0:00:01.110,0:00:03.310 What is a function? 0:00:03.970,0:00:08.470 Now, one definition of a[br]function is that a function is a 0:00:08.470,0:00:12.970 rule that Maps 1 unique number[br]to another unique number. So In 0:00:12.970,0:00:17.845 other words, If I start off with[br]a number and I apply my 0:00:17.845,0:00:21.595 function, I finish up with[br]another unique number. So for 0:00:21.595,0:00:25.345 example, let's suppose that my[br]function added three to any 0:00:25.345,0:00:30.595 number I could start off with a[br]#2. I apply my function and I 0:00:30.595,0:00:32.845 finish up with the number 5. 0:00:33.440,0:00:37.318 Start off with the number 8 and[br]I apply my function and I finish 0:00:37.318,0:00:38.703 up with the number 11. 0:00:39.370,0:00:45.730 And if I start off with a number[br]X and I apply my function, I 0:00:45.730,0:00:50.818 finish up with the number X +3[br]and we can show this 0:00:50.818,0:00:56.330 mathematically by writing F of X[br]equals X plus three, where X is 0:00:56.330,0:01:00.570 our inputs, which we often[br]called the arguments of the 0:01:00.570,0:01:06.506 function and X +3 is our output.[br]Now suppose I had an argument of 0:01:06.506,0:01:09.898 two, I could write down F of two 0:01:09.898,0:01:14.220 equals. 2 + 3, which gives us an[br]output of five. 0:01:15.080,0:01:21.112 Suppose I had an argument of[br]eight. I could write down F of 0:01:21.112,0:01:26.680 eight equals 8 + 3, which gives[br]me an output of 11. 0:01:27.330,0:01:33.322 And suppose I had an argument of[br]minus six. I could write F of 0:01:33.322,0:01:39.804 minus 6. Equals minus 6 + 3,[br]which would give me minus three 0:01:39.804,0:01:41.136 for my output. 0:01:42.030,0:01:48.810 And if I had an argument of zed,[br]I could write down F of said 0:01:48.810,0:01:50.618 equals zed plus 3. 0:01:51.600,0:01:57.264 And likewise, if I had an[br]argument of X squared, I could 0:01:57.264,0:02:01.984 write down F of X squared equals[br]X squared +3. 0:02:02.720,0:02:07.062 Now it is with me pointing out[br]here that's a lower first sight. 0:02:07.062,0:02:10.736 It might appear that we can[br]choose any value for argument. 0:02:10.736,0:02:14.744 That's not always the case, as[br]we shall see later, but because 0:02:14.744,0:02:19.086 we do have some choice on what[br]number we can pick the argument 0:02:19.086,0:02:22.092 to be, this is sometimes called[br]the independent variable. 0:02:22.710,0:02:26.910 And because output depends on[br]our choice of arguments, the 0:02:26.910,0:02:29.010 output is sometimes called the 0:02:29.010,0:02:35.650 dependent variable. Now let's[br]have a look at an example 0:02:35.650,0:02:39.430 F of X equals 3 X 0:02:39.430,0:02:42.510 squared. Minus 4. 0:02:43.080,0:02:48.274 Now, as we said before, X is our[br]input which we call the argument 0:02:48.274,0:02:50.129 and that is the independent 0:02:50.129,0:02:54.791 variable. And our output which[br]is 3 X squared minus four is 0:02:54.791,0:02:55.814 our dependent variable. 0:02:56.860,0:03:01.720 Now we can choose different[br]values for arguments, which is 0:03:01.720,0:03:08.524 often a good place to start when[br]we get function like this. So F 0:03:08.524,0:03:15.814 of zero will give us 3 * 0[br]squared takeaway 4, which is 0 - 0:03:15.814,0:03:18.730 4, which is just minus 4. 0:03:19.540,0:03:21.927 If we start off with an argument 0:03:21.927,0:03:28.820 of one. We get F of one[br]equals 3 * 1 squared takeaway 4, 0:03:28.820,0:03:34.045 which is just three takeaway[br]four which gives us minus one. 0:03:34.310,0:03:41.240 If we have an argument of two F[br]of two equals 3 * 2 squared 0:03:41.240,0:03:46.784 takeaway, four switches 3 * 4,[br]which is 12 takeaway. Four gives 0:03:46.784,0:03:52.264 us 8. And as I said before,[br]there's no reason why we can't 0:03:52.264,0:03:56.736 include and negative arguments.