
What is a function?

Now, one definition of a
function is that a function is a

rule that Maps 1 unique number
to another unique number. So In

other words, If I start off with
a number and I apply my

function, I finish up with
another unique number. So for

example, let's suppose that my
function added three to any

number I could start off with a
#2. I apply my function and I

finish up with the number 5.

Start off with the number 8 and
I apply my function and I finish

up with the number 11.

And if I start off with a number
X and I apply my function, I

finish up with the number X +3
and we can show this

mathematically by writing F of X
equals X plus three, where X is

our inputs, which we often
called the arguments of the

function and X +3 is our output.
Now suppose I had an argument of

two, I could write down F of two

equals. 2 + 3, which gives us an
output of five.

Suppose I had an argument of
eight. I could write down F of

eight equals 8 + 3, which gives
me an output of 11.

And suppose I had an argument of
minus six. I could write F of

minus 6. Equals minus 6 + 3,
which would give me minus three

for my output.

And if I had an argument of zed,
I could write down F of said

equals zed plus 3.

And likewise, if I had an
argument of X squared, I could

write down F of X squared equals
X squared +3.

Now it is with me pointing out
here that's a lower first sight.

It might appear that we can
choose any value for argument.

That's not always the case, as
we shall see later, but because

we do have some choice on what
number we can pick the argument

to be, this is sometimes called
the independent variable.

And because output depends on
our choice of arguments, the

output is sometimes called the

dependent variable. Now let's
have a look at an example

F of X equals 3 X

squared. Minus 4.

Now, as we said before, X is our
input which we call the argument

and that is the independent

variable. And our output which
is 3 X squared minus four is

our dependent variable.

Now we can choose different
values for arguments, which is

often a good place to start when
we get function like this. So F

of zero will give us 3 * 0
squared takeaway 4, which is 0 

4, which is just minus 4.

If we start off with an argument

of one. We get F of one
equals 3 * 1 squared takeaway 4,

which is just three takeaway
four which gives us minus one.

If we have an argument of two F
of two equals 3 * 2 squared

takeaway, four switches 3 * 4,
which is 12 takeaway. Four gives

us 8. And as I said before,
there's no reason why we can't

include and negative arguments.
So if I put F of minus one in.

The three times minus one
squared takeaway 4, which is 3 *

1 gives us three takeaway. Four
gives us minus one.

And F of minus 2.

Would give us three times minus
2 squared takeaway 4 which is 3

* 4 gives us 12 and take away
four will give us 8.

So what are we going to do with
these results? Well, one thing

we can do is put them into a
table to help us plot the graph

of the function. So if we do a
table of X&F of X.

The values were chosen for RX
column, which is. Our arguments

are minus 1  2 zero one and

two. So minus 2
 1 zero 12.

And the corresponding outputs
are 8  1.

Minus 4  1 and

8. And we can use this as I
said to help us plot the

graph of the function.

So just to copy the table down
again. So we've got handy.

We have minus 2
 1 zero 12.

8  1  4
 1 and 8.

So we plot our graph.

Of F of X on the vertical
axis, so output and arguments.

On the horizontal axis.

So we've got

one too. Minus 1 
2.

And we've got.

Minus 1  2  3  4.

I'm going off.

Up to 81234567 and eight. So
our first point is minus 2

eight so we can put the

appear. Our second point, minus
1  1 should appear down here.

0  4.

Here one and minus one.

Yeah, but up on two and eight
which will appear over here

and we can see we can draw a
smooth curve through these

points, which will be the
graph of the function.

OK, now why are we drawing a
graph of a function? Because

this is quite useful to us
because we can now read off the

output of a function for any
given arguments straight off the

graph without the need to do any

calculations. So for example, if
we look at two and arguments of

two, we know that's going to
give us 8 before I do work that

out. But if we looked and we
wanted to figure out.

But the output would be when the
argument was one point 5. If we

follow our lineup and across you
can see that that gives us a

value between 2:00 and 3:00 for
the output, and if we substitute

in 1.5 back into our original
expression for the function, you

can see actually gives us an
exact value of 2.75.

Now earlier on when I discussed
uniqueness, I said that a unique

inputs had to give us a unique
output and by that what we mean

is that for any given argument
we should get only one output.

One of the benefits of having a
graph of a function is that we

can check this using our ruler.

If we line our ruler up
vertically and we move it left

and right across the graph.

We can make sure that the rule I
only have across is the graph

wants at any point.

And as we can see, that's
clearly the case in this

example. And when that happens,
the graph is a valid function.

