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What is a function?
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Now, one definition of a
function is that a function is a
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rule that Maps 1 unique number
to another unique number. So In
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other words, If I start off with
a number and I apply my
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function, I finish up with
another unique number. So for
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example, let's suppose that my
function added three to any
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number I could start off with a
#2. I apply my function and I
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finish up with the number 5.
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Start off with the number 8 and
I apply my function and I finish
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up with the number 11.
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And if I start off with a number
X and I apply my function, I
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finish up with the number X +3
and we can show this
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mathematically by writing F of X
equals X plus three, where X is
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our inputs, which we often
called the arguments of the
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function and X +3 is our output.
Now suppose I had an argument of
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two, I could write down F of two
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equals. 2 + 3, which gives us an
output of five.
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Suppose I had an argument of
eight. I could write down F of
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eight equals 8 + 3, which gives
me an output of 11.
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And suppose I had an argument of
minus six. I could write F of
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minus 6. Equals minus 6 + 3,
which would give me minus three
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for my output.
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And if I had an argument of zed,
I could write down F of said
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equals zed plus 3.
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And likewise, if I had an
argument of X squared, I could
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write down F of X squared equals
X squared +3.
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Now it is with me pointing out
here that's a lower first sight.
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It might appear that we can
choose any value for argument.
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That's not always the case, as
we shall see later, but because
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we do have some choice on what
number we can pick the argument
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to be, this is sometimes called
the independent variable.
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And because output depends on
our choice of arguments, the
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output is sometimes called the
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dependent variable. Now let's
have a look at an example
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F of X equals 3 X
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squared. Minus 4.
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Now, as we said before, X is our
input which we call the argument
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and that is the independent
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variable. And our output which
is 3 X squared minus four is
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our dependent variable.
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Now we can choose different
values for arguments, which is
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often a good place to start when
we get function like this. So F
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of zero will give us 3 * 0
squared takeaway 4, which is 0 -
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4, which is just minus 4.
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If we start off with an argument
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of one. We get F of one
equals 3 * 1 squared takeaway 4,
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which is just three takeaway
four which gives us minus one.
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If we have an argument of two F
of two equals 3 * 2 squared
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takeaway, four switches 3 * 4,
which is 12 takeaway. Four gives
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us 8. And as I said before,
there's no reason why we can't
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include and negative arguments.
So if I put F of minus one in.
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The three times minus one
squared takeaway 4, which is 3 *
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1 gives us three takeaway. Four
gives us minus one.
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And F of minus 2.
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Would give us three times minus
2 squared takeaway 4 which is 3
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* 4 gives us 12 and take away
four will give us 8.
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So what are we going to do with
these results? Well, one thing
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we can do is put them into a
table to help us plot the graph
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of the function. So if we do a
table of X&F of X.
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The values were chosen for RX
column, which is. Our arguments
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are minus 1 - 2 zero one and
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two. So minus 2
- 1 zero 12.
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And the corresponding outputs
are 8 - 1.
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Minus 4 - 1 and
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8. And we can use this as I
said to help us plot the
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graph of the function.
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So just to copy the table down
again. So we've got handy.
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We have minus 2
- 1 zero 12.
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8 - 1 - 4
- 1 and 8.
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So we plot our graph.
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Of F of X on the vertical
axis, so output and arguments.
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On the horizontal axis.
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So we've got
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one too. Minus 1 -
2.
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And we've got.
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Minus 1 - 2 - 3 - 4.
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I'm going off.
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Up to 81234567 and eight. So
our first point is minus 2
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eight so we can put the
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appear. Our second point, minus
1 - 1 should appear down here.
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0 - 4.
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Here one and minus one.
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Yeah, but up on two and eight
which will appear over here
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and we can see we can draw a
smooth curve through these
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points, which will be the
graph of the function.
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OK, now why are we drawing a
graph of a function? Because
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this is quite useful to us
because we can now read off the
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output of a function for any
given arguments straight off the
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graph without the need to do any
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calculations. So for example, if
we look at two and arguments of
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two, we know that's going to
give us 8 before I do work that
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out. But if we looked and we
wanted to figure out.