[br]So if I put F of minus one in. 0:03:56.750,0:04:03.122 The three times minus one[br]squared takeaway 4, which is 3 * 0:04:03.122,0:04:08.432 1 gives us three takeaway. Four[br]gives us minus one. 0:04:08.450,0:04:11.950 And F of minus 2. 0:04:12.010,0:04:17.639 Would give us three times minus[br]2 squared takeaway 4 which is 3 0:04:17.639,0:04:23.268 * 4 gives us 12 and take away[br]four will give us 8. 0:04:24.120,0:04:28.436 So what are we going to do with[br]these results? Well, one thing 0:04:28.436,0:04:33.416 we can do is put them into a[br]table to help us plot the graph 0:04:33.416,0:04:37.732 of the function. So if we do a[br]table of X&F of X. 0:04:38.560,0:04:43.455 The values were chosen for RX[br]column, which is. Our arguments 0:04:43.455,0:04:47.015 are minus 1 - 2 zero one and 0:04:47.015,0:04:54.697 two. So minus 2[br]- 1 zero 12. 0:04:55.430,0:05:00.374 And the corresponding outputs[br]are 8 - 1. 0:05:01.500,0:05:05.180 Minus 4 - 1 and 0:05:05.180,0:05:09.624 8. And we can use this as I[br]said to help us plot the 0:05:09.624,0:05:10.576 graph of the function. 0:05:11.680,0:05:18.316 So just to copy the table down[br]again. So we've got handy. 0:05:18.990,0:05:25.750 We have minus 2[br]- 1 zero 12. 0:05:26.420,0:05:32.468 8 - 1 - 4[br]- 1 and 8. 0:05:33.140,0:05:35.360 So we plot our graph. 0:05:36.240,0:05:42.540 Of F of X on the vertical[br]axis, so output and arguments. 0:05:43.690,0:05:47.058 On the horizontal axis. 0:05:47.060,0:05:50.108 So we've got 0:05:50.108,0:05:54.240 one too. Minus 1 -[br]2. 0:05:55.250,0:05:57.749 And we've got. 0:05:57.750,0:06:00.830 Minus 1 - 2 - 3 - 4. 0:06:01.820,0:06:04.298 I'm going off. 0:06:05.560,0:06:12.640 Up to 81234567 and eight. So[br]our first point is minus 2 0:06:12.640,0:06:16.180 eight so we can put the 0:06:16.180,0:06:22.318 appear. Our second point, minus[br]1 - 1 should appear down here. 0:06:23.480,0:06:26.459 0 - 4. 0:06:26.460,0:06:29.220 Here one and minus one. 0:06:30.480,0:06:34.752 Yeah, but up on two and eight[br]which will appear over here 0:06:34.752,0:06:39.024 and we can see we can draw a[br]smooth curve through these 0:06:39.024,0:06:42.228 points, which will be the[br]graph of the function. 0:06:49.090,0:06:53.398 OK, now why are we drawing a[br]graph of a function? Because 0:06:53.398,0:06:58.065 this is quite useful to us[br]because we can now read off the 0:06:58.065,0:07:02.014 output of a function for any[br]given arguments straight off the 0:07:02.014,0:07:04.527 graph without the need to do any 0:07:04.527,0:07:08.025 calculations. So for example, if[br]we look at two and arguments of 0:07:08.025,0:07:11.175 two, we know that's going to[br]give us 8 before I do work that 0:07:11.175,0:07:15.570 out. But if we looked and we[br]wanted to figure out. 0:07:16.360,0:07:21.148 But the output would be when the[br]argument was one point 5. If we 0:07:21.148,0:07:25.594 follow our lineup and across you[br]can see that that gives us a 0:07:25.594,0:07:29.698 value between 2:00 and 3:00 for[br]the output, and if we substitute 0:07:29.698,0:07:33.460 in 1.5 back into our original[br]expression for the function, you 0:07:33.460,0:07:36.880 can see actually gives us an[br]exact value of 2.75. 0:07:37.820,0:07:42.008 Now earlier on when I discussed[br]uniqueness, I said that a unique 0:07:42.008,0:07:46.894 inputs had to give us a unique[br]output and by that what we mean 0:07:46.894,0:07:51.082 is that for any given argument[br]we should get only one output. 