[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.11,0:00:03.31,Default,,0000,0000,0000,,What is a function?
Dialogue: 0,0:00:03.97,0:00:08.47,Default,,0000,0000,0000,,Now, one definition of a\Nfunction is that a function is a
Dialogue: 0,0:00:08.47,0:00:12.97,Default,,0000,0000,0000,,rule that Maps 1 unique number\Nto another unique number. So In
Dialogue: 0,0:00:12.97,0:00:17.84,Default,,0000,0000,0000,,other words, If I start off with\Na number and I apply my
Dialogue: 0,0:00:17.84,0:00:21.60,Default,,0000,0000,0000,,function, I finish up with\Nanother unique number. So for
Dialogue: 0,0:00:21.60,0:00:25.34,Default,,0000,0000,0000,,example, let's suppose that my\Nfunction added three to any
Dialogue: 0,0:00:25.34,0:00:30.60,Default,,0000,0000,0000,,number I could start off with a\N#2. I apply my function and I
Dialogue: 0,0:00:30.60,0:00:32.84,Default,,0000,0000,0000,,finish up with the number 5.
Dialogue: 0,0:00:33.44,0:00:37.32,Default,,0000,0000,0000,,Start off with the number 8 and\NI apply my function and I finish
Dialogue: 0,0:00:37.32,0:00:38.70,Default,,0000,0000,0000,,up with the number 11.
Dialogue: 0,0:00:39.37,0:00:45.73,Default,,0000,0000,0000,,And if I start off with a number\NX and I apply my function, I
Dialogue: 0,0:00:45.73,0:00:50.82,Default,,0000,0000,0000,,finish up with the number X +3\Nand we can show this
Dialogue: 0,0:00:50.82,0:00:56.33,Default,,0000,0000,0000,,mathematically by writing F of X\Nequals X plus three, where X is
Dialogue: 0,0:00:56.33,0:01:00.57,Default,,0000,0000,0000,,our inputs, which we often\Ncalled the arguments of the
Dialogue: 0,0:01:00.57,0:01:06.51,Default,,0000,0000,0000,,function and X +3 is our output.\NNow suppose I had an argument of
Dialogue: 0,0:01:06.51,0:01:09.90,Default,,0000,0000,0000,,two, I could write down F of two
Dialogue: 0,0:01:09.90,0:01:14.22,Default,,0000,0000,0000,,equals. 2 + 3, which gives us an\Noutput of five.
Dialogue: 0,0:01:15.08,0:01:21.11,Default,,0000,0000,0000,,Suppose I had an argument of\Neight. I could write down F of
Dialogue: 0,0:01:21.11,0:01:26.68,Default,,0000,0000,0000,,eight equals 8 + 3, which gives\Nme an output of 11.
Dialogue: 0,0:01:27.33,0:01:33.32,Default,,0000,0000,0000,,And suppose I had an argument of\Nminus six. I could write F of
Dialogue: 0,0:01:33.32,0:01:39.80,Default,,0000,0000,0000,,minus 6. Equals minus 6 + 3,\Nwhich would give me minus three
Dialogue: 0,0:01:39.80,0:01:41.14,Default,,0000,0000,0000,,for my output.
Dialogue: 0,0:01:42.03,0:01:48.81,Default,,0000,0000,0000,,And if I had an argument of zed,\NI could write down F of said
Dialogue: 0,0:01:48.81,0:01:50.62,Default,,0000,0000,0000,,equals zed plus 3.
Dialogue: 0,0:01:51.60,0:01:57.26,Default,,0000,0000,0000,,And likewise, if I had an\Nargument of X squared, I could
Dialogue: 0,0:01:57.26,0:02:01.98,Default,,0000,0000,0000,,write down F of X squared equals\NX squared +3.
Dialogue: 0,0:02:02.72,0:02:07.06,Default,,0000,0000,0000,,Now it is with me pointing out\Nhere that's a lower first sight.
Dialogue: 0,0:02:07.06,0:02:10.74,Default,,0000,0000,0000,,It might appear that we can\Nchoose any value for argument.
Dialogue: 0,0:02:10.74,0:02:14.74,Default,,0000,0000,0000,,That's not always the case, as\Nwe shall see later, but because
Dialogue: 0,0:02:14.74,0:02:19.09,Default,,0000,0000,0000,,we do have some choice on what\Nnumber we can pick the argument
Dialogue: 0,0:02:19.09,0:02:22.09,Default,,0000,0000,0000,,to be, this is sometimes called\Nthe independent variable.
Dialogue: 0,0:02:22.71,0:02:26.91,Default,,0000,0000,0000,,And because output depends on\Nour choice of arguments, the
Dialogue: 0,0:02:26.91,0:02:29.01,Default,,0000,0000,0000,,output is sometimes called the
Dialogue: 0,0:02:29.01,0:02:35.65,Default,,0000,0000,0000,,dependent variable. Now let's\Nhave a look at an example
Dialogue: 0,0:02:35.65,0:02:39.43,Default,,0000,0000,0000,,F of X equals 3 X
Dialogue: 0,0:02:39.43,0:02:42.51,Default,,0000,0000,0000,,squared. Minus 4.
Dialogue: 0,0:02:43.08,0:02:48.27,Default,,0000,0000,0000,,Now, as we said before, X is our\Ninput which we call the argument
Dialogue: 0,0:02:48.27,0:02:50.13,Default,,0000,0000,0000,,and that is the independent
Dialogue: 0,0:02:50.13,0:02:54.79,Default,,0000,0000,0000,,variable. And our output which\Nis 3 X squared minus four is
Dialogue: 0,0:02:54.79,0:02:55.81,Default,,0000,0000,0000,,our dependent variable.
Dialogue: 0,0:02:56.86,0:03:01.72,Default,,0000,0000,0000,,Now we can choose different\Nvalues for arguments, which is
Dialogue: 0,0:03:01.72,0:03:08.52,Default,,0000,0000,0000,,often a good place to start when\Nwe get function like this. So F
Dialogue: 0,0:03:08.52,0:03:15.81,Default,,0000,0000,0000,,of zero will give us 3 * 0\Nsquared takeaway 4, which is 0 -
Dialogue: 0,0:03:15.81,0:03:18.73,Default,,0000,0000,0000,,4, which is just minus 4.
Dialogue: 0,0:03:19.54,0:03:21.93,Default,,0000,0000,0000,,If we start off with an argument
Dialogue: 0,0:03:21.93,0:03:28.82,Default,,0000,0000,0000,,of one. We get F of one\Nequals 3 * 1 squared takeaway 4,
Dialogue: 0,0:03:28.82,0:03:34.04,Default,,0000,0000,0000,,which is just three takeaway\Nfour which gives us minus one.
Dialogue: 0,0:03:34.31,0:03:41.24,Default,,0000,0000,0000,,If we have an argument of two F\Nof two equals 3 * 2 squared
Dialogue: 0,0:03:41.24,0:03:46.78,Default,,0000,0000,0000,,takeaway, four switches 3 * 4,\Nwhich is 12 takeaway. Four gives
Dialogue: 0,0:03:46.78,0:03:52.26,Default,,0000,0000,0000,,us 8. And as I said before,\Nthere's no reason why we can't
Dialogue: 0,0:03:52.26,0:03:56.74,Default,,0000,0000,0000,,include and negative arguments.\NSo if I put F of minus one in.
Dialogue: 0,0:03:56.75,0:04:03.12,Default,,0000,0000,0000,,The three times minus one\Nsquared takeaway 4, which is 3 *
Dialogue: 0,0:04:03.12,0:04:08.43,Default,,0000,0000,0000,,1 gives us three takeaway. Four\Ngives us minus one.