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But the output would be when the
argument was one point 5. If we
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follow our lineup and across you
can see that that gives us a
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value between 2:00 and 3:00 for
the output, and if we substitute
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in 1.5 back into our original
expression for the function, you
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can see actually gives us an
exact value of 2.75.
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Now earlier on when I discussed
uniqueness, I said that a unique
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inputs had to give us a unique
output and by that what we mean
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is that for any given argument
we should get only one output.
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One of the benefits of having a
graph of a function is that we
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can check this using our ruler.
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If we line our ruler up
vertically and we move it left
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and right across the graph.
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We can make sure that the rule I
only have across is the graph
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wants at any point.
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And as we can see, that's
clearly the case in this
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example. And when that happens,
the graph is a valid function.
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Now, if we had the example.
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F of X.
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Equals root X.
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A good place to start is always
to substitute in some values for
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the arguments, so F of 0.
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This gives us the square root of
0, which is 0.
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Half of one's own arguments of
one will give us plus or minus
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one for the square root.
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F of two.
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Will give us plus or minus
1.4 just to one decimal place
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there. Half of 3.
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Which gives us the square root
of 3 gives us plus or minus 1.7.
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An F4. Will give us
square root of 4 which is just
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plus or minus 2.
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Now. If we try
to put in any negative
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arguments here you can see
that we're going to run
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into trouble because we
have to try and calculate
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the square root of a
negative number and we'll
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come back to this problem
in a second. But for now,
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let's plot the points that
we've got so far.
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So if we take out F of X
axis vertical again.
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And our arguments access X
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horizontal. We've
got
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1234.
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And there are vertical axis we
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have minus one. Minus 2.
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Plus one. Plus two points.
We've got zero and zero.
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One on plus one.
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And also 1A minus one.
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We've got two and
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positive 1.4. So round
about that and also to negative
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1.4. We've got three and
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positive 1.7. And we've
got three and negative 1.7.
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And finally we have four and +2.
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And four and negative 2.
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OK, and we've got enough points
here that we can draw a smooth
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curve through these points.
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At something it looks like.
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This.
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OK. Now.
As usual, we will apply our
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ruler test to make sure that the
function is valid and you can
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see straight away that when we
line up all the vertically and
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move it across for any given
positive arguments, we're
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getting two outputs. So
obviously we need to do
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something about this to make the
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function valid. One way to get
around this problem is by
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defining route X to take only
positive values or 0.
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This is sometimes called the
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positive square root. So in
effect we lose the bottom
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negative half of this graph.
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And obviously we also have the
issue of the negative arguments,
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and since we can't take the
square root of a negative
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number, we also have to exclude
these from the X axis. Now when
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we start talking about these
kind of restrictions, it's
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important that we use the right
kind of mathematical language.
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So the set of possible inputs is
what we call the domain, and the
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set of possible outputs is what
we call the range.
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So in this case, when we've got
RF of X equals the square root
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of X. We need to restrict our
domain to be X is more than or
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equal to 0, 'cause we only
wanted the positive values and
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zero. But we also notice that
now because we've got rid of the
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bottom half of the graph.
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The only part of the range which
are included are also more than
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or equal to 0. So range is
defined by F of X more than or
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equal to 0. So now we have a
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valid function. So what will do
now is just look at a couple
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more examples to pull together
everything that we've done so
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far and will start with this
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one. Let's look at the function
F of X equals 2 X squared
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minus three X +5.
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No, as usual, a good place to
start when you get a function is
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to substitute in some values for
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the arguments. So let's start
with that. So now arguments of 0
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would give us F of 0, which is 2
* 0 squared.
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Minus 3 * 0 +
5 which is just zero
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takeaway 0 + 5.
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So we get now pose A5 that if we
had an argument of one.
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We get 2 * 1
squared takeaway 3 * 1
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+ 5. Which gives us 2
* 1 here, which is 2.
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Take away 3 * 1 which is take
away three and plus five. So two
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takeaway three is minus 1 + 5
gives us 4.