0:07:51.670,0:07:55.044 One of the benefits of having a[br]graph of a function is that we 0:07:55.044,0:07:56.490 can check this using our ruler. 0:07:57.460,0:08:00.916 If we line our ruler up[br]vertically and we move it left 0:08:00.916,0:08:02.356 and right across the graph. 0:08:03.350,0:08:07.074 We can make sure that the rule I[br]only have across is the graph 0:08:07.074,0:08:08.138 wants at any point. 0:08:09.050,0:08:13.329 And as we can see, that's[br]clearly the case in this 0:08:13.329,0:08:17.608 example. And when that happens,[br]the graph is a valid function. 0:08:19.300,0:08:22.978 Now, if we had the example. 0:08:23.530,0:08:25.609 F of X. 0:08:26.310,0:08:28.968 Equals root X. 0:08:30.040,0:08:36.449 A good place to start is always[br]to substitute in some values for 0:08:36.449,0:08:39.407 the arguments, so F of 0. 0:08:39.420,0:08:43.061 This gives us the square root of[br]0, which is 0. 0:08:43.740,0:08:48.459 Half of one's own arguments of[br]one will give us plus or minus 0:08:48.459,0:08:50.274 one for the square root. 0:08:51.010,0:08:53.410 F of two. 0:08:53.990,0:09:01.070 Will give us plus or minus[br]1.4 just to one decimal place 0:09:01.070,0:09:04.780 there. Half of 3. 0:09:05.320,0:09:10.948 Which gives us the square root[br]of 3 gives us plus or minus 1.7. 0:09:11.700,0:09:18.190 An F4. Will give us[br]square root of 4 which is just 0:09:18.190,0:09:19.670 plus or minus 2. 0:09:20.240,0:09:25.486 Now. If we try[br]to put in any negative 0:09:25.486,0:09:28.356 arguments here you can see[br]that we're going to run 0:09:28.356,0:09:30.939 into trouble because we[br]have to try and calculate 0:09:30.939,0:09:33.522 the square root of a[br]negative number and we'll 0:09:33.522,0:09:36.679 come back to this problem[br]in a second. But for now, 0:09:36.679,0:09:39.262 let's plot the points that[br]we've got so far. 0:09:40.520,0:09:45.668 So if we take out F of X[br]axis vertical again. 0:09:45.670,0:09:48.585 And our arguments access X 0:09:48.585,0:09:54.416 horizontal. We've[br]got 0:09:54.416,0:09:56.729 1234. 0:10:00.630,0:10:02.172 And there are vertical axis we 0:10:02.172,0:10:04.940 have minus one. Minus 2. 0:10:06.200,0:10:13.618 Plus one. Plus two points.[br]We've got zero and zero. 0:10:14.870,0:10:16.298 One on plus one. 0:10:17.230,0:10:19.750 And also 1A minus one. 0:10:21.280,0:10:24.504 We've got two and 0:10:24.504,0:10:31.716 positive 1.4. So round[br]about that and also to negative 0:10:31.716,0:10:35.150 1.4. We've got three and 0:10:35.150,0:10:42.645 positive 1.7. And we've[br]got three and negative 1.7. 0:10:42.740,0:10:45.428 And finally we have four and +2. 0:10:46.660,0:10:48.950 And four and negative 2. 0:10:49.630,0:10:54.700 OK, and we've got enough points[br]here that we can draw a smooth 0:10:54.700,0:10:56.260 curve through these points. 0:10:59.560,0:11:01.260 At something it looks like. 0:11:02.510,0:11:03.250 This. 0:11:05.200,0:11:11.152 OK. Now.[br]As usual, we will apply our 0:11:11.152,0:11:15.663 ruler test to make sure that the[br]function is valid and you can 0:11:15.663,0:11:19.827 see straight away that when we[br]line up all the vertically and 0:11:19.827,0:11:22.950 move it across for any given[br]positive arguments, we're 0:11:22.950,0:11:26.073 getting two outputs. So[br]obviously we need to do 0:11:26.073,0:11:28.155 something about this to make the 0:11:28.155,0:11:32.867 function valid. One way to get[br]around this problem is by 0:11:32.867,0:11:36.397 defining route X to take only[br]positive values or 0. 0:11:37.230,0:11:38.860 This is sometimes called the 0:11:38.860,0:11:42.606 positive square root. So in[br]effect we lose the bottom 0:11:42.606,0:11:44.146 negative half of this graph. 