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?