Dialogue: 0,0:04:08.45,0:04:11.95,Default,,0000,0000,0000,,And F of minus 2.
Dialogue: 0,0:04:12.01,0:04:17.64,Default,,0000,0000,0000,,Would give us three times minus\N2 squared takeaway 4 which is 3
Dialogue: 0,0:04:17.64,0:04:23.27,Default,,0000,0000,0000,,* 4 gives us 12 and take away\Nfour will give us 8.
Dialogue: 0,0:04:24.12,0:04:28.44,Default,,0000,0000,0000,,So what are we going to do with\Nthese results? Well, one thing
Dialogue: 0,0:04:28.44,0:04:33.42,Default,,0000,0000,0000,,we can do is put them into a\Ntable to help us plot the graph
Dialogue: 0,0:04:33.42,0:04:37.73,Default,,0000,0000,0000,,of the function. So if we do a\Ntable of X&F of X.
Dialogue: 0,0:04:38.56,0:04:43.46,Default,,0000,0000,0000,,The values were chosen for RX\Ncolumn, which is. Our arguments
Dialogue: 0,0:04:43.46,0:04:47.02,Default,,0000,0000,0000,,are minus 1 - 2 zero one and
Dialogue: 0,0:04:47.02,0:04:54.70,Default,,0000,0000,0000,,two. So minus 2\N- 1 zero 12.
Dialogue: 0,0:04:55.43,0:05:00.37,Default,,0000,0000,0000,,And the corresponding outputs\Nare 8 - 1.
Dialogue: 0,0:05:01.50,0:05:05.18,Default,,0000,0000,0000,,Minus 4 - 1 and
Dialogue: 0,0:05:05.18,0:05:09.62,Default,,0000,0000,0000,,8. And we can use this as I\Nsaid to help us plot the
Dialogue: 0,0:05:09.62,0:05:10.58,Default,,0000,0000,0000,,graph of the function.
Dialogue: 0,0:05:11.68,0:05:18.32,Default,,0000,0000,0000,,So just to copy the table down\Nagain. So we've got handy.
Dialogue: 0,0:05:18.99,0:05:25.75,Default,,0000,0000,0000,,We have minus 2\N- 1 zero 12.
Dialogue: 0,0:05:26.42,0:05:32.47,Default,,0000,0000,0000,,8 - 1 - 4\N- 1 and 8.
Dialogue: 0,0:05:33.14,0:05:35.36,Default,,0000,0000,0000,,So we plot our graph.
Dialogue: 0,0:05:36.24,0:05:42.54,Default,,0000,0000,0000,,Of F of X on the vertical\Naxis, so output and arguments.
Dialogue: 0,0:05:43.69,0:05:47.06,Default,,0000,0000,0000,,On the horizontal axis.
Dialogue: 0,0:05:47.06,0:05:50.11,Default,,0000,0000,0000,,So we've got
Dialogue: 0,0:05:50.11,0:05:54.24,Default,,0000,0000,0000,,one too. Minus 1 -\N2.
Dialogue: 0,0:05:55.25,0:05:57.75,Default,,0000,0000,0000,,And we've got.
Dialogue: 0,0:05:57.75,0:06:00.83,Default,,0000,0000,0000,,Minus 1 - 2 - 3 - 4.
Dialogue: 0,0:06:01.82,0:06:04.30,Default,,0000,0000,0000,,I'm going off.
Dialogue: 0,0:06:05.56,0:06:12.64,Default,,0000,0000,0000,,Up to 81234567 and eight. So\Nour first point is minus 2
Dialogue: 0,0:06:12.64,0:06:16.18,Default,,0000,0000,0000,,eight so we can put the
Dialogue: 0,0:06:16.18,0:06:22.32,Default,,0000,0000,0000,,appear. Our second point, minus\N1 - 1 should appear down here.
Dialogue: 0,0:06:23.48,0:06:26.46,Default,,0000,0000,0000,,0 - 4.
Dialogue: 0,0:06:26.46,0:06:29.22,Default,,0000,0000,0000,,Here one and minus one.
Dialogue: 0,0:06:30.48,0:06:34.75,Default,,0000,0000,0000,,Yeah, but up on two and eight\Nwhich will appear over here
Dialogue: 0,0:06:34.75,0:06:39.02,Default,,0000,0000,0000,,and we can see we can draw a\Nsmooth curve through these
Dialogue: 0,0:06:39.02,0:06:42.23,Default,,0000,0000,0000,,points, which will be the\Ngraph of the function.
Dialogue: 0,0:06:49.09,0:06:53.40,Default,,0000,0000,0000,,OK, now why are we drawing a\Ngraph of a function? Because
Dialogue: 0,0:06:53.40,0:06:58.06,Default,,0000,0000,0000,,this is quite useful to us\Nbecause we can now read off the
Dialogue: 0,0:06:58.06,0:07:02.01,Default,,0000,0000,0000,,output of a function for any\Ngiven arguments straight off the
Dialogue: 0,0:07:02.01,0:07:04.53,Default,,0000,0000,0000,,graph without the need to do any
Dialogue: 0,0:07:04.53,0:07:08.02,Default,,0000,0000,0000,,calculations. So for example, if\Nwe look at two and arguments of
Dialogue: 0,0:07:08.02,0:07:11.18,Default,,0000,0000,0000,,two, we know that's going to\Ngive us 8 before I do work that
Dialogue: 0,0:07:11.18,0:07:15.57,Default,,0000,0000,0000,,out. But if we looked and we\Nwanted to figure out.
Dialogue: 0,0:07:16.36,0:07:21.15,Default,,0000,0000,0000,,But the output would be when the\Nargument was one point 5. If we
Dialogue: 0,0:07:21.15,0:07:25.59,Default,,0000,0000,0000,,follow our lineup and across you\Ncan see that that gives us a
Dialogue: 0,0:07:25.59,0:07:29.70,Default,,0000,0000,0000,,value between 2:00 and 3:00 for\Nthe output, and if we substitute
Dialogue: 0,0:07:29.70,0:07:33.46,Default,,0000,0000,0000,,in 1.5 back into our original\Nexpression for the function, you
Dialogue: 0,0:07:33.46,0:07:36.88,Default,,0000,0000,0000,,can see actually gives us an\Nexact value of 2.75.
Dialogue: 0,0:07:37.82,0:07:42.01,Default,,0000,0000,0000,,Now earlier on when I discussed\Nuniqueness, I said that a unique
Dialogue: 0,0:07:42.01,0:07:46.89,Default,,0000,0000,0000,,inputs had to give us a unique\Noutput and by that what we mean
Dialogue: 0,0:07:46.89,0:07:51.08,Default,,0000,0000,0000,,is that for any given argument\Nwe should get only one output.
Dialogue: 0,0:07:51.67,0:07:55.04,Default,,0000,0000,0000,,One of the benefits of having a\Ngraph of a function is that we
Dialogue: 0,0:07:55.04,0:07:56.49,Default,,0000,0000,0000,,can check this using our ruler.
Dialogue: 0,0:07:57.46,0:08:00.92,Default,,0000,0000,0000,,If we line our ruler up\Nvertically and we move it left
Dialogue: 0,0:08:00.92,0:08:02.36,Default,,0000,0000,0000,,and right across the graph.
Dialogue: 0,0:08:03.35,0:08:07.07,Default,,0000,0000,0000,,We can make sure that the rule I\Nonly have across is the graph
Dialogue: 0,0:08:07.07,0:08:08.14,Default,,0000,0000,0000,,wants at any point.