0:11:44.750,0:11:48.226 And obviously we also have the[br]issue of the negative arguments, 0:11:48.226,0:11:51.702 and since we can't take the[br]square root of a negative 0:11:51.702,0:11:55.810 number, we also have to exclude[br]these from the X axis. Now when 0:11:55.810,0:11:58.654 we start talking about these[br]kind of restrictions, it's 0:11:58.654,0:12:01.814 important that we use the right[br]kind of mathematical language. 0:12:02.420,0:12:07.110 So the set of possible inputs is[br]what we call the domain, and the 0:12:07.110,0:12:10.460 set of possible outputs is what[br]we call the range. 0:12:11.120,0:12:16.356 So in this case, when we've got[br]RF of X equals the square root 0:12:16.356,0:12:22.592 of X. We need to restrict our[br]domain to be X is more than or 0:12:22.592,0:12:26.706 equal to 0, 'cause we only[br]wanted the positive values and 0:12:26.706,0:12:31.568 zero. But we also notice that[br]now because we've got rid of the 0:12:31.568,0:12:33.438 bottom half of the graph. 0:12:34.340,0:12:39.228 The only part of the range which[br]are included are also more than 0:12:39.228,0:12:44.868 or equal to 0. So range is[br]defined by F of X more than or 0:12:44.868,0:12:47.876 equal to 0. So now we have a 0:12:47.876,0:12:53.253 valid function. So what will do[br]now is just look at a couple 0:12:53.253,0:12:56.683 more examples to pull together[br]everything that we've done so 0:12:56.683,0:12:58.741 far and will start with this 0:12:58.741,0:13:06.128 one. Let's look at the function[br]F of X equals 2 X squared 0:13:06.128,0:13:08.224 minus three X +5. 0:13:08.770,0:13:12.970 No, as usual, a good place to[br]start when you get a function is 0:13:12.970,0:13:14.770 to substitute in some values for 0:13:14.770,0:13:19.730 the arguments. So let's start[br]with that. So now arguments of 0 0:13:19.730,0:13:24.110 would give us F of 0, which is 2[br]* 0 squared. 0:13:24.680,0:13:31.480 Minus 3 * 0 +[br]5 which is just zero 0:13:31.480,0:13:34.200 takeaway 0 + 5. 0:13:34.210,0:13:38.760 So we get now pose A5 that if we[br]had an argument of one. 0:13:39.470,0:13:46.810 We get 2 * 1[br]squared takeaway 3 * 1 0:13:46.810,0:13:53.000 + 5. Which gives us 2[br]* 1 here, which is 2. 0:13:53.880,0:14:00.090 Take away 3 * 1 which is take[br]away three and plus five. So two 0:14:00.090,0:14:04.230 takeaway three is minus 1 + 5[br]gives us 4. 0:14:05.120,0:14:08.464 OK, we look at an[br]argument of two. 0:14:09.530,0:14:13.118 We get 2 * 2 squared. 0:14:13.730,0:14:15.880 Take away 3 * 2. 0:14:16.380,0:14:23.911 I'm plus five which gives us 2 *[br]4, which is 8 and take away 3 * 0:14:23.911,0:14:30.556 2 which is take away 6 and then[br]forget our plus five at the end. 0:14:30.556,0:14:33.214 So eight takeaway six is 2. 0:14:33.810,0:14:36.310 +5 gives us 7. 0:14:36.900,0:14:40.399 OK. If we look at an argument of 0:14:40.399,0:14:46.084 three. Half of three gives us[br]2 * 3 squared. 0:14:47.420,0:14:49.290 Take away 3 * 3. 0:14:49.810,0:14:51.300 And +5. 0:14:52.470,0:14:56.026 So this is 2 * 9 here 0:14:56.026,0:15:01.604 18. Take away 3 * 3 which is[br]take away nine and +5. 0:15:02.310,0:15:07.205 So 18 takeaway 9 is 9 + 5[br]gives us 14. 0:15:08.120,0:15:12.905 And last but not least, we can[br]also include a negative 0:15:12.905,0:15:18.125 arguments, so we'll put negative[br]arguments of minus one. So F of 0:15:18.125,0:15:22.040 minus one gives us two times[br]minus 1 squared. 0:15:22.690,0:15:26.050 Take away three times minus one. 0:15:27.370,0:15:30.220 And of course, our +5. 0:15:30.790,0:15:34.790 So two times minus one squared[br]just gives us 2. 0:15:36.160,0:15:39.706 Takeaway minus sorry takeaway[br]three times negative one which 0:15:39.706,0:15:42.070 just gives us a plus 3. 0:15:42.950,0:15:45.128 And then we've got a +5. 