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OK, we look at an
argument of two.
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We get 2 * 2 squared.
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Take away 3 * 2.
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I'm plus five which gives us 2 *
4, which is 8 and take away 3 *
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2 which is take away 6 and then
forget our plus five at the end.
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So eight takeaway six is 2.
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+5 gives us 7.
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OK. If we look at an argument of
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three. Half of three gives us
2 * 3 squared.
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Take away 3 * 3.
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And +5.
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So this is 2 * 9 here
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18. Take away 3 * 3 which is
take away nine and +5.
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So 18 takeaway 9 is 9 + 5
gives us 14.
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And last but not least, we can
also include a negative
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arguments, so we'll put negative
arguments of minus one. So F of
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minus one gives us two times
minus 1 squared.
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Take away three times minus one.
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And of course, our +5.
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So two times minus one squared
just gives us 2.
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Takeaway minus sorry takeaway
three times negative one which
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just gives us a plus 3.
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And then we've got a +5.
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2 + 3 + 5 just gives us 10.
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And what we can do is as before,
just put this into a table to
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make it nice and easy to make a
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graph of the function. So we put
it into a table of X.
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And F of X.
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For arguments we
had minus 10123.
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For Outputs, we had 10 five.
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4, Seven and
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14. OK, so let's see the graph
of this function then.
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You start off with our.
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F of X on the vertical axis as
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before. An argument.
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X on the horizontal axis.
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And we've got over 2 - 1, The
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One. 2. And
three, and on the vertical axis.
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Go to 15.
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OK, so our first point to plot
is minus one and 10 which will
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give us something there zero and
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five. There's a point here. One
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and four. It gives the points
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here. Two and Seven.
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Should be around here and
three and 14 which will be
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about here so we can see the
kind of shape that this
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00:17:19,407 --> 00:17:23,715
function is starting to take
here. And we can draw in the
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graph.
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What we want is to say that
every input gives us only one
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single output, so we can get our
ruler again and just quickly
240
00:17:42,775 --> 00:17:47,297
check by going along and we can
see that as we go along. Our
241
00:17:47,297 --> 00:17:49,235
rule across is the curve once
242
00:17:49,235 --> 00:17:53,388
and once only. Which means that
this function is valid.
243
00:17:53,940 --> 00:17:59,044
However, an interesting point to
note is this point here. The
244
00:17:59,044 --> 00:18:03,684
minimum point which actually
occurs when X is North .75.
245
00:18:04,580 --> 00:18:11,184
So with X value of North .75
are outputs can take a minimum
246
00:18:11,184 --> 00:18:12,708
value of 3.875.
247
00:18:13,320 --> 00:18:17,254
So this means when we look at
our domain and range, we need to
248
00:18:17,254 --> 00:18:20,064
make no restrictions on the
domain because our function was
249
00:18:20,064 --> 00:18:27,040
valid. However. Our range has
a minimum of 3.875, so we write
250
00:18:27,040 --> 00:18:30,510
this. As F of X equals.
251
00:18:31,290 --> 00:18:34,797
Two X squared minus three X +5.
252
00:18:36,040 --> 00:18:42,982
And the range F of X has
always been more than or equal
253
00:18:42,982 --> 00:18:48,970
to 3.875. So for
the next example.
254
00:18:50,020 --> 00:18:54,376
What would happen if we had a
function F defined by?
255
00:18:54,960 --> 00:18:59,160
F of X equals
one over X.
256
00:19:00,370 --> 00:19:04,495
Well, that's always the first
stage is to substitute in some
257
00:19:04,495 --> 00:19:05,995
values for the arguments.
258
00:19:06,920 --> 00:19:09,040
So for F of one.
259
00:19:09,870 --> 00:19:13,687
The argument is one is 1 / 1
just gives US1.
260
00:19:14,980 --> 00:19:16,450
For Port F of two.
261
00:19:17,030 --> 00:19:18,730
We just get one half.
262
00:19:19,760 --> 00:19:22,358
F of three gives us 1/3.