Dialogue: 0,0:08:09.05,0:08:13.33,Default,,0000,0000,0000,,And as we can see, that's\Nclearly the case in this
Dialogue: 0,0:08:13.33,0:08:17.61,Default,,0000,0000,0000,,example. And when that happens,\Nthe graph is a valid function.
Dialogue: 0,0:08:19.30,0:08:22.98,Default,,0000,0000,0000,,Now, if we had the example.
Dialogue: 0,0:08:23.53,0:08:25.61,Default,,0000,0000,0000,,F of X.
Dialogue: 0,0:08:26.31,0:08:28.97,Default,,0000,0000,0000,,Equals root X.
Dialogue: 0,0:08:30.04,0:08:36.45,Default,,0000,0000,0000,,A good place to start is always\Nto substitute in some values for
Dialogue: 0,0:08:36.45,0:08:39.41,Default,,0000,0000,0000,,the arguments, so F of 0.
Dialogue: 0,0:08:39.42,0:08:43.06,Default,,0000,0000,0000,,This gives us the square root of\N0, which is 0.
Dialogue: 0,0:08:43.74,0:08:48.46,Default,,0000,0000,0000,,Half of one's own arguments of\None will give us plus or minus
Dialogue: 0,0:08:48.46,0:08:50.27,Default,,0000,0000,0000,,one for the square root.
Dialogue: 0,0:08:51.01,0:08:53.41,Default,,0000,0000,0000,,F of two.
Dialogue: 0,0:08:53.99,0:09:01.07,Default,,0000,0000,0000,,Will give us plus or minus\N1.4 just to one decimal place
Dialogue: 0,0:09:01.07,0:09:04.78,Default,,0000,0000,0000,,there. Half of 3.
Dialogue: 0,0:09:05.32,0:09:10.95,Default,,0000,0000,0000,,Which gives us the square root\Nof 3 gives us plus or minus 1.7.
Dialogue: 0,0:09:11.70,0:09:18.19,Default,,0000,0000,0000,,An F4. Will give us\Nsquare root of 4 which is just
Dialogue: 0,0:09:18.19,0:09:19.67,Default,,0000,0000,0000,,plus or minus 2.
Dialogue: 0,0:09:20.24,0:09:25.49,Default,,0000,0000,0000,,Now. If we try\Nto put in any negative
Dialogue: 0,0:09:25.49,0:09:28.36,Default,,0000,0000,0000,,arguments here you can see\Nthat we're going to run
Dialogue: 0,0:09:28.36,0:09:30.94,Default,,0000,0000,0000,,into trouble because we\Nhave to try and calculate
Dialogue: 0,0:09:30.94,0:09:33.52,Default,,0000,0000,0000,,the square root of a\Nnegative number and we'll
Dialogue: 0,0:09:33.52,0:09:36.68,Default,,0000,0000,0000,,come back to this problem\Nin a second. But for now,
Dialogue: 0,0:09:36.68,0:09:39.26,Default,,0000,0000,0000,,let's plot the points that\Nwe've got so far.
Dialogue: 0,0:09:40.52,0:09:45.67,Default,,0000,0000,0000,,So if we take out F of X\Naxis vertical again.
Dialogue: 0,0:09:45.67,0:09:48.58,Default,,0000,0000,0000,,And our arguments access X
Dialogue: 0,0:09:48.58,0:09:54.42,Default,,0000,0000,0000,,horizontal. We've\Ngot
Dialogue: 0,0:09:54.42,0:09:56.73,Default,,0000,0000,0000,,1234.
Dialogue: 0,0:10:00.63,0:10:02.17,Default,,0000,0000,0000,,And there are vertical axis we
Dialogue: 0,0:10:02.17,0:10:04.94,Default,,0000,0000,0000,,have minus one. Minus 2.
Dialogue: 0,0:10:06.20,0:10:13.62,Default,,0000,0000,0000,,Plus one. Plus two points.\NWe've got zero and zero.
Dialogue: 0,0:10:14.87,0:10:16.30,Default,,0000,0000,0000,,One on plus one.
Dialogue: 0,0:10:17.23,0:10:19.75,Default,,0000,0000,0000,,And also 1A minus one.
Dialogue: 0,0:10:21.28,0:10:24.50,Default,,0000,0000,0000,,We've got two and
Dialogue: 0,0:10:24.50,0:10:31.72,Default,,0000,0000,0000,,positive 1.4. So round\Nabout that and also to negative
Dialogue: 0,0:10:31.72,0:10:35.15,Default,,0000,0000,0000,,1.4. We've got three and
Dialogue: 0,0:10:35.15,0:10:42.64,Default,,0000,0000,0000,,positive 1.7. And we've\Ngot three and negative 1.7.
Dialogue: 0,0:10:42.74,0:10:45.43,Default,,0000,0000,0000,,And finally we have four and +2.
Dialogue: 0,0:10:46.66,0:10:48.95,Default,,0000,0000,0000,,And four and negative 2.
Dialogue: 0,0:10:49.63,0:10:54.70,Default,,0000,0000,0000,,OK, and we've got enough points\Nhere that we can draw a smooth
Dialogue: 0,0:10:54.70,0:10:56.26,Default,,0000,0000,0000,,curve through these points.
Dialogue: 0,0:10:59.56,0:11:01.26,Default,,0000,0000,0000,,At something it looks like.
Dialogue: 0,0:11:02.51,0:11:03.25,Default,,0000,0000,0000,,This.
Dialogue: 0,0:11:05.20,0:11:11.15,Default,,0000,0000,0000,,OK. Now.\NAs usual, we will apply our
Dialogue: 0,0:11:11.15,0:11:15.66,Default,,0000,0000,0000,,ruler test to make sure that the\Nfunction is valid and you can
Dialogue: 0,0:11:15.66,0:11:19.83,Default,,0000,0000,0000,,see straight away that when we\Nline up all the vertically and
Dialogue: 0,0:11:19.83,0:11:22.95,Default,,0000,0000,0000,,move it across for any given\Npositive arguments, we're
Dialogue: 0,0:11:22.95,0:11:26.07,Default,,0000,0000,0000,,getting two outputs. So\Nobviously we need to do
Dialogue: 0,0:11:26.07,0:11:28.16,Default,,0000,0000,0000,,something about this to make the
Dialogue: 0,0:11:28.16,0:11:32.87,Default,,0000,0000,0000,,function valid. One way to get\Naround this problem is by
Dialogue: 0,0:11:32.87,0:11:36.40,Default,,0000,0000,0000,,defining route X to take only\Npositive values or 0.
Dialogue: 0,0:11:37.23,0:11:38.86,Default,,0000,0000,0000,,This is sometimes called the
Dialogue: 0,0:11:38.86,0:11:42.61,Default,,0000,0000,0000,,positive square root. So in\Neffect we lose the bottom
Dialogue: 0,0:11:42.61,0:11:44.15,Default,,0000,0000,0000,,negative half of this graph.
Dialogue: 0,0:11:44.75,0:11:48.23,Default,,0000,0000,0000,,And obviously we also have the\Nissue of the negative arguments,
Dialogue: 0,0:11:48.23,0:11:51.70,Default,,0000,0000,0000,,and since we can't take the\Nsquare root of a negative
Dialogue: 0,0:11:51.70,0:11:55.81,Default,,0000,0000,0000,,number, we also have to exclude\Nthese from the X axis. Now when
Dialogue: 0,0:11:55.81,0:11:58.65,Default,,0000,0000,0000,,we start talking about these\Nkind of restrictions, it's
Dialogue: 0,0:11:58.65,0:12:01.81,Default,,0000,0000,0000,,important that we use the right\Nkind of mathematical language.