0:15:45.930,0:15:49.188 2 + 3 + 5 just gives us 10. 0:15:49.790,0:15:53.390 And what we can do is as before,[br]just put this into a table to 0:15:53.390,0:15:55.310 make it nice and easy to make a 0:15:55.310,0:16:01.595 graph of the function. So we put[br]it into a table of X. 0:16:02.160,0:16:03.940 And F of X. 0:16:04.680,0:16:12.078 For arguments we[br]had minus 10123. 0:16:12.940,0:16:16.780 For Outputs, we had 10 five. 0:16:17.380,0:16:20.605 4, Seven and 0:16:20.605,0:16:24.380 14. OK, so let's see the graph[br]of this function then. 0:16:26.530,0:16:27.740 You start off with our. 0:16:28.430,0:16:31.446 F of X on the vertical axis as 0:16:31.446,0:16:33.820 before. An argument. 0:16:34.330,0:16:36.540 X on the horizontal axis. 0:16:37.680,0:16:41.104 And we've got over 2 - 1, The 0:16:41.104,0:16:47.986 One. 2. And[br]three, and on the vertical axis. 0:16:49.820,0:16:52.748 Go to 15. 0:16:53.540,0:16:59.616 OK, so our first point to plot[br]is minus one and 10 which will 0:16:59.616,0:17:02.220 give us something there zero and 0:17:02.220,0:17:05.810 five. There's a point here. One 0:17:05.810,0:17:08.532 and four. It gives the points 0:17:08.532,0:17:10.389 here. Two and Seven. 0:17:11.150,0:17:15.099 Should be around here and[br]three and 14 which will be 0:17:15.099,0:17:19.407 about here so we can see the[br]kind of shape that this 0:17:19.407,0:17:23.715 function is starting to take[br]here. And we can draw in the 0:17:23.715,0:17:24.074 graph. 0:17:34.700,0:17:38.899 What we want is to say that[br]every input gives us only one 0:17:38.899,0:17:42.775 single output, so we can get our[br]ruler again and just quickly 0:17:42.775,0:17:47.297 check by going along and we can[br]see that as we go along. Our 0:17:47.297,0:17:49.235 rule across is the curve once 0:17:49.235,0:17:53.388 and once only. Which means that[br]this function is valid. 0:17:53.940,0:17:59.044 However, an interesting point to[br]note is this point here. The 0:17:59.044,0:18:03.684 minimum point which actually[br]occurs when X is North .75. 0:18:04.580,0:18:11.184 So with X value of North .75[br]are outputs can take a minimum 0:18:11.184,0:18:12.708 value of 3.875. 0:18:13.320,0:18:17.254 So this means when we look at[br]our domain and range, we need to 0:18:17.254,0:18:20.064 make no restrictions on the[br]domain because our function was 0:18:20.064,0:18:27.040 valid. However. Our range has[br]a minimum of 3.875, so we write 0:18:27.040,0:18:30.510 this. As F of X equals. 0:18:31.290,0:18:34.797 Two X squared minus three X +5. 0:18:36.040,0:18:42.982 And the range F of X has[br]always been more than or equal 0:18:42.982,0:18:48.970 to 3.875. So for[br]the next example. 0:18:50.020,0:18:54.376 What would happen if we had a[br]function F defined by? 0:18:54.960,0:18:59.160 F of X equals[br]one over X. 0:19:00.370,0:19:04.495 Well, that's always the first[br]stage is to substitute in some 0:19:04.495,0:19:05.995 values for the arguments. 0:19:06.920,0:19:09.040 So for F of one. 0:19:09.870,0:19:13.687 The argument is one is 1 / 1[br]just gives US1. 0:19:14.980,0:19:16.450 For Port F of two. 0:19:17.030,0:19:18.730 We just get one half. 0:19:19.760,0:19:22.358 F of three gives us 1/3. 0:19:22.910,0:19:29.498 And F4 will[br]give us 1/4. 0:19:30.520,0:19:35.030 And as before, we can also look[br]at some negative arguments. 0:19:35.710,0:19:37.735 So if I look at F of minus one. 0:19:38.480,0:19:42.528 Skip 1 divided by minus one,[br]which is just minus one. 0:19:43.350,0:19:45.190 F of minus 2. 0:19:45.740,0:19:50.396 Is 1 divided by minus two, which[br]just gives us minus 1/2? 0:19:51.880,0:19:56.401 F of minus three. Same thing[br]will give us minus 1/3. 