 00:07:42,008 --> 00:07:46,894 inputs had to give us a unique output and by that what we mean 100 00:07:46,894 --> 00:07:51,082 is that for any given argument we should get only one output. 101 00:07:51,670 --> 00:07:55,044 One of the benefits of having a graph of a function is that we 102 00:07:55,044 --> 00:07:56,490 can check this using our ruler. 103 00:07:57,460 --> 00:08:00,916 If we line our ruler up vertically and we move it left 104 00:08:00,916 --> 00:08:02,356 and right across the graph. 105 00:08:03,350 --> 00:08:07,074 We can make sure that the rule I only have across is the graph 106 00:08:07,074 --> 00:08:08,138 wants at any point. 107 00:08:09,050 --> 00:08:13,329 And as we can see, that's clearly the case in this 108 00:08:13,329 --> 00:08:17,608 example. And when that happens, the graph is a valid function. 109 00:08:19,300 --> 00:08:22,978 Now, if we had the example. 110 00:08:23,530 --> 00:08:25,609 F of X. 111 00:08:26,310 --> 00:08:28,968 Equals root X. 112 00:08:30,040 --> 00:08:36,449 A good place to start is always to substitute in some values for 113 00:08:36,449 --> 00:08:39,407 the arguments, so F of 0. 114 00:08:39,420 --> 00:08:43,061 This gives us the square root of 0, which is 0. 115 00:08:43,740 --> 00:08:48,459 Half of one's own arguments of one will give us plus or minus 116 00:08:48,459 --> 00:08:50,274 one for the square root. 117 00:08:51,010 --> 00:08:53,410 F of two. 118 00:08:53,990 --> 00:09:01,070 Will give us plus or minus 1.4 just to one decimal place 119 00:09:01,070 --> 00:09:04,780 there. Half of 3. 120 00:09:05,320 --> 00:09:10,948 Which gives us the square root of 3 gives us plus or minus 1.7. 121 00:09:11,700 --> 00:09:18,190 An F4. Will give us square root of 4 which is just 122 00:09:18,190 --> 00:09:19,670 plus or minus 2. 123 00:09:20,240 --> 00:09:25,486 Now. If we try to put in any negative 124 00:09:25,486 --> 00:09:28,356 arguments here you can see that we're going to run 125 00:09:28,356 --> 00:09:30,939 into trouble because we have to try and calculate 126 00:09:30,939 --> 00:09:33,522 the square root of a negative number and we'll 127 00:09:33,522 --> 00:09:36,679 come back to this problem in a second. But for now, 128 00:09:36,679 --> 00:09:39,262 let's plot the points that we've got so far. 129 00:09:40,520 --> 00:09:45,668 So if we take out F of X axis vertical again. 130 00:09:45,670 --> 00:09:48,585 And our arguments access X 131 00:09:48,585 --> 00:09:54,416 horizontal. We've got 132 00:09:54,416 --> 00:09:56,729 1234. 133 00:10:00,630 --> 00:10:02,172 And there are vertical axis we 134 00:10:02,172 --> 00:10:04,940 have minus one. Minus 2. 135 00:10:06,200 --> 00:10:13,618 Plus one. Plus two points. We've got zero and zero. 136 00:10:14,870 --> 00:10:16,298 One on plus one. 137 00:10:17,230 --> 00:10:19,750 And also 1A minus one. 138 00:10:21,280 --> 00:10:24,504 We've got two and 139 00:10:24,504 --> 00:10:31,716 positive 1.4. So round about that and also to negative 140 00:10:31,716 --> 00:10:35,150 1.4. We've got three and 141 00:10:35,150 --> 00:10:42,645 positive 1.7. And we've got three and negative 1.7. 142 00:10:42,740 --> 00:10:45,428 And finally we have four and +2. 143 00:10:46,660 --> 00:10:48,950 And four and negative 2. 144 00:10:49,630 --> 00:10:54,700 OK, and we've got enough points here that we can draw a smooth 145 00:10:54,700 --> 00:10:56,260 curve through these points. 