Dialogue: 0,0:12:02.42,0:12:07.11,Default,,0000,0000,0000,,So the set of possible inputs is\Nwhat we call the domain, and the
Dialogue: 0,0:12:07.11,0:12:10.46,Default,,0000,0000,0000,,set of possible outputs is what\Nwe call the range.
Dialogue: 0,0:12:11.12,0:12:16.36,Default,,0000,0000,0000,,So in this case, when we've got\NRF of X equals the square root
Dialogue: 0,0:12:16.36,0:12:22.59,Default,,0000,0000,0000,,of X. We need to restrict our\Ndomain to be X is more than or
Dialogue: 0,0:12:22.59,0:12:26.71,Default,,0000,0000,0000,,equal to 0, 'cause we only\Nwanted the positive values and
Dialogue: 0,0:12:26.71,0:12:31.57,Default,,0000,0000,0000,,zero. But we also notice that\Nnow because we've got rid of the
Dialogue: 0,0:12:31.57,0:12:33.44,Default,,0000,0000,0000,,bottom half of the graph.
Dialogue: 0,0:12:34.34,0:12:39.23,Default,,0000,0000,0000,,The only part of the range which\Nare included are also more than
Dialogue: 0,0:12:39.23,0:12:44.87,Default,,0000,0000,0000,,or equal to 0. So range is\Ndefined by F of X more than or
Dialogue: 0,0:12:44.87,0:12:47.88,Default,,0000,0000,0000,,equal to 0. So now we have a
Dialogue: 0,0:12:47.88,0:12:53.25,Default,,0000,0000,0000,,valid function. So what will do\Nnow is just look at a couple
Dialogue: 0,0:12:53.25,0:12:56.68,Default,,0000,0000,0000,,more examples to pull together\Neverything that we've done so
Dialogue: 0,0:12:56.68,0:12:58.74,Default,,0000,0000,0000,,far and will start with this
Dialogue: 0,0:12:58.74,0:13:06.13,Default,,0000,0000,0000,,one. Let's look at the function\NF of X equals 2 X squared
Dialogue: 0,0:13:06.13,0:13:08.22,Default,,0000,0000,0000,,minus three X +5.
Dialogue: 0,0:13:08.77,0:13:12.97,Default,,0000,0000,0000,,No, as usual, a good place to\Nstart when you get a function is
Dialogue: 0,0:13:12.97,0:13:14.77,Default,,0000,0000,0000,,to substitute in some values for
Dialogue: 0,0:13:14.77,0:13:19.73,Default,,0000,0000,0000,,the arguments. So let's start\Nwith that. So now arguments of 0
Dialogue: 0,0:13:19.73,0:13:24.11,Default,,0000,0000,0000,,would give us F of 0, which is 2\N* 0 squared.
Dialogue: 0,0:13:24.68,0:13:31.48,Default,,0000,0000,0000,,Minus 3 * 0 +\N5 which is just zero
Dialogue: 0,0:13:31.48,0:13:34.20,Default,,0000,0000,0000,,takeaway 0 + 5.
Dialogue: 0,0:13:34.21,0:13:38.76,Default,,0000,0000,0000,,So we get now pose A5 that if we\Nhad an argument of one.
Dialogue: 0,0:13:39.47,0:13:46.81,Default,,0000,0000,0000,,We get 2 * 1\Nsquared takeaway 3 * 1
Dialogue: 0,0:13:46.81,0:13:53.00,Default,,0000,0000,0000,,+ 5. Which gives us 2\N* 1 here, which is 2.
Dialogue: 0,0:13:53.88,0:14:00.09,Default,,0000,0000,0000,,Take away 3 * 1 which is take\Naway three and plus five. So two
Dialogue: 0,0:14:00.09,0:14:04.23,Default,,0000,0000,0000,,takeaway three is minus 1 + 5\Ngives us 4.
Dialogue: 0,0:14:05.12,0:14:08.46,Default,,0000,0000,0000,,OK, we look at an\Nargument of two.
Dialogue: 0,0:14:09.53,0:14:13.12,Default,,0000,0000,0000,,We get 2 * 2 squared.
Dialogue: 0,0:14:13.73,0:14:15.88,Default,,0000,0000,0000,,Take away 3 * 2.
Dialogue: 0,0:14:16.38,0:14:23.91,Default,,0000,0000,0000,,I'm plus five which gives us 2 *\N4, which is 8 and take away 3 *
Dialogue: 0,0:14:23.91,0:14:30.56,Default,,0000,0000,0000,,2 which is take away 6 and then\Nforget our plus five at the end.
Dialogue: 0,0:14:30.56,0:14:33.21,Default,,0000,0000,0000,,So eight takeaway six is 2.
Dialogue: 0,0:14:33.81,0:14:36.31,Default,,0000,0000,0000,,+5 gives us 7.
Dialogue: 0,0:14:36.90,0:14:40.40,Default,,0000,0000,0000,,OK. If we look at an argument of
Dialogue: 0,0:14:40.40,0:14:46.08,Default,,0000,0000,0000,,three. Half of three gives us\N2 * 3 squared.
Dialogue: 0,0:14:47.42,0:14:49.29,Default,,0000,0000,0000,,Take away 3 * 3.
Dialogue: 0,0:14:49.81,0:14:51.30,Default,,0000,0000,0000,,And +5.
Dialogue: 0,0:14:52.47,0:14:56.03,Default,,0000,0000,0000,,So this is 2 * 9 here
Dialogue: 0,0:14:56.03,0:15:01.60,Default,,0000,0000,0000,,18. Take away 3 * 3 which is\Ntake away nine and +5.
Dialogue: 0,0:15:02.31,0:15:07.20,Default,,0000,0000,0000,,So 18 takeaway 9 is 9 + 5\Ngives us 14.
Dialogue: 0,0:15:08.12,0:15:12.90,Default,,0000,0000,0000,,And last but not least, we can\Nalso include a negative
Dialogue: 0,0:15:12.90,0:15:18.12,Default,,0000,0000,0000,,arguments, so we'll put negative\Narguments of minus one. So F of
Dialogue: 0,0:15:18.12,0:15:22.04,Default,,0000,0000,0000,,minus one gives us two times\Nminus 1 squared.
Dialogue: 0,0:15:22.69,0:15:26.05,Default,,0000,0000,0000,,Take away three times minus one.
Dialogue: 0,0:15:27.37,0:15:30.22,Default,,0000,0000,0000,,And of course, our +5.
Dialogue: 0,0:15:30.79,0:15:34.79,Default,,0000,0000,0000,,So two times minus one squared\Njust gives us 2.
Dialogue: 0,0:15:36.16,0:15:39.71,Default,,0000,0000,0000,,Takeaway minus sorry takeaway\Nthree times negative one which
Dialogue: 0,0:15:39.71,0:15:42.07,Default,,0000,0000,0000,,just gives us a plus 3.
Dialogue: 0,0:15:42.95,0:15:45.13,Default,,0000,0000,0000,,And then we've got a +5.
Dialogue: 0,0:15:45.93,0:15:49.19,Default,,0000,0000,0000,,2 + 3 + 5 just gives us 10.
Dialogue: 0,0:15:49.79,0:15:53.39,Default,,0000,0000,0000,,And what we can do is as before,\Njust put this into a table to
Dialogue: 0,0:15:53.39,0:15:55.31,Default,,0000,0000,0000,,make it nice and easy to make a
Dialogue: 0,0:15:55.31,0:16:01.60,Default,,0000,0000,0000,,graph of the function. So we put\Nit into a table of X.