0:19:57.080,0:19:59.380 And F of minus 4. 0:19:59.940,0:20:02.200 Will give us minus 1/4? 0:20:02.770,0:20:05.714 Now if we look at F of 0. 0:20:05.960,0:20:09.896 We have 1 / 0:20:09.896,0:20:13.270 0. Which is obviously a problem 0:20:13.270,0:20:17.208 for us. Because of this[br]problem, we have to restrict 0:20:17.208,0:20:21.720 our domain so that it does not[br]include the arguments X equals 0:20:21.720,0:20:26.232 0. So let's have a look at[br]what the graph of this 0:20:26.232,0:20:27.736 function actually looks like. 0:20:29.230,0:20:33.169 So let's be 4F of X and are[br]vertical axis for the Outputs. 0:20:34.240,0:20:38.174 An argument sax on[br]the horizontal axis. 0:20:39.640,0:20:46.115 OK, so we've gone[br]over here 1234. 0:20:47.790,0:20:54.698 And minus 1 - 2 -[br]3 - 4 over here. 0:20:54.810,0:20:56.430 As we go. 0:20:57.030,0:20:58.990 All the web to one down to minus 0:20:58.990,0:21:05.260 one. Has it as well? So[br]we've got one and one. 0:21:06.390,0:21:09.456 We've got 2 0:21:09.456,0:21:12.510 1/2. 3 0:21:12.510,0:21:16.046 1/3. 4 0:21:16.046,0:21:23.556 one quarter.[br]We've got minus 1 - 1. 0:21:23.650,0:21:27.046 Minus 2 - 0:21:27.046,0:21:33.750 1/2. Minus[br]3 - 1/3. 0:21:33.750,0:21:37.218 A minus four and minus 1/4. 0:21:37.770,0:21:42.114 OK, and obviously we've excluded[br]0 from our domain. As we said 0:21:42.114,0:21:44.286 before. So if we join these 0:21:44.286,0:21:47.030 points up. And a smooth curve. 0:21:48.830,0:21:53.288 Get something that[br]looks like this. 0:21:54.910,0:21:59.618 Now, obviously we've excluded X[br]equals 0 from our domain, but 0:21:59.618,0:22:03.898 it's also worth noticing here.[br]Thought there's nothing at the 0:22:03.898,0:22:09.890 output of F of X equals 0, so[br]that also ends up being excluded 0:22:09.890,0:22:11.174 from the range. 0:22:11.790,0:22:16.548 So we actually end up with F of[br]X equals one over X. 0:22:17.450,0:22:21.212 And we've got X not equal to 0[br]from the domain. 0:22:21.870,0:22:26.690 And also. In the range F of X[br]never equals 0 either. 0:22:28.910,0:22:30.595 But what's actually happening at 0:22:30.595,0:22:35.408 this point? X equals 0 when the[br]arguments is zero. What is going 0:22:35.408,0:22:40.000 on? Well, let's have a look and[br]will start off by having a look 0:22:40.000,0:22:43.280 what happens as we get closer[br]and closer to 0. 0:22:43.820,0:22:45.910 Now. 0:22:47.040,0:22:52.352 If we start off with a value of[br]1/2 of one and remember F of X 0:22:52.352,0:22:54.676 was just equal to one over X. 0:22:55.340,0:23:01.085 Half of 1 just gives US1, so if[br]I get closer to 0 again, let's 0:23:01.085,0:23:03.000 look at half of 1/2. 0:23:03.000,0:23:05.628 At 1 / 1/2. 0:23:06.200,0:23:11.294 This one over 1/2 which just[br]gives us 2. 0:23:11.300,0:23:15.292 So about F of 0:23:15.292,0:23:21.168 110th. It just gives us 1 / 1[br]tenth, which gives us 10. 0:23:21.820,0:23:29.512 Half of one over 1000 will[br]just give us 1 / 1 0:23:29.512,0:23:33.314 over 1000. Which gives 0:23:33.314,0:23:40.236 us 1000. What about[br]one over 1,000,000, so F of 0:23:40.236,0:23:42.564 one over a million? 0:23:42.570,0:23:47.550 It's actually just in the same[br]way as before, just going to 0:23:47.550,0:23:49.210 give us a million. 0:23:49.220,0:23:52.068 So we can see. 0:23:52.710,0:23:57.316 The US we get closer and closer[br]to zero from the right hand side 0:23:57.316,0:23:59.619 as we saw on our graph before. 0:23:59.620,0:24:03.230 We're getting closer and closer[br]to positive Infinity to the 0:24:03.230,0:24:05.035 graph goes off to positive 0:24:05.035,0:24:09.135 Infinity that. What happens when[br]we approach zero from the left 0:24:09.135,0:24:10.965 hand side? Well, let's have a 0:24:10.965,0:24:17.692 look. This is minus one, just[br]gives us 1 divided by minus one, 0:24:17.692,0:24:19.556 which is minus one. 0:24:20.500,0:24:23.988 F of minus 1/2. 