146 00:10:59,560 --> 00:11:01,260 At something it looks like. 147 00:11:02,510 --> 00:11:03,250 This. 148 00:11:05,200 --> 00:11:11,152 OK. Now. As usual, we will apply our 149 00:11:11,152 --> 00:11:15,663 ruler test to make sure that the function is valid and you can 150 00:11:15,663 --> 00:11:19,827 see straight away that when we line up all the vertically and 151 00:11:19,827 --> 00:11:22,950 move it across for any given positive arguments, we're 152 00:11:22,950 --> 00:11:26,073 getting two outputs. So obviously we need to do 153 00:11:26,073 --> 00:11:28,155 something about this to make the 154 00:11:28,155 --> 00:11:32,867 function valid. One way to get around this problem is by 155 00:11:32,867 --> 00:11:36,397 defining route X to take only positive values or 0. 156 00:11:37,230 --> 00:11:38,860 This is sometimes called the 157 00:11:38,860 --> 00:11:42,606 positive square root. So in effect we lose the bottom 158 00:11:42,606 --> 00:11:44,146 negative half of this graph. 159 00:11:44,750 --> 00:11:48,226 And obviously we also have the issue of the negative arguments, 160 00:11:48,226 --> 00:11:51,702 and since we can't take the square root of a negative 161 00:11:51,702 --> 00:11:55,810 number, we also have to exclude these from the X axis. Now when 162 00:11:55,810 --> 00:11:58,654 we start talking about these kind of restrictions, it's 163 00:11:58,654 --> 00:12:01,814 important that we use the right kind of mathematical language. 164 00:12:02,420 --> 00:12:07,110 So the set of possible inputs is what we call the domain, and the 165 00:12:07,110 --> 00:12:10,460 set of possible outputs is what we call the range. 166 00:12:11,120 --> 00:12:16,356 So in this case, when we've got RF of X equals the square root 167 00:12:16,356 --> 00:12:22,592 of X. We need to restrict our domain to be X is more than or 168 00:12:22,592 --> 00:12:26,706 equal to 0, 'cause we only wanted the positive values and 169 00:12:26,706 --> 00:12:31,568 zero. But we also notice that now because we've got rid of the 170 00:12:31,568 --> 00:12:33,438 bottom half of the graph. 171 00:12:34,340 --> 00:12:39,228 The only part of the range which are included are also more than 172 00:12:39,228 --> 00:12:44,868 or equal to 0. So range is defined by F of X more than or 173 00:12:44,868 --> 00:12:47,876 equal to 0. So now we have a 174 00:12:47,876 --> 00:12:53,253 valid function. So what will do now is just look at a couple 175 00:12:53,253 --> 00:12:56,683 more examples to pull together everything that we've done so 176 00:12:56,683 --> 00:12:58,741 far and will start with this 177 00:12:58,741 --> 00:13:06,128 one. Let's look at the function F of X equals 2 X squared 178 00:13:06,128 --> 00:13:08,224 minus three X +5. 179 00:13:08,770 --> 00:13:12,970 No, as usual, a good place to start when you get a function is 180 00:13:12,970 --> 00:13:14,770 to substitute in some values for 181 00:13:14,770 --> 00:13:19,730 the arguments. So let's start with that. So now arguments of 0 182 00:13:19,730 --> 00:13:24,110 would give us F of 0, which is 2 * 0 squared. 183 00:13:24,680 --> 00:13:31,480 Minus 3 * 0 + 5 which is just zero 184 00:13:31,480 --> 00:13:34,200 takeaway 0 + 5. 185 00:13:34,210 --> 00:13:38,760 So we get now pose A5 that if we had an argument of one. 186 00:13:39,470 --> 00:13:46,810 We get 2 * 1 squared takeaway 3 * 1 187 00:13:46,810 --> 00:13:53,000 + 5. Which gives us 2 * 1 here, which is 2. 188 00:13:53,880 --> 00:14:00,090 Take away 3 * 1 which is take away three and plus five. So two 189 00:14:00,090 --> 00:14:04,230 takeaway three is minus 1 + 5 gives us 4. 190 00:14:05,120 --> 00:14:08,464 OK, we look at an argument of two. 191 00:14:09,530 --> 00:14:13,118 We get 2 * 2 squared. 192 00:14:13,730 --> 00:14:15,880 Take away 3 * 2. 