Dialogue: 0,0:16:02.16,0:16:03.94,Default,,0000,0000,0000,,And F of X.
Dialogue: 0,0:16:04.68,0:16:12.08,Default,,0000,0000,0000,,For arguments we\Nhad minus 10123.
Dialogue: 0,0:16:12.94,0:16:16.78,Default,,0000,0000,0000,,For Outputs, we had 10 five.
Dialogue: 0,0:16:17.38,0:16:20.60,Default,,0000,0000,0000,,4, Seven and
Dialogue: 0,0:16:20.60,0:16:24.38,Default,,0000,0000,0000,,14. OK, so let's see the graph\Nof this function then.
Dialogue: 0,0:16:26.53,0:16:27.74,Default,,0000,0000,0000,,You start off with our.
Dialogue: 0,0:16:28.43,0:16:31.45,Default,,0000,0000,0000,,F of X on the vertical axis as
Dialogue: 0,0:16:31.45,0:16:33.82,Default,,0000,0000,0000,,before. An argument.
Dialogue: 0,0:16:34.33,0:16:36.54,Default,,0000,0000,0000,,X on the horizontal axis.
Dialogue: 0,0:16:37.68,0:16:41.10,Default,,0000,0000,0000,,And we've got over 2 - 1, The
Dialogue: 0,0:16:41.10,0:16:47.99,Default,,0000,0000,0000,,One. 2. And\Nthree, and on the vertical axis.
Dialogue: 0,0:16:49.82,0:16:52.75,Default,,0000,0000,0000,,Go to 15.
Dialogue: 0,0:16:53.54,0:16:59.62,Default,,0000,0000,0000,,OK, so our first point to plot\Nis minus one and 10 which will
Dialogue: 0,0:16:59.62,0:17:02.22,Default,,0000,0000,0000,,give us something there zero and
Dialogue: 0,0:17:02.22,0:17:05.81,Default,,0000,0000,0000,,five. There's a point here. One
Dialogue: 0,0:17:05.81,0:17:08.53,Default,,0000,0000,0000,,and four. It gives the points
Dialogue: 0,0:17:08.53,0:17:10.39,Default,,0000,0000,0000,,here. Two and Seven.
Dialogue: 0,0:17:11.15,0:17:15.10,Default,,0000,0000,0000,,Should be around here and\Nthree and 14 which will be
Dialogue: 0,0:17:15.10,0:17:19.41,Default,,0000,0000,0000,,about here so we can see the\Nkind of shape that this
Dialogue: 0,0:17:19.41,0:17:23.72,Default,,0000,0000,0000,,function is starting to take\Nhere. And we can draw in the
Dialogue: 0,0:17:23.72,0:17:24.07,Default,,0000,0000,0000,,graph.
Dialogue: 0,0:17:34.70,0:17:38.90,Default,,0000,0000,0000,,What we want is to say that\Nevery input gives us only one
Dialogue: 0,0:17:38.90,0:17:42.78,Default,,0000,0000,0000,,single output, so we can get our\Nruler again and just quickly
Dialogue: 0,0:17:42.78,0:17:47.30,Default,,0000,0000,0000,,check by going along and we can\Nsee that as we go along. Our
Dialogue: 0,0:17:47.30,0:17:49.24,Default,,0000,0000,0000,,rule across is the curve once
Dialogue: 0,0:17:49.24,0:17:53.39,Default,,0000,0000,0000,,and once only. Which means that\Nthis function is valid.
Dialogue: 0,0:17:53.94,0:17:59.04,Default,,0000,0000,0000,,However, an interesting point to\Nnote is this point here. The
Dialogue: 0,0:17:59.04,0:18:03.68,Default,,0000,0000,0000,,minimum point which actually\Noccurs when X is North .75.
Dialogue: 0,0:18:04.58,0:18:11.18,Default,,0000,0000,0000,,So with X value of North .75\Nare outputs can take a minimum
Dialogue: 0,0:18:11.18,0:18:12.71,Default,,0000,0000,0000,,value of 3.875.
Dialogue: 0,0:18:13.32,0:18:17.25,Default,,0000,0000,0000,,So this means when we look at\Nour domain and range, we need to
Dialogue: 0,0:18:17.25,0:18:20.06,Default,,0000,0000,0000,,make no restrictions on the\Ndomain because our function was
Dialogue: 0,0:18:20.06,0:18:27.04,Default,,0000,0000,0000,,valid. However. Our range has\Na minimum of 3.875, so we write
Dialogue: 0,0:18:27.04,0:18:30.51,Default,,0000,0000,0000,,this. As F of X equals.
Dialogue: 0,0:18:31.29,0:18:34.80,Default,,0000,0000,0000,,Two X squared minus three X +5.
Dialogue: 0,0:18:36.04,0:18:42.98,Default,,0000,0000,0000,,And the range F of X has\Nalways been more than or equal
Dialogue: 0,0:18:42.98,0:18:48.97,Default,,0000,0000,0000,,to 3.875. So for\Nthe next example.
Dialogue: 0,0:18:50.02,0:18:54.38,Default,,0000,0000,0000,,What would happen if we had a\Nfunction F defined by?
Dialogue: 0,0:18:54.96,0:18:59.16,Default,,0000,0000,0000,,F of X equals\None over X.
Dialogue: 0,0:19:00.37,0:19:04.50,Default,,0000,0000,0000,,Well, that's always the first\Nstage is to substitute in some
Dialogue: 0,0:19:04.50,0:19:05.100,Default,,0000,0000,0000,,values for the arguments.
Dialogue: 0,0:19:06.92,0:19:09.04,Default,,0000,0000,0000,,So for F of one.
Dialogue: 0,0:19:09.87,0:19:13.69,Default,,0000,0000,0000,,The argument is one is 1 / 1\Njust gives US1.
Dialogue: 0,0:19:14.98,0:19:16.45,Default,,0000,0000,0000,,For Port F of two.
Dialogue: 0,0:19:17.03,0:19:18.73,Default,,0000,0000,0000,,We just get one half.
Dialogue: 0,0:19:19.76,0:19:22.36,Default,,0000,0000,0000,,F of three gives us 1/3.
Dialogue: 0,0:19:22.91,0:19:29.50,Default,,0000,0000,0000,,And F4 will\Ngive us 1/4.
Dialogue: 0,0:19:30.52,0:19:35.03,Default,,0000,0000,0000,,And as before, we can also look\Nat some negative arguments.
Dialogue: 0,0:19:35.71,0:19:37.74,Default,,0000,0000,0000,,So if I look at F of minus one.
Dialogue: 0,0:19:38.48,0:19:42.53,Default,,0000,0000,0000,,Skip 1 divided by minus one,\Nwhich is just minus one.
Dialogue: 0,0:19:43.35,0:19:45.19,Default,,0000,0000,0000,,F of minus 2.
Dialogue: 0,0:19:45.74,0:19:50.40,Default,,0000,0000,0000,,Is 1 divided by minus two, which\Njust gives us minus 1/2?
Dialogue: 0,0:19:51.88,0:19:56.40,Default,,0000,0000,0000,,F of minus three. Same thing\Nwill give us minus 1/3.
Dialogue: 0,0:19:57.08,0:19:59.38,Default,,0000,0000,0000,,And F of minus 4.
Dialogue: 0,0:19:59.94,0:20:02.20,Default,,0000,0000,0000,,Will give us minus 1/4?
Dialogue: 0,0:20:02.77,0:20:05.71,Default,,0000,0000,0000,,Now if we look at F of 0.