263
00:19:22,910 --> 00:19:29,498
And F4 will
give us 1/4.
264
00:19:30,520 --> 00:19:35,030
And as before, we can also look
at some negative arguments.
265
00:19:35,710 --> 00:19:37,735
So if I look at F of minus one.
266
00:19:38,480 --> 00:19:42,528
Skip 1 divided by minus one,
which is just minus one.
267
00:19:43,350 --> 00:19:45,190
F of minus 2.
268
00:19:45,740 --> 00:19:50,396
Is 1 divided by minus two, which
just gives us minus 1/2?
269
00:19:51,880 --> 00:19:56,401
F of minus three. Same thing
will give us minus 1/3.
270
00:19:57,080 --> 00:19:59,380
And F of minus 4.
271
00:19:59,940 --> 00:20:02,200
Will give us minus 1/4?
272
00:20:02,770 --> 00:20:05,714
Now if we look at F of 0.
273
00:20:05,960 --> 00:20:09,896
We have 1 /
274
00:20:09,896 --> 00:20:13,270
0. Which is obviously a problem
275
00:20:13,270 --> 00:20:17,208
for us. Because of this
problem, we have to restrict
276
00:20:17,208 --> 00:20:21,720
our domain so that it does not
include the arguments X equals
277
00:20:21,720 --> 00:20:26,232
0. So let's have a look at
what the graph of this
278
00:20:26,232 --> 00:20:27,736
function actually looks like.
279
00:20:29,230 --> 00:20:33,169
So let's be 4F of X and are
vertical axis for the Outputs.
280
00:20:34,240 --> 00:20:38,174
An argument sax on
the horizontal axis.
281
00:20:39,640 --> 00:20:46,115
OK, so we've gone
over here 1234.
282
00:20:47,790 --> 00:20:54,698
And minus 1 - 2 -
3 - 4 over here.
283
00:20:54,810 --> 00:20:56,430
As we go.
284
00:20:57,030 --> 00:20:58,990
All the web to one down to minus
285
00:20:58,990 --> 00:21:05,260
one. Has it as well? So
we've got one and one.
286
00:21:06,390 --> 00:21:09,456
We've got 2
287
00:21:09,456 --> 00:21:12,510
1/2. 3
288
00:21:12,510 --> 00:21:16,046
1/3. 4
289
00:21:16,046 --> 00:21:23,556
one quarter.
We've got minus 1 - 1.
290
00:21:23,650 --> 00:21:27,046
Minus 2 -
291
00:21:27,046 --> 00:21:33,750
1/2. Minus
3 - 1/3.
292
00:21:33,750 --> 00:21:37,218
A minus four and minus 1/4.
293
00:21:37,770 --> 00:21:42,114
OK, and obviously we've excluded
0 from our domain. As we said
294
00:21:42,114 --> 00:21:44,286
before. So if we join these
295
00:21:44,286 --> 00:21:47,030
points up. And a smooth curve.
296
00:21:48,830 --> 00:21:53,288
Get something that
looks like this.
297
00:21:54,910 --> 00:21:59,618
Now, obviously we've excluded X
equals 0 from our domain, but
298
00:21:59,618 --> 00:22:03,898
it's also worth noticing here.
Thought there's nothing at the
299
00:22:03,898 --> 00:22:09,890
output of F of X equals 0, so
that also ends up being excluded
300
00:22:09,890 --> 00:22:11,174
from the range.
301
00:22:11,790 --> 00:22:16,548
So we actually end up with F of
X equals one over X.
302
00:22:17,450 --> 00:22:21,212
And we've got X not equal to 0
from the domain.
303
00:22:21,870 --> 00:22:26,690
And also. In the range F of X
never equals 0 either.
304
00:22:28,910 --> 00:22:30,595
But what's actually happening at
305
00:22:30,595 --> 00:22:35,408
this point? X equals 0 when the
arguments is zero. What is going
306
00:22:35,408 --> 00:22:40,000
on? Well, let's have a look and
will start off by having a look
307
00:22:40,000 --> 00:22:43,280
what happens as we get closer
and closer to 0.