0:24:23.990,0:24:27.790 It's going to be just one[br]divided by minus 1/2. 0:24:28.330,0:24:30.110 Just give his minus 2. 0:24:31.290,0:24:34.401 Half of minus 0:24:34.401,0:24:40.840 110th. Is 1 divided[br]by minus 110th? 0:24:40.910,0:24:47.150 It just gives us minus 10 and we[br]can kind of see a pattern here. 0:24:47.150,0:24:50.478 OK so F of minus one over 1000. 0:24:50.510,0:24:52.760 Will actually give us minus 0:24:52.760,0:24:59.384 1000. And F of[br]minus one over 1,000,000. 0:24:59.390,0:25:05.542 Actually gives[br]us minus 0:25:05.542,0:25:11.467 1,000,000. So you can see that[br]as we approach zero from the 0:25:11.467,0:25:12.895 left or outputs approaches 0:25:12.895,0:25:17.606 negative Infinity. And as we[br]approach zero from the right 0:25:17.606,0:25:21.876 hand side are output approaches[br]positive Infinity, and these are 0:25:21.876,0:25:25.292 very different things. OK, for[br]the last example. 0:25:25.940,0:25:30.241 I just like to look at the[br]function F defined by. 0:25:30.250,0:25:37.576 F of X equals one over[br]X minus two all squared. 0:25:38.310,0:25:41.770 So as always with the examples[br]we've done, it's worthwhile 0:25:41.770,0:25:45.230 started off by looking at some[br]different values for the 0:25:45.230,0:25:51.234 arguments. So we start off with[br]an argument of minus two, so 0:25:51.234,0:25:56.876 half of minus two gives us one[br]over minus 2 - 2 squared. 0:25:57.420,0:26:03.803 Which gives us one over minus 4[br]squared, which is one over 16. 0:26:03.900,0:26:07.388 F of minus one. 0:26:07.390,0:26:14.628 Will give us one over minus 1[br]- 2 all squared, which gives us 0:26:14.628,0:26:19.798 one over minus 3 squared which[br]is one over 9. 0:26:19.820,0:26:26.450 Now, FO arguments of zero will[br]give us one over 0 - 2 0:26:26.450,0:26:32.570 all squared, which is one over[br]minus 2 squared, which works out 0:26:32.570,0:26:34.100 as one quarter. 0:26:34.960,0:26:41.932 And F of one will give[br]us one over 1 - 2 0:26:41.932,0:26:46.880 squared. Which is just one over[br]minus one squared, which gives 0:26:46.880,0:26:48.008 us just one. 0:26:48.580,0:26:55.990 OK. Half[br]of two gives us one over. 0:26:56.690,0:27:01.730 2 - 2 or squared, which gives us[br]one over 0 which presents us 0:27:01.730,0:27:06.050 with exactly the same problem we[br]had in the previous example when 0:27:06.050,0:27:11.810 we had one over X and so we have[br]to exclude X equals 2 from the 0:27:11.810,0:27:19.370 domain. Half of three gives us[br]one over 3 - 2 all squared. 0:27:20.220,0:27:25.225 Which is one over 1 squared is[br]just gives US1 again. 0:27:25.230,0:27:32.440 After four gives us one over 4[br]- 2 or squared, which gives us 0:27:32.440,0:27:33.985 one over 4. 0:27:34.110,0:27:42.080 After 5.[br]Gives us one over 5 - 2 or 0:27:42.080,0:27:48.515 squared which gives us one over[br]3 squared which is 1 ninth and 0:27:48.515,0:27:55.940 finally F of six gives us one[br]over 6 - 2 all squared which is 0:27:55.940,0:28:01.385 one over 4 squared which works[br]out as one over 16. 0:28:01.940,0:28:06.659 Now if we want to plot the graph[br]of this function will probably 0:28:06.659,0:28:09.200 need to put this into a table 0:28:09.200,0:28:15.520 first. So as usual, do our[br]table of X&F of X. 0:28:16.370,0:28:23.104 OK, and we went from minus 2[br]- 1 zero all the way. 0:28:23.880,0:28:26.298 Up to and arguments of sex. 0:28:27.160,0:28:28.896 And the values we got for the 0:28:28.896,0:28:31.932 Outputs. One over 0:28:31.932,0:28:38.160 16. One 9th,[br]one quarter and 0:28:38.160,0:28:44.292 1:01. One quarter,[br]1 ninth and 0:28:44.292,0:28:47.358 one over 16. 