Dialogue: 0,0:20:05.96,0:20:09.90,Default,,0000,0000,0000,,We have 1 /
Dialogue: 0,0:20:09.90,0:20:13.27,Default,,0000,0000,0000,,0. Which is obviously a problem
Dialogue: 0,0:20:13.27,0:20:17.21,Default,,0000,0000,0000,,for us. Because of this\Nproblem, we have to restrict
Dialogue: 0,0:20:17.21,0:20:21.72,Default,,0000,0000,0000,,our domain so that it does not\Ninclude the arguments X equals
Dialogue: 0,0:20:21.72,0:20:26.23,Default,,0000,0000,0000,,0. So let's have a look at\Nwhat the graph of this
Dialogue: 0,0:20:26.23,0:20:27.74,Default,,0000,0000,0000,,function actually looks like.
Dialogue: 0,0:20:29.23,0:20:33.17,Default,,0000,0000,0000,,So let's be 4F of X and are\Nvertical axis for the Outputs.
Dialogue: 0,0:20:34.24,0:20:38.17,Default,,0000,0000,0000,,An argument sax on\Nthe horizontal axis.
Dialogue: 0,0:20:39.64,0:20:46.12,Default,,0000,0000,0000,,OK, so we've gone\Nover here 1234.
Dialogue: 0,0:20:47.79,0:20:54.70,Default,,0000,0000,0000,,And minus 1 - 2 -\N3 - 4 over here.
Dialogue: 0,0:20:54.81,0:20:56.43,Default,,0000,0000,0000,,As we go.
Dialogue: 0,0:20:57.03,0:20:58.99,Default,,0000,0000,0000,,All the web to one down to minus
Dialogue: 0,0:20:58.99,0:21:05.26,Default,,0000,0000,0000,,one. Has it as well? So\Nwe've got one and one.
Dialogue: 0,0:21:06.39,0:21:09.46,Default,,0000,0000,0000,,We've got 2
Dialogue: 0,0:21:09.46,0:21:12.51,Default,,0000,0000,0000,,1/2. 3
Dialogue: 0,0:21:12.51,0:21:16.05,Default,,0000,0000,0000,,1/3. 4
Dialogue: 0,0:21:16.05,0:21:23.56,Default,,0000,0000,0000,,one quarter.\NWe've got minus 1 - 1.
Dialogue: 0,0:21:23.65,0:21:27.05,Default,,0000,0000,0000,,Minus 2 -
Dialogue: 0,0:21:27.05,0:21:33.75,Default,,0000,0000,0000,,1/2. Minus\N3 - 1/3.
Dialogue: 0,0:21:33.75,0:21:37.22,Default,,0000,0000,0000,,A minus four and minus 1/4.
Dialogue: 0,0:21:37.77,0:21:42.11,Default,,0000,0000,0000,,OK, and obviously we've excluded\N0 from our domain. As we said
Dialogue: 0,0:21:42.11,0:21:44.29,Default,,0000,0000,0000,,before. So if we join these
Dialogue: 0,0:21:44.29,0:21:47.03,Default,,0000,0000,0000,,points up. And a smooth curve.
Dialogue: 0,0:21:48.83,0:21:53.29,Default,,0000,0000,0000,,Get something that\Nlooks like this.
Dialogue: 0,0:21:54.91,0:21:59.62,Default,,0000,0000,0000,,Now, obviously we've excluded X\Nequals 0 from our domain, but
Dialogue: 0,0:21:59.62,0:22:03.90,Default,,0000,0000,0000,,it's also worth noticing here.\NThought there's nothing at the
Dialogue: 0,0:22:03.90,0:22:09.89,Default,,0000,0000,0000,,output of F of X equals 0, so\Nthat also ends up being excluded
Dialogue: 0,0:22:09.89,0:22:11.17,Default,,0000,0000,0000,,from the range.
Dialogue: 0,0:22:11.79,0:22:16.55,Default,,0000,0000,0000,,So we actually end up with F of\NX equals one over X.
Dialogue: 0,0:22:17.45,0:22:21.21,Default,,0000,0000,0000,,And we've got X not equal to 0\Nfrom the domain.
Dialogue: 0,0:22:21.87,0:22:26.69,Default,,0000,0000,0000,,And also. In the range F of X\Nnever equals 0 either.
Dialogue: 0,0:22:28.91,0:22:30.60,Default,,0000,0000,0000,,But what's actually happening at
Dialogue: 0,0:22:30.60,0:22:35.41,Default,,0000,0000,0000,,this point? X equals 0 when the\Narguments is zero. What is going
Dialogue: 0,0:22:35.41,0:22:40.00,Default,,0000,0000,0000,,on? Well, let's have a look and\Nwill start off by having a look
Dialogue: 0,0:22:40.00,0:22:43.28,Default,,0000,0000,0000,,what happens as we get closer\Nand closer to 0.
Dialogue: 0,0:22:43.82,0:22:45.91,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:22:47.04,0:22:52.35,Default,,0000,0000,0000,,If we start off with a value of\N1/2 of one and remember F of X
Dialogue: 0,0:22:52.35,0:22:54.68,Default,,0000,0000,0000,,was just equal to one over X.
Dialogue: 0,0:22:55.34,0:23:01.08,Default,,0000,0000,0000,,Half of 1 just gives US1, so if\NI get closer to 0 again, let's
Dialogue: 0,0:23:01.08,0:23:03.00,Default,,0000,0000,0000,,look at half of 1/2.
Dialogue: 0,0:23:03.00,0:23:05.63,Default,,0000,0000,0000,,At 1 / 1/2.
Dialogue: 0,0:23:06.20,0:23:11.29,Default,,0000,0000,0000,,This one over 1/2 which just\Ngives us 2.
Dialogue: 0,0:23:11.30,0:23:15.29,Default,,0000,0000,0000,,So about F of
Dialogue: 0,0:23:15.29,0:23:21.17,Default,,0000,0000,0000,,110th. It just gives us 1 / 1\Ntenth, which gives us 10.
Dialogue: 0,0:23:21.82,0:23:29.51,Default,,0000,0000,0000,,Half of one over 1000 will\Njust give us 1 / 1
Dialogue: 0,0:23:29.51,0:23:33.31,Default,,0000,0000,0000,,over 1000. Which gives
Dialogue: 0,0:23:33.31,0:23:40.24,Default,,0000,0000,0000,,us 1000. What about\None over 1,000,000, so F of
Dialogue: 0,0:23:40.24,0:23:42.56,Default,,0000,0000,0000,,one over a million?
Dialogue: 0,0:23:42.57,0:23:47.55,Default,,0000,0000,0000,,It's actually just in the same\Nway as before, just going to
Dialogue: 0,0:23:47.55,0:23:49.21,Default,,0000,0000,0000,,give us a million.
Dialogue: 0,0:23:49.22,0:23:52.07,Default,,0000,0000,0000,,So we can see.
Dialogue: 0,0:23:52.71,0:23:57.32,Default,,0000,0000,0000,,The US we get closer and closer\Nto zero from the right hand side
Dialogue: 0,0:23:57.32,0:23:59.62,Default,,0000,0000,0000,,as we saw on our graph before.
Dialogue: 0,0:23:59.62,0:24:03.23,Default,,0000,0000,0000,,We're getting closer and closer\Nto positive Infinity to the
Dialogue: 0,0:24:03.23,0:24:05.04,Default,,0000,0000,0000,,graph goes off to positive
Dialogue: 0,0:24:05.04,0:24:09.14,Default,,0000,0000,0000,,Infinity that. What happens when\Nwe approach zero from the left
Dialogue: 0,0:24:09.14,0:24:10.96,Default,,0000,0000,0000,,hand side? Well, let's have a
Dialogue: 0,0:24:10.96,0:24:17.69,Default,,0000,0000,0000,,look. This is minus one, just\Ngives us 1 divided by minus one,
Dialogue: 0,0:24:17.69,0:24:19.56,Default,,0000,0000,0000,,which is minus one.