193 00:14:16,380 --> 00:14:23,911 I'm plus five which gives us 2 * 4, which is 8 and take away 3 * 194 00:14:23,911 --> 00:14:30,556 2 which is take away 6 and then forget our plus five at the end. 195 00:14:30,556 --> 00:14:33,214 So eight takeaway six is 2. 196 00:14:33,810 --> 00:14:36,310 +5 gives us 7. 197 00:14:36,900 --> 00:14:40,399 OK. If we look at an argument of 198 00:14:40,399 --> 00:14:46,084 three. Half of three gives us 2 * 3 squared. 199 00:14:47,420 --> 00:14:49,290 Take away 3 * 3. 200 00:14:49,810 --> 00:14:51,300 And +5. 201 00:14:52,470 --> 00:14:56,026 So this is 2 * 9 here 202 00:14:56,026 --> 00:15:01,604 18. Take away 3 * 3 which is take away nine and +5. 203 00:15:02,310 --> 00:15:07,205 So 18 takeaway 9 is 9 + 5 gives us 14. 204 00:15:08,120 --> 00:15:12,905 And last but not least, we can also include a negative 205 00:15:12,905 --> 00:15:18,125 arguments, so we'll put negative arguments of minus one. So F of 206 00:15:18,125 --> 00:15:22,040 minus one gives us two times minus 1 squared. 207 00:15:22,690 --> 00:15:26,050 Take away three times minus one. 208 00:15:27,370 --> 00:15:30,220 And of course, our +5. 209 00:15:30,790 --> 00:15:34,790 So two times minus one squared just gives us 2. 210 00:15:36,160 --> 00:15:39,706 Takeaway minus sorry takeaway three times negative one which 211 00:15:39,706 --> 00:15:42,070 just gives us a plus 3. 212 00:15:42,950 --> 00:15:45,128 And then we've got a +5. 213 00:15:45,930 --> 00:15:49,188 2 + 3 + 5 just gives us 10. 214 00:15:49,790 --> 00:15:53,390 And what we can do is as before, just put this into a table to 215 00:15:53,390 --> 00:15:55,310 make it nice and easy to make a 216 00:15:55,310 --> 00:16:01,595 graph of the function. So we put it into a table of X. 217 00:16:02,160 --> 00:16:03,940 And F of X. 218 00:16:04,680 --> 00:16:12,078 For arguments we had minus 10123. 219 00:16:12,940 --> 00:16:16,780 For Outputs, we had 10 five. 220 00:16:17,380 --> 00:16:20,605 4, Seven and 221 00:16:20,605 --> 00:16:24,380 14. OK, so let's see the graph of this function then. 222 00:16:26,530 --> 00:16:27,740 You start off with our. 223 00:16:28,430 --> 00:16:31,446 F of X on the vertical axis as 224 00:16:31,446 --> 00:16:33,820 before. An argument. 225 00:16:34,330 --> 00:16:36,540 X on the horizontal axis. 226 00:16:37,680 --> 00:16:41,104 And we've got over 2 - 1, The 227 00:16:41,104 --> 00:16:47,986 One. 2. And three, and on the vertical axis. 228 00:16:49,820 --> 00:16:52,748 Go to 15. 229 00:16:53,540 --> 00:16:59,616 OK, so our first point to plot is minus one and 10 which will 230 00:16:59,616 --> 00:17:02,220 give us something there zero and 231 00:17:02,220 --> 00:17:05,810 five. There's a point here. One 232 00:17:05,810 --> 00:17:08,532 and four. It gives the points 233 00:17:08,532 --> 00:17:10,389 here. Two and Seven. 234 00:17:11,150 --> 00:17:15,099 Should be around here and three and 14 which will be 235 00:17:15,099 --> 00:17:19,407 about here so we can see the kind of shape that this 236 00:17:19,407 --> 00:17:23,715 function is starting to take here. And we can draw in the 237 00:17:23,715 --> 00:17:24,074 graph. 238 00:17:34,700 --> 00:17:38,899 What we want is to say that every input gives us only one 239 00:17:38,899 --> 00:17:42,775 single output, so we can get our ruler again and just quickly 240 00:17:42,775 --> 00:17:47,297 check by going along and we can see that as we go along. Our 241 00:17:47,297 --> 00:17:49,235 rule across is the curve once 242 00:17:49,235 --> 00:17:53,388 and once only. Which means that this function is valid. 243 00:17:53,940 --> 00:17:59,044 However, an interesting point to note is this point here. The 244 00:17:59,044 --> 00:18:03,684 minimum point which actually occurs when X is North .75. 