308
00:22:43,820 --> 00:22:45,910
Now.
309
00:22:47,040 --> 00:22:52,352
If we start off with a value of
1/2 of one and remember F of X
310
00:22:52,352 --> 00:22:54,676
was just equal to one over X.
311
00:22:55,340 --> 00:23:01,085
Half of 1 just gives US1, so if
I get closer to 0 again, let's
312
00:23:01,085 --> 00:23:03,000
look at half of 1/2.
313
00:23:03,000 --> 00:23:05,628
At 1 / 1/2.
314
00:23:06,200 --> 00:23:11,294
This one over 1/2 which just
gives us 2.
315
00:23:11,300 --> 00:23:15,292
So about F of
316
00:23:15,292 --> 00:23:21,168
110th. It just gives us 1 / 1
tenth, which gives us 10.
317
00:23:21,820 --> 00:23:29,512
Half of one over 1000 will
just give us 1 / 1
318
00:23:29,512 --> 00:23:33,314
over 1000. Which gives
319
00:23:33,314 --> 00:23:40,236
us 1000. What about
one over 1,000,000, so F of
320
00:23:40,236 --> 00:23:42,564
one over a million?
321
00:23:42,570 --> 00:23:47,550
It's actually just in the same
way as before, just going to
322
00:23:47,550 --> 00:23:49,210
give us a million.
323
00:23:49,220 --> 00:23:52,068
So we can see.
324
00:23:52,710 --> 00:23:57,316
The US we get closer and closer
to zero from the right hand side
325
00:23:57,316 --> 00:23:59,619
as we saw on our graph before.
326
00:23:59,620 --> 00:24:03,230
We're getting closer and closer
to positive Infinity to the
327
00:24:03,230 --> 00:24:05,035
graph goes off to positive
328
00:24:05,035 --> 00:24:09,135
Infinity that. What happens when
we approach zero from the left
329
00:24:09,135 --> 00:24:10,965
hand side? Well, let's have a
330
00:24:10,965 --> 00:24:17,692
look. This is minus one, just
gives us 1 divided by minus one,
331
00:24:17,692 --> 00:24:19,556
which is minus one.
332
00:24:20,500 --> 00:24:23,988
F of minus 1/2.
333
00:24:23,990 --> 00:24:27,790
It's going to be just one
divided by minus 1/2.
334
00:24:28,330 --> 00:24:30,110
Just give his minus 2.
335
00:24:31,290 --> 00:24:34,401
Half of minus
336
00:24:34,401 --> 00:24:40,840
110th. Is 1 divided
by minus 110th?
337
00:24:40,910 --> 00:24:47,150
It just gives us minus 10 and we
can kind of see a pattern here.
338
00:24:47,150 --> 00:24:50,478
OK so F of minus one over 1000.
339
00:24:50,510 --> 00:24:52,760
Will actually give us minus
340
00:24:52,760 --> 00:24:59,384
1000. And F of
minus one over 1,000,000.
341
00:24:59,390 --> 00:25:05,542
Actually gives
us minus
342
00:25:05,542 --> 00:25:11,467
1,000,000. So you can see that
as we approach zero from the
343
00:25:11,467 --> 00:25:12,895
left or outputs approaches
344
00:25:12,895 --> 00:25:17,606
negative Infinity. And as we
approach zero from the right
345
00:25:17,606 --> 00:25:21,876
hand side are output approaches
positive Infinity, and these are
346
00:25:21,876 --> 00:25:25,292
very different things. OK, for
the last example.
347
00:25:25,940 --> 00:25:30,241
I just like to look at the
function F defined by.
348
00:25:30,250 --> 00:25:37,576
F of X equals one over
X minus two all squared.
349
00:25:38,310 --> 00:25:41,770
So as always with the examples
we've done, it's worthwhile
350
00:25:41,770 --> 00:25:45,230
started off by looking at some
different values for the
351
00:25:45,230 --> 00:25:51,234
arguments. So we start off with
an argument of minus two, so
352
00:25:51,234 --> 00:25:56,876
half of minus two gives us one
over minus 2 - 2 squared.