Dialogue: 0,0:24:20.50,0:24:23.99,Default,,0000,0000,0000,,F of minus 1/2.
Dialogue: 0,0:24:23.99,0:24:27.79,Default,,0000,0000,0000,,It's going to be just one\Ndivided by minus 1/2.
Dialogue: 0,0:24:28.33,0:24:30.11,Default,,0000,0000,0000,,Just give his minus 2.
Dialogue: 0,0:24:31.29,0:24:34.40,Default,,0000,0000,0000,,Half of minus
Dialogue: 0,0:24:34.40,0:24:40.84,Default,,0000,0000,0000,,110th. Is 1 divided\Nby minus 110th?
Dialogue: 0,0:24:40.91,0:24:47.15,Default,,0000,0000,0000,,It just gives us minus 10 and we\Ncan kind of see a pattern here.
Dialogue: 0,0:24:47.15,0:24:50.48,Default,,0000,0000,0000,,OK so F of minus one over 1000.
Dialogue: 0,0:24:50.51,0:24:52.76,Default,,0000,0000,0000,,Will actually give us minus
Dialogue: 0,0:24:52.76,0:24:59.38,Default,,0000,0000,0000,,1000. And F of\Nminus one over 1,000,000.
Dialogue: 0,0:24:59.39,0:25:05.54,Default,,0000,0000,0000,,Actually gives\Nus minus
Dialogue: 0,0:25:05.54,0:25:11.47,Default,,0000,0000,0000,,1,000,000. So you can see that\Nas we approach zero from the
Dialogue: 0,0:25:11.47,0:25:12.90,Default,,0000,0000,0000,,left or outputs approaches
Dialogue: 0,0:25:12.90,0:25:17.61,Default,,0000,0000,0000,,negative Infinity. And as we\Napproach zero from the right
Dialogue: 0,0:25:17.61,0:25:21.88,Default,,0000,0000,0000,,hand side are output approaches\Npositive Infinity, and these are
Dialogue: 0,0:25:21.88,0:25:25.29,Default,,0000,0000,0000,,very different things. OK, for\Nthe last example.
Dialogue: 0,0:25:25.94,0:25:30.24,Default,,0000,0000,0000,,I just like to look at the\Nfunction F defined by.
Dialogue: 0,0:25:30.25,0:25:37.58,Default,,0000,0000,0000,,F of X equals one over\NX minus two all squared.
Dialogue: 0,0:25:38.31,0:25:41.77,Default,,0000,0000,0000,,So as always with the examples\Nwe've done, it's worthwhile
Dialogue: 0,0:25:41.77,0:25:45.23,Default,,0000,0000,0000,,started off by looking at some\Ndifferent values for the
Dialogue: 0,0:25:45.23,0:25:51.23,Default,,0000,0000,0000,,arguments. So we start off with\Nan argument of minus two, so
Dialogue: 0,0:25:51.23,0:25:56.88,Default,,0000,0000,0000,,half of minus two gives us one\Nover minus 2 - 2 squared.
Dialogue: 0,0:25:57.42,0:26:03.80,Default,,0000,0000,0000,,Which gives us one over minus 4\Nsquared, which is one over 16.
Dialogue: 0,0:26:03.90,0:26:07.39,Default,,0000,0000,0000,,F of minus one.
Dialogue: 0,0:26:07.39,0:26:14.63,Default,,0000,0000,0000,,Will give us one over minus 1\N- 2 all squared, which gives us
Dialogue: 0,0:26:14.63,0:26:19.80,Default,,0000,0000,0000,,one over minus 3 squared which\Nis one over 9.
Dialogue: 0,0:26:19.82,0:26:26.45,Default,,0000,0000,0000,,Now, FO arguments of zero will\Ngive us one over 0 - 2
Dialogue: 0,0:26:26.45,0:26:32.57,Default,,0000,0000,0000,,all squared, which is one over\Nminus 2 squared, which works out
Dialogue: 0,0:26:32.57,0:26:34.10,Default,,0000,0000,0000,,as one quarter.
Dialogue: 0,0:26:34.96,0:26:41.93,Default,,0000,0000,0000,,And F of one will give\Nus one over 1 - 2
Dialogue: 0,0:26:41.93,0:26:46.88,Default,,0000,0000,0000,,squared. Which is just one over\Nminus one squared, which gives
Dialogue: 0,0:26:46.88,0:26:48.01,Default,,0000,0000,0000,,us just one.
Dialogue: 0,0:26:48.58,0:26:55.99,Default,,0000,0000,0000,,OK. Half\Nof two gives us one over.
Dialogue: 0,0:26:56.69,0:27:01.73,Default,,0000,0000,0000,,2 - 2 or squared, which gives us\None over 0 which presents us
Dialogue: 0,0:27:01.73,0:27:06.05,Default,,0000,0000,0000,,with exactly the same problem we\Nhad in the previous example when
Dialogue: 0,0:27:06.05,0:27:11.81,Default,,0000,0000,0000,,we had one over X and so we have\Nto exclude X equals 2 from the
Dialogue: 0,0:27:11.81,0:27:19.37,Default,,0000,0000,0000,,domain. Half of three gives us\None over 3 - 2 all squared.
Dialogue: 0,0:27:20.22,0:27:25.22,Default,,0000,0000,0000,,Which is one over 1 squared is\Njust gives US1 again.
Dialogue: 0,0:27:25.23,0:27:32.44,Default,,0000,0000,0000,,After four gives us one over 4\N- 2 or squared, which gives us
Dialogue: 0,0:27:32.44,0:27:33.98,Default,,0000,0000,0000,,one over 4.
Dialogue: 0,0:27:34.11,0:27:42.08,Default,,0000,0000,0000,,After 5.\NGives us one over 5 - 2 or
Dialogue: 0,0:27:42.08,0:27:48.52,Default,,0000,0000,0000,,squared which gives us one over\N3 squared which is 1 ninth and
Dialogue: 0,0:27:48.52,0:27:55.94,Default,,0000,0000,0000,,finally F of six gives us one\Nover 6 - 2 all squared which is
Dialogue: 0,0:27:55.94,0:28:01.38,Default,,0000,0000,0000,,one over 4 squared which works\Nout as one over 16.
Dialogue: 0,0:28:01.94,0:28:06.66,Default,,0000,0000,0000,,Now if we want to plot the graph\Nof this function will probably
Dialogue: 0,0:28:06.66,0:28:09.20,Default,,0000,0000,0000,,need to put this into a table
Dialogue: 0,0:28:09.20,0:28:15.52,Default,,0000,0000,0000,,first. So as usual, do our\Ntable of X&F of X.
Dialogue: 0,0:28:16.37,0:28:23.10,Default,,0000,0000,0000,,OK, and we went from minus 2\N- 1 zero all the way.
Dialogue: 0,0:28:23.88,0:28:26.30,Default,,0000,0000,0000,,Up to and arguments of sex.
Dialogue: 0,0:28:27.16,0:28:28.90,Default,,0000,0000,0000,,And the values we got for the
Dialogue: 0,0:28:28.90,0:28:31.93,Default,,0000,0000,0000,,Outputs. One over
Dialogue: 0,0:28:31.93,0:28:38.16,Default,,0000,0000,0000,,16. One 9th,\None quarter and
Dialogue: 0,0:28:38.16,0:28:44.29,Default,,0000,0000,0000,,1:01. One quarter,\N1 ninth and
Dialogue: 0,0:28:44.29,0:28:47.36,Default,,0000,0000,0000,,one over 16.