245 00:18:04,580 --> 00:18:11,184 So with X value of North .75 are outputs can take a minimum 246 00:18:11,184 --> 00:18:12,708 value of 3.875. 247 00:18:13,320 --> 00:18:17,254 So this means when we look at our domain and range, we need to 248 00:18:17,254 --> 00:18:20,064 make no restrictions on the domain because our function was 249 00:18:20,064 --> 00:18:27,040 valid. However. Our range has a minimum of 3.875, so we write 250 00:18:27,040 --> 00:18:30,510 this. As F of X equals. 251 00:18:31,290 --> 00:18:34,797 Two X squared minus three X +5. 252 00:18:36,040 --> 00:18:42,982 And the range F of X has always been more than or equal 253 00:18:42,982 --> 00:18:48,970 to 3.875. So for the next example. 254 00:18:50,020 --> 00:18:54,376 What would happen if we had a function F defined by? 255 00:18:54,960 --> 00:18:59,160 F of X equals one over X. 256 00:19:00,370 --> 00:19:04,495 Well, that's always the first stage is to substitute in some 257 00:19:04,495 --> 00:19:05,995 values for the arguments. 258 00:19:06,920 --> 00:19:09,040 So for F of one. 259 00:19:09,870 --> 00:19:13,687 The argument is one is 1 / 1 just gives US1. 260 00:19:14,980 --> 00:19:16,450 For Port F of two. 261 00:19:17,030 --> 00:19:18,730 We just get one half. 262 00:19:19,760 --> 00:19:22,358 F of three gives us 1/3. 263 00:19:22,910 --> 00:19:29,498 And F4 will give us 1/4. 264 00:19:30,520 --> 00:19:35,030 And as before, we can also look at some negative arguments. 265 00:19:35,710 --> 00:19:37,735 So if I look at F of minus one. 266 00:19:38,480 --> 00:19:42,528 Skip 1 divided by minus one, which is just minus one. 267 00:19:43,350 --> 00:19:45,190 F of minus 2. 268 00:19:45,740 --> 00:19:50,396 Is 1 divided by minus two, which just gives us minus 1/2? 269 00:19:51,880 --> 00:19:56,401 F of minus three. Same thing will give us minus 1/3. 270 00:19:57,080 --> 00:19:59,380 And F of minus 4. 271 00:19:59,940 --> 00:20:02,200 Will give us minus 1/4? 272 00:20:02,770 --> 00:20:05,714 Now if we look at F of 0. 273 00:20:05,960 --> 00:20:09,896 We have 1 / 274 00:20:09,896 --> 00:20:13,270 0. Which is obviously a problem 275 00:20:13,270 --> 00:20:17,208 for us. Because of this problem, we have to restrict 276 00:20:17,208 --> 00:20:21,720 our domain so that it does not include the arguments X equals 277 00:20:21,720 --> 00:20:26,232 0. So let's have a look at what the graph of this 278 00:20:26,232 --> 00:20:27,736 function actually looks like. 279 00:20:29,230 --> 00:20:33,169 So let's be 4F of X and are vertical axis for the Outputs. 280 00:20:34,240 --> 00:20:38,174 An argument sax on the horizontal axis. 281 00:20:39,640 --> 00:20:46,115 OK, so we've gone over here 1234. 282 00:20:47,790 --> 00:20:54,698 And minus 1 - 2 - 3 - 4 over here. 283 00:20:54,810 --> 00:20:56,430 As we go. 284 00:20:57,030 --> 00:20:58,990 All the web to one down to minus 285 00:20:58,990 --> 00:21:05,260 one. Has it as well? So we've got one and one. 286 00:21:06,390 --> 00:21:09,456 We've got 2 287 00:21:09,456 --> 00:21:12,510 1/2. 3 288 00:21:12,510 --> 00:21:16,046 1/3. 4 289 00:21:16,046 --> 00:21:23,556 one quarter. We've got minus 1 - 1. 290 00:21:23,650 --> 00:21:27,046 Minus 2 - 291 00:21:27,046 --> 00:21:33,750 1/2. Minus 3 - 1/3. 292 00:21:33,750 --> 00:21:37,218 A minus four and minus 1/4. 293 00:21:37,770 --> 00:21:42,114 OK, and obviously we've excluded 0 from our domain. As we said 294 00:21:42,114 --> 00:21:44,286 before. So if we join these 295 00:21:44,286 --> 00:21:47,030 points up. And a smooth curve. 