353
00:25:57,420 --> 00:26:03,803
Which gives us one over minus 4
squared, which is one over 16.
354
00:26:03,900 --> 00:26:07,388
F of minus one.
355
00:26:07,390 --> 00:26:14,628
Will give us one over minus 1
- 2 all squared, which gives us
356
00:26:14,628 --> 00:26:19,798
one over minus 3 squared which
is one over 9.
357
00:26:19,820 --> 00:26:26,450
Now, FO arguments of zero will
give us one over 0 - 2
358
00:26:26,450 --> 00:26:32,570
all squared, which is one over
minus 2 squared, which works out
359
00:26:32,570 --> 00:26:34,100
as one quarter.
360
00:26:34,960 --> 00:26:41,932
And F of one will give
us one over 1 - 2
361
00:26:41,932 --> 00:26:46,880
squared. Which is just one over
minus one squared, which gives
362
00:26:46,880 --> 00:26:48,008
us just one.
363
00:26:48,580 --> 00:26:55,990
OK. Half
of two gives us one over.
364
00:26:56,690 --> 00:27:01,730
2 - 2 or squared, which gives us
one over 0 which presents us
365
00:27:01,730 --> 00:27:06,050
with exactly the same problem we
had in the previous example when
366
00:27:06,050 --> 00:27:11,810
we had one over X and so we have
to exclude X equals 2 from the
367
00:27:11,810 --> 00:27:19,370
domain. Half of three gives us
one over 3 - 2 all squared.
368
00:27:20,220 --> 00:27:25,225
Which is one over 1 squared is
just gives US1 again.
369
00:27:25,230 --> 00:27:32,440
After four gives us one over 4
- 2 or squared, which gives us
370
00:27:32,440 --> 00:27:33,985
one over 4.
371
00:27:34,110 --> 00:27:42,080
After 5.
Gives us one over 5 - 2 or
372
00:27:42,080 --> 00:27:48,515
squared which gives us one over
3 squared which is 1 ninth and
373
00:27:48,515 --> 00:27:55,940
finally F of six gives us one
over 6 - 2 all squared which is
374
00:27:55,940 --> 00:28:01,385
one over 4 squared which works
out as one over 16.
375
00:28:01,940 --> 00:28:06,659
Now if we want to plot the graph
of this function will probably
376
00:28:06,659 --> 00:28:09,200
need to put this into a table
377
00:28:09,200 --> 00:28:15,520
first. So as usual, do our
table of X&F of X.
378
00:28:16,370 --> 00:28:23,104
OK, and we went from minus 2
- 1 zero all the way.
379
00:28:23,880 --> 00:28:26,298
Up to and arguments of sex.
380
00:28:27,160 --> 00:28:28,896
And the values we got for the
381
00:28:28,896 --> 00:28:31,932
Outputs. One over
382
00:28:31,932 --> 00:28:38,160
16. One 9th,
one quarter and
383
00:28:38,160 --> 00:28:44,292
1:01. One quarter,
1 ninth and
384
00:28:44,292 --> 00:28:47,358
one over 16.
385
00:28:48,270 --> 00:28:51,550
So we plot that onto.
386
00:28:52,100 --> 00:28:54,120
The graph as before.
387
00:28:57,060 --> 00:29:00,606
So we have arguments going along
the horizontal axis.
388
00:29:01,290 --> 00:29:03,600
And Outputs going along
the vertical axis.
389
00:29:04,720 --> 00:29:12,112
We've gone from minus
1 - 2 over
390
00:29:12,112 --> 00:29:15,808
there 123456 along this
391
00:29:15,808 --> 00:29:18,810
way. And then going off, we've
392
00:29:18,810 --> 00:29:24,960
gone too. One up here, so put in
a few of the marks 1/2.
393
00:29:25,540 --> 00:29:27,250
It's put in 1/4 that.