Dialogue: 0,0:28:48.27,0:28:51.55,Default,,0000,0000,0000,,So we plot that onto.
Dialogue: 0,0:28:52.10,0:28:54.12,Default,,0000,0000,0000,,The graph as before.
Dialogue: 0,0:28:57.06,0:29:00.61,Default,,0000,0000,0000,,So we have arguments going along\Nthe horizontal axis.
Dialogue: 0,0:29:01.29,0:29:03.60,Default,,0000,0000,0000,,And Outputs going along\Nthe vertical axis.
Dialogue: 0,0:29:04.72,0:29:12.11,Default,,0000,0000,0000,,We've gone from minus\N1 - 2 over
Dialogue: 0,0:29:12.11,0:29:15.81,Default,,0000,0000,0000,,there 123456 along this
Dialogue: 0,0:29:15.81,0:29:18.81,Default,,0000,0000,0000,,way. And then going off, we've
Dialogue: 0,0:29:18.81,0:29:24.96,Default,,0000,0000,0000,,gone too. One up here, so put in\Na few of the marks 1/2.
Dialogue: 0,0:29:25.54,0:29:27.25,Default,,0000,0000,0000,,It's put in 1/4 that.
Dialogue: 0,0:29:28.02,0:29:34.73,Default,,0000,0000,0000,,Put in 3/4. OK, so we've got\Nminus two and 116th, which is
Dialogue: 0,0:29:34.73,0:29:37.82,Default,,0000,0000,0000,,going to come in down here.
Dialogue: 0,0:29:38.34,0:29:40.58,Default,,0000,0000,0000,,Minus one and one 9th.
Dialogue: 0,0:29:41.38,0:29:47.11,Default,,0000,0000,0000,,For coming over here zero\Nand one quarter.
Dialogue: 0,0:29:47.11,0:29:49.59,Default,,0000,0000,0000,,Over here. 1 on one.
Dialogue: 0,0:29:50.93,0:29:52.41,Default,,0000,0000,0000,,Right, the way up here?
Dialogue: 0,0:29:53.20,0:29:57.23,Default,,0000,0000,0000,,To an. Obviously this was the\Ndivide by zero, so we couldn't
Dialogue: 0,0:29:57.23,0:30:00.59,Default,,0000,0000,0000,,do anything with that. We've\Nexcluded, uh, from our domain.
Dialogue: 0,0:30:01.69,0:30:03.04,Default,,0000,0000,0000,,Three and one.
Dialogue: 0,0:30:03.62,0:30:07.28,Default,,0000,0000,0000,,Pay up. Four and one quarter.
Dialogue: 0,0:30:08.84,0:30:16.63,Default,,0000,0000,0000,,I'm here. Five and one,\N9th and six and 116th.
Dialogue: 0,0:30:16.63,0:30:21.64,Default,,0000,0000,0000,,Because we've excluded X equals\N2 from our domain.
Dialogue: 0,0:30:23.37,0:30:26.41,Default,,0000,0000,0000,,Put dotted line there,\Nso that's an asymptotes.
Dialogue: 0,0:30:27.82,0:30:29.63,Default,,0000,0000,0000,,And we can draw our curve.
Dialogue: 0,0:30:31.97,0:30:33.30,Default,,0000,0000,0000,,Up through the points.
Dialogue: 0,0:30:33.98,0:30:36.77,Default,,0000,0000,0000,,On this side.
Dialogue: 0,0:30:38.19,0:30:42.49,Default,,0000,0000,0000,,And we can see differently to\Nthe other example where F of X
Dialogue: 0,0:30:42.49,0:30:44.48,Default,,0000,0000,0000,,is one over X this time.
Dialogue: 0,0:30:45.09,0:30:49.96,Default,,0000,0000,0000,,As we get approach to from both\Nthe left and from the right,
Dialogue: 0,0:30:49.96,0:30:54.09,Default,,0000,0000,0000,,both of the outputs are heading\Ntowards positive Infinity, so a
Dialogue: 0,0:30:54.09,0:30:57.84,Default,,0000,0000,0000,,little bit different, and also\Nbecause we've excluded X equals
Dialogue: 0,0:30:57.84,0:31:00.46,Default,,0000,0000,0000,,2 from the domain of function is
Dialogue: 0,0:31:00.46,0:31:05.47,Default,,0000,0000,0000,,now valid. But most also notice\Nthat our range is never zero,
Dialogue: 0,0:31:05.47,0:31:07.32,Default,,0000,0000,0000,,and it's also never negative.
Dialogue: 0,0:31:07.82,0:31:10.83,Default,,0000,0000,0000,,So to write this out properly.
Dialogue: 0,0:31:10.83,0:31:16.39,Default,,0000,0000,0000,,Our function F of X equals one\Nover X minus two all squared.
Dialogue: 0,0:31:17.71,0:31:18.83,Default,,0000,0000,0000,,And we said.
Dialogue: 0,0:31:19.34,0:31:20.80,Default,,0000,0000,0000,,They are domain is restricted so
Dialogue: 0,0:31:20.80,0:31:26.51,Default,,0000,0000,0000,,it doesn't include two. And our\Nrange is always more than 0.
Dialogue: 0,0:31:27.64,0:31:31.32,Default,,0000,0000,0000,,OK, so let's just recap on what\Nwe've done in this unit.
Dialogue: 0,0:31:32.31,0:31:34.45,Default,,0000,0000,0000,,So firstly, the definition of a
Dialogue: 0,0:31:34.45,0:31:39.48,Default,,0000,0000,0000,,function. And that was that. A\Nfunction is a rule that Maps are
Dialogue: 0,0:31:39.48,0:31:42.88,Default,,0000,0000,0000,,unique number X to another\Nunique number F of X.
Dialogue: 0,0:31:44.23,0:31:48.05,Default,,0000,0000,0000,,Secondly, was the idea that\Nan argument is exactly the
Dialogue: 0,0:31:48.05,0:31:49.58,Default,,0000,0000,0000,,same as an input.
Dialogue: 0,0:31:51.63,0:31:55.95,Default,,0000,0000,0000,,Thirdly, we looked at the idea\Nof independent and dependent
Dialogue: 0,0:31:55.95,0:32:00.44,Default,,0000,0000,0000,,variables. And we said that\Nthe input axe was the
Dialogue: 0,0:32:00.44,0:32:03.54,Default,,0000,0000,0000,,independent variable and the\Noutput was the dependent
Dialogue: 0,0:32:03.54,0:32:03.93,Default,,0000,0000,0000,,variable.
Dialogue: 0,0:32:05.33,0:32:10.23,Default,,0000,0000,0000,,4th, we looked at the domain and\Nwe said that the domain was the
Dialogue: 0,0:32:10.23,0:32:11.63,Default,,0000,0000,0000,,set of possible inputs.
Dialogue: 0,0:32:12.62,0:32:16.55,Default,,0000,0000,0000,,And finally we looked at the\Nrange and we said that the range
Dialogue: 0,0:32:16.55,0:32:18.36,Default,,0000,0000,0000,,was the set of possible outputs.
Dialogue: 0,0:32:19.61,0:32:21.51,Default,,0000,0000,0000,,So now you know how to define a
Dialogue: 0,0:32:21.51,0:32:24.100,Default,,0000,0000,0000,,function. And how to find\Nthe outputs of a function
Dialogue: 0,0:32:24.100,0:32:26.21,Default,,0000,0000,0000,,for a given argument?