296 00:21:48,830 --> 00:21:53,288 Get something that looks like this. 297 00:21:54,910 --> 00:21:59,618 Now, obviously we've excluded X equals 0 from our domain, but 298 00:21:59,618 --> 00:22:03,898 it's also worth noticing here. Thought there's nothing at the 299 00:22:03,898 --> 00:22:09,890 output of F of X equals 0, so that also ends up being excluded 300 00:22:09,890 --> 00:22:11,174 from the range. 301 00:22:11,790 --> 00:22:16,548 So we actually end up with F of X equals one over X. 302 00:22:17,450 --> 00:22:21,212 And we've got X not equal to 0 from the domain. 303 00:22:21,870 --> 00:22:26,690 And also. In the range F of X never equals 0 either. 304 00:22:28,910 --> 00:22:30,595 But what's actually happening at 305 00:22:30,595 --> 00:22:35,408 this point? X equals 0 when the arguments is zero. What is going 306 00:22:35,408 --> 00:22:40,000 on? Well, let's have a look and will start off by having a look 307 00:22:40,000 --> 00:22:43,280 what happens as we get closer and closer to 0. 308 00:22:43,820 --> 00:22:45,910 Now. 309 00:22:47,040 --> 00:22:52,352 If we start off with a value of 1/2 of one and remember F of X 310 00:22:52,352 --> 00:22:54,676 was just equal to one over X. 311 00:22:55,340 --> 00:23:01,085 Half of 1 just gives US1, so if I get closer to 0 again, let's 312 00:23:01,085 --> 00:23:03,000 look at half of 1/2. 313 00:23:03,000 --> 00:23:05,628 At 1 / 1/2. 314 00:23:06,200 --> 00:23:11,294 This one over 1/2 which just gives us 2. 315 00:23:11,300 --> 00:23:15,292 So about F of 316 00:23:15,292 --> 00:23:21,168 110th. It just gives us 1 / 1 tenth, which gives us 10. 317 00:23:21,820 --> 00:23:29,512 Half of one over 1000 will just give us 1 / 1 318 00:23:29,512 --> 00:23:33,314 over 1000. Which gives 319 00:23:33,314 --> 00:23:40,236 us 1000. What about one over 1,000,000, so F of 320 00:23:40,236 --> 00:23:42,564 one over a million? 321 00:23:42,570 --> 00:23:47,550 It's actually just in the same way as before, just going to 322 00:23:47,550 --> 00:23:49,210 give us a million. 323 00:23:49,220 --> 00:23:52,068 So we can see. 324 00:23:52,710 --> 00:23:57,316 The US we get closer and closer to zero from the right hand side 325 00:23:57,316 --> 00:23:59,619 as we saw on our graph before. 326 00:23:59,620 --> 00:24:03,230 We're getting closer and closer to positive Infinity to the 327 00:24:03,230 --> 00:24:05,035 graph goes off to positive 328 00:24:05,035 --> 00:24:09,135 Infinity that. What happens when we approach zero from the left 329 00:24:09,135 --> 00:24:10,965 hand side? Well, let's have a 330 00:24:10,965 --> 00:24:17,692 look. This is minus one, just gives us 1 divided by minus one, 331 00:24:17,692 --> 00:24:19,556 which is minus one. 332 00:24:20,500 --> 00:24:23,988 F of minus 1/2. 333 00:24:23,990 --> 00:24:27,790 It's going to be just one divided by minus 1/2. 334 00:24:28,330 --> 00:24:30,110 Just give his minus 2. 335 00:24:31,290 --> 00:24:34,401 Half of minus 336 00:24:34,401 --> 00:24:40,840 110th. Is 1 divided by minus 110th? 337 00:24:40,910 --> 00:24:47,150 It just gives us minus 10 and we can kind of see a pattern here. 338 00:24:47,150 --> 00:24:50,478 OK so F of minus one over 1000. 339 00:24:50,510 --> 00:24:52,760 Will actually give us minus 340 00:24:52,760 --> 00:24:59,384 1000. And F of minus one over 1,000,000. 341 00:24:59,390 --> 00:25:05,542 Actually gives us minus 342 00:25:05,542 --> 00:25:11,467 1,000,000. So you can see that as we approach zero from the 343 00:25:11,467 --> 00:25:12,895 left or outputs approaches 344 00:25:12,895 --> 00:25:17,606 negative Infinity. And as we approach zero from the right 345 00:25:17,606 --> 00:25:21,876 hand side are output approaches positive Infinity, and these are 346 00:25:21,876 --> 00:25:25,292 very different things. OK, for the last example. 