394
00:29:28,020 --> 00:29:34,728
Put in 3/4. OK, so we've got
minus two and 116th, which is
395
00:29:34,728 --> 00:29:37,824
going to come in down here.
396
00:29:38,340 --> 00:29:40,580
Minus one and one 9th.
397
00:29:41,380 --> 00:29:47,108
For coming over here zero
and one quarter.
398
00:29:47,110 --> 00:29:49,589
Over here. 1 on one.
399
00:29:50,930 --> 00:29:52,410
Right, the way up here?
400
00:29:53,200 --> 00:29:57,232
To an. Obviously this was the
divide by zero, so we couldn't
401
00:29:57,232 --> 00:30:00,592
do anything with that. We've
excluded, uh, from our domain.
402
00:30:01,690 --> 00:30:03,040
Three and one.
403
00:30:03,620 --> 00:30:07,280
Pay up. Four and one quarter.
404
00:30:08,840 --> 00:30:16,628
I'm here. Five and one,
9th and six and 116th.
405
00:30:16,630 --> 00:30:21,643
Because we've excluded X equals
2 from our domain.
406
00:30:23,370 --> 00:30:26,410
Put dotted line there,
so that's an asymptotes.
407
00:30:27,820 --> 00:30:29,626
And we can draw our curve.
408
00:30:31,970 --> 00:30:33,298
Up through the points.
409
00:30:33,980 --> 00:30:36,770
On this side.
410
00:30:38,190 --> 00:30:42,493
And we can see differently to
the other example where F of X
411
00:30:42,493 --> 00:30:44,479
is one over X this time.
412
00:30:45,090 --> 00:30:49,965
As we get approach to from both
the left and from the right,
413
00:30:49,965 --> 00:30:54,090
both of the outputs are heading
towards positive Infinity, so a
414
00:30:54,090 --> 00:30:57,840
little bit different, and also
because we've excluded X equals
415
00:30:57,840 --> 00:31:00,465
2 from the domain of function is
416
00:31:00,465 --> 00:31:05,470
now valid. But most also notice
that our range is never zero,
417
00:31:05,470 --> 00:31:07,315
and it's also never negative.
418
00:31:07,820 --> 00:31:10,826
So to write this out properly.
419
00:31:10,830 --> 00:31:16,394
Our function F of X equals one
over X minus two all squared.
420
00:31:17,710 --> 00:31:18,829
And we said.
421
00:31:19,340 --> 00:31:20,804
They are domain is restricted so
422
00:31:20,804 --> 00:31:26,506
it doesn't include two. And our
range is always more than 0.
423
00:31:27,640 --> 00:31:31,324
OK, so let's just recap on what
we've done in this unit.
424
00:31:32,310 --> 00:31:34,452
So firstly, the definition of a
425
00:31:34,452 --> 00:31:39,480
function. And that was that. A
function is a rule that Maps are
426
00:31:39,480 --> 00:31:42,880
unique number X to another
unique number F of X.
427
00:31:44,230 --> 00:31:48,050
Secondly, was the idea that
an argument is exactly the
428
00:31:48,050 --> 00:31:49,578
same as an input.
429
00:31:51,630 --> 00:31:55,950
Thirdly, we looked at the idea
of independent and dependent
430
00:31:55,950 --> 00:32:00,443
variables. And we said that
the input axe was the
431
00:32:00,443 --> 00:32:03,539
independent variable and the
output was the dependent
432
00:32:03,539 --> 00:32:03,926
variable.
433
00:32:05,330 --> 00:32:10,230
4th, we looked at the domain and
we said that the domain was the
434
00:32:10,230 --> 00:32:11,630
set of possible inputs.
435
00:32:12,620 --> 00:32:16,546
And finally we looked at the
range and we said that the range
436
00:32:16,546 --> 00:32:18,358
was the set of possible outputs.
437
00:32:19,610 --> 00:32:21,514
So now you know how to define a
438
00:32:21,514 --> 00:32:24,997
function. And how to find
the outputs of a function
439
00:32:24,997 --> 00:32:26,209
for a given argument?