0:28:48.270,0:28:51.550 So we plot that onto. 0:28:52.100,0:28:54.120 The graph as before. 0:28:57.060,0:29:00.606 So we have arguments going along[br]the horizontal axis. 0:29:01.290,0:29:03.600 And Outputs going along[br]the vertical axis. 0:29:04.720,0:29:12.112 We've gone from minus[br]1 - 2 over 0:29:12.112,0:29:15.808 there 123456 along this 0:29:15.808,0:29:18.810 way. And then going off, we've 0:29:18.810,0:29:24.960 gone too. One up here, so put in[br]a few of the marks 1/2. 0:29:25.540,0:29:27.250 It's put in 1/4 that. 0:29:28.020,0:29:34.728 Put in 3/4. OK, so we've got[br]minus two and 116th, which is 0:29:34.728,0:29:37.824 going to come in down here. 0:29:38.340,0:29:40.580 Minus one and one 9th. 0:29:41.380,0:29:47.108 For coming over here zero[br]and one quarter. 0:29:47.110,0:29:49.589 Over here. 1 on one. 0:29:50.930,0:29:52.410 Right, the way up here? 0:29:53.200,0:29:57.232 To an. Obviously this was the[br]divide by zero, so we couldn't 0:29:57.232,0:30:00.592 do anything with that. We've[br]excluded, uh, from our domain. 0:30:01.690,0:30:03.040 Three and one. 0:30:03.620,0:30:07.280 Pay up. Four and one quarter. 0:30:08.840,0:30:16.628 I'm here. Five and one,[br]9th and six and 116th. 0:30:16.630,0:30:21.643 Because we've excluded X equals[br]2 from our domain. 0:30:23.370,0:30:26.410 Put dotted line there,[br]so that's an asymptotes. 0:30:27.820,0:30:29.626 And we can draw our curve. 0:30:31.970,0:30:33.298 Up through the points. 0:30:33.980,0:30:36.770 On this side. 0:30:38.190,0:30:42.493 And we can see differently to[br]the other example where F of X 0:30:42.493,0:30:44.479 is one over X this time. 0:30:45.090,0:30:49.965 As we get approach to from both[br]the left and from the right, 0:30:49.965,0:30:54.090 both of the outputs are heading[br]towards positive Infinity, so a 0:30:54.090,0:30:57.840 little bit different, and also[br]because we've excluded X equals 0:30:57.840,0:31:00.465 2 from the domain of function is 0:31:00.465,0:31:05.470 now valid. But most also notice[br]that our range is never zero, 0:31:05.470,0:31:07.315 and it's also never negative. 0:31:07.820,0:31:10.826 So to write this out properly. 0:31:10.830,0:31:16.394 Our function F of X equals one[br]over X minus two all squared. 0:31:17.710,0:31:18.829 And we said. 0:31:19.340,0:31:20.804 They are domain is restricted so 0:31:20.804,0:31:26.506 it doesn't include two. And our[br]range is always more than 0. 0:31:27.640,0:31:31.324 OK, so let's just recap on what[br]we've done in this unit. 0:31:32.310,0:31:34.452 So firstly, the definition of a 0:31:34.452,0:31:39.480 function. And that was that. A[br]function is a rule that Maps are 0:31:39.480,0:31:42.880 unique number X to another[br]unique number F of X. 0:31:44.230,0:31:48.050 Secondly, was the idea that[br]an argument is exactly the 0:31:48.050,0:31:49.578 same as an input. 0:31:51.630,0:31:55.950 Thirdly, we looked at the idea[br]of independent and dependent 0:31:55.950,0:32:00.443 variables. And we said that[br]the input axe was the 0:32:00.443,0:32:03.539 independent variable and the[br]output was the dependent 0:32:03.539,0:32:03.926 variable. 0:32:05.330,0:32:10.230 4th, we looked at the domain and[br]we said that the domain was the 0:32:10.230,0:32:11.630 set of possible inputs. 0:32:12.620,0:32:16.546 And finally we looked at the[br]range and we said that the range 0:32:16.546,0:32:18.358 was the set of possible outputs. 0:32:19.610,0:32:21.514 So now you know how to define a 0:32:21.514,0:32:24.997 function. And how to find[br]the outputs of a function 0:32:24.997,0:32:26.209 for a given argument?