347 00:25:25,940 --> 00:25:30,241 I just like to look at the function F defined by. 348 00:25:30,250 --> 00:25:37,576 F of X equals one over X minus two all squared. 349 00:25:38,310 --> 00:25:41,770 So as always with the examples we've done, it's worthwhile 350 00:25:41,770 --> 00:25:45,230 started off by looking at some different values for the 351 00:25:45,230 --> 00:25:51,234 arguments. So we start off with an argument of minus two, so 352 00:25:51,234 --> 00:25:56,876 half of minus two gives us one over minus 2 - 2 squared. 353 00:25:57,420 --> 00:26:03,803 Which gives us one over minus 4 squared, which is one over 16. 354 00:26:03,900 --> 00:26:07,388 F of minus one. 355 00:26:07,390 --> 00:26:14,628 Will give us one over minus 1 - 2 all squared, which gives us 356 00:26:14,628 --> 00:26:19,798 one over minus 3 squared which is one over 9. 357 00:26:19,820 --> 00:26:26,450 Now, FO arguments of zero will give us one over 0 - 2 358 00:26:26,450 --> 00:26:32,570 all squared, which is one over minus 2 squared, which works out 359 00:26:32,570 --> 00:26:34,100 as one quarter. 360 00:26:34,960 --> 00:26:41,932 And F of one will give us one over 1 - 2 361 00:26:41,932 --> 00:26:46,880 squared. Which is just one over minus one squared, which gives 362 00:26:46,880 --> 00:26:48,008 us just one. 363 00:26:48,580 --> 00:26:55,990 OK. Half of two gives us one over. 364 00:26:56,690 --> 00:27:01,730 2 - 2 or squared, which gives us one over 0 which presents us 365 00:27:01,730 --> 00:27:06,050 with exactly the same problem we had in the previous example when 366 00:27:06,050 --> 00:27:11,810 we had one over X and so we have to exclude X equals 2 from the 367 00:27:11,810 --> 00:27:19,370 domain. Half of three gives us one over 3 - 2 all squared. 368 00:27:20,220 --> 00:27:25,225 Which is one over 1 squared is just gives US1 again. 369 00:27:25,230 --> 00:27:32,440 After four gives us one over 4 - 2 or squared, which gives us 370 00:27:32,440 --> 00:27:33,985 one over 4. 371 00:27:34,110 --> 00:27:42,080 After 5. Gives us one over 5 - 2 or 372 00:27:42,080 --> 00:27:48,515 squared which gives us one over 3 squared which is 1 ninth and 373 00:27:48,515 --> 00:27:55,940 finally F of six gives us one over 6 - 2 all squared which is 374 00:27:55,940 --> 00:28:01,385 one over 4 squared which works out as one over 16. 375 00:28:01,940 --> 00:28:06,659 Now if we want to plot the graph of this function will probably 376 00:28:06,659 --> 00:28:09,200 need to put this into a table 377 00:28:09,200 --> 00:28:15,520 first. So as usual, do our table of X&F of X. 378 00:28:16,370 --> 00:28:23,104 OK, and we went from minus 2 - 1 zero all the way. 379 00:28:23,880 --> 00:28:26,298 Up to and arguments of sex. 380 00:28:27,160 --> 00:28:28,896 And the values we got for the 381 00:28:28,896 --> 00:28:31,932 Outputs. One over 382 00:28:31,932 --> 00:28:38,160 16. One 9th, one quarter and 383 00:28:38,160 --> 00:28:44,292 1:01. One quarter, 1 ninth and 384 00:28:44,292 --> 00:28:47,358 one over 16. 385 00:28:48,270 --> 00:28:51,550 So we plot that onto. 386 00:28:52,100 --> 00:28:54,120 The graph as before. 387 00:28:57,060 --> 00:29:00,606 So we have arguments going along the horizontal axis. 388 00:29:01,290 --> 00:29:03,600 And Outputs going along the vertical axis. 389 00:29:04,720 --> 00:29:12,112 We've gone from minus 1 - 2 over 390 00:29:12,112 --> 00:29:15,808 there 123456 along this 391 00:29:15,808 --> 00:29:18,810 way. And then going off, we've 392 00:29:18,810 --> 00:29:24,960 gone too. One up here, so put in a few of the marks 1/2. 393 00:29:25,540 --> 00:29:27,250 It's put in 1/4 that. 394 00:29:28,020 --> 00:29:34,728 Put in 3/4. OK, so we've got minus two and 116th, which is 395 00:29:34,728 --> 00:29:37,824 going to come in down here. 396 00:29:38,340 --> 00:29:40,580 Minus one and one 9th. 397 00:29:41,380 --> 00:29:47,108 For coming over here zero and one quarter. 398 00:29:47,110 --> 00:29:49,589 Over here. 1 on one. 399 00:29:50,930 --> 00:29:52,410 Right, the way up here? 400 00:29:53,200 --> 00:29:57,232 To an. Obviously this was the divide by zero, so we couldn't 401 00:29:57,232 --> 00:30:00,592 do anything with that. We've excluded, uh, from our domain. 402 00:30:01,690 --> 00:30:03,040 Three and one. 403 00:30:03,620 --> 00:30:07,280 Pay up. Four and one quarter. 404 00:30:08,840 --> 00:30:16,628 I'm here. Five and one, 9th and six and 116th. 405 00:30:16,630 --> 00:30:21,643 Because we've excluded X equals 2 from our domain. 406 00:30:23,370 --> 00:30:26,410 Put dotted line there, so that's an asymptotes. 407 00:30:27,820 --> 00:30:29,626 And we can draw our curve. 408 00:30:31,970 --> 00:30:33,298 Up through the points. 409 00:30:33,980 --> 00:30:36,770 On this side. 410 00:30:38,190 --> 00:30:42,493 And we can see differently to the other example where F of X 411 00:30:42,493 --> 00:30:44,479 is one over X this time. 412 00:30:45,090 --> 00:30:49,965 As we get approach to from both the left and from the right, 413 00:30:49,965 --> 00:30:54,090 both of the outputs are heading towards positive Infinity, so a 414 00:30:54,090 --> 00:30:57,840 little bit different, and also because we've excluded X equals 415 00:30:57,840 --> 00:31:00,465 2 from the domain of function is 416 00:31:00,465 --> 00:31:05,470 now valid. But most also notice that our range is never zero, 417 00:31:05,470 --> 00:31:07,315 and it's also never negative. 418 00:31:07,820 --> 00:31:10,826 So to write this out properly. 419 00:31:10,830 --> 00:31:16,394 Our function F of X equals one over X minus two all squared. 420 00:31:17,710 --> 00:31:18,829 And we said. 421 00:31:19,340 --> 00:31:20,804 They are domain is restricted so 422 00:31:20,804 --> 00:31:26,506 it doesn't include two. And our range is always more than 0. 423 00:31:27,640 --> 00:31:31,324 OK, so let's just recap on what we've done in this unit. 424 00:31:32,310 --> 00:31:34,452 So firstly, the definition of a 425 00:31:34,452 --> 00:31:39,480 function. And that was that. A function is a rule that Maps are 426 00:31:39,480 --> 00:31:42,880 unique number X to another unique number F of X. 427 00:31:44,230 --> 00:31:48,050 Secondly, was the idea that an argument is exactly the 428 00:31:48,050 --> 00:31:49,578 same as an input. 429 00:31:51,630 --> 00:31:55,950 Thirdly, we looked at the idea of independent and dependent 430 00:31:55,950 --> 00:32:00,443 variables. And we said that the input axe was the 431 00:32:00,443 --> 00:32:03,539 independent variable and the output was the dependent 432 00:32:03,539 --> 00:32:03,926 variable. 433 00:32:05,330 --> 00:32:10,230 4th, we looked at the domain and we said that the domain was the 434 00:32:10,230 --> 00:32:11,630 set of possible inputs. 435 00:32:12,620 --> 00:32:16,546 And finally we looked at the range and we said that the range 436 00:32:16,546 --> 00:32:18,358 was the set of possible outputs. 437 00:32:19,610 --> 00:32:21,514 So now you know how to define a 438 00:32:21,514 --> 00:32:24,997 function. And how to find the outputs of a function 439 00:32:24,997 --> 00:32:26,209 for a given argument?