Now, if we had the example.

F of X.

Equals root X.

A good place to start is always
to substitute in some values for

the arguments, so F of 0.

This gives us the square root of
0, which is 0.

Half of one's own arguments of
one will give us plus or minus

one for the square root.

F of two.

Will give us plus or minus
1.4 just to one decimal place

there. Half of 3.

Which gives us the square root
of 3 gives us plus or minus 1.7.

An F4. Will give us
square root of 4 which is just

plus or minus 2.

Now. If we try
to put in any negative

arguments here you can see
that we're going to run

into trouble because we
have to try and calculate

the square root of a
negative number and we'll

come back to this problem
in a second. But for now,

let's plot the points that
we've got so far.

So if we take out F of X
axis vertical again.

And our arguments access X

horizontal. We've
got

1234.

And there are vertical axis we

have minus one. Minus 2.

Plus one. Plus two points.
We've got zero and zero.

One on plus one.

And also 1A minus one.

We've got two and

positive 1.4. So round
about that and also to negative

1.4. We've got three and

positive 1.7. And we've
got three and negative 1.7.

And finally we have four and +2.

And four and negative 2.

OK, and we've got enough points
here that we can draw a smooth

curve through these points.

At something it looks like.

This.

OK. Now.
As usual, we will apply our

ruler test to make sure that the
function is valid and you can

see straight away that when we
line up all the vertically and

move it across for any given
positive arguments, we're

getting two outputs. So
obviously we need to do

something about this to make the

function valid. One way to get
around this problem is by

defining route X to take only
positive values or 0.

This is sometimes called the

positive square root. So in
effect we lose the bottom

negative half of this graph.

And obviously we also have the
issue of the negative arguments,

and since we can't take the
square root of a negative

number, we also have to exclude
these from the X axis. Now when

we start talking about these
kind of restrictions, it's

important that we use the right
kind of mathematical language.

So the set of possible inputs is
what we call the domain, and the

set of possible outputs is what
we call the range.

So in this case, when we've got
RF of X equals the square root

of X. We need to restrict our
domain to be X is more than or

equal to 0, 'cause we only
wanted the positive values and

zero. But we also notice that
now because we've got rid of the

bottom half of the graph.

The only part of the range which
are included are also more than

or equal to 0. So range is
defined by F of X more than or

equal to 0. So now we have a

valid function. So what will do
now is just look at a couple

more examples to pull together
everything that we've done so

far and will start with this

one. Let's look at the function
F of X equals 2 X squared

minus three X +5.

No, as usual, a good place to
start when you get a function is

to substitute in some values for

the arguments. So let's start
with that. So now arguments of 0

would give us F of 0, which is 2
* 0 squared.

Minus 3 * 0 +
5 which is just zero

takeaway 0 + 5.

So we get now pose A5 that if we
had an argument of one.

We get 2 * 1
squared takeaway 3 * 1

+ 5. Which gives us 2
* 1 here, which is 2.

Take away 3 * 1 which is take
away three and plus five. So two

takeaway three is minus 1 + 5
gives us 4.

OK, we look at an
argument of two.

We get 2 * 2 squared.

Take away 3 * 2.

I'm plus five which gives us 2 *
4, which is 8 and take away 3 *

2 which is take away 6 and then
forget our plus five at the end.

So eight takeaway six is 2.

+5 gives us 7.

OK. If we look at an argument of

three. Half of three gives us
2 * 3 squared.

Take away 3 * 3.

And +5.

So this is 2 * 9 here

18. Take away 3 * 3 which is
take away nine and +5.

So 18 takeaway 9 is 9 + 5
gives us 14.

And last but not least, we can
also include a negative

arguments, so we'll put negative
arguments of minus one. So F of

minus one gives us two times
minus 1 squared.

Take away three times minus one.

And of course, our +5.

So two times minus one squared
just gives us 2.

Takeaway minus sorry takeaway
three times negative one which

just gives us a plus 3.

And then we've got a +5.

2 + 3 + 5 just gives us 10.

And what we can do is as before,
just put this into a table to

make it nice and easy to make a

graph of the function. So we put
it into a table of X.

And F of X.

For arguments we
had minus 10123.

For Outputs, we had 10 five.

4, Seven and

14. OK, so let's see the graph
of this function then.

You start off with our.

F of X on the vertical axis as

before. An argument.

X on the horizontal axis.

And we've got over 2  1, The

One. 2. And
three, and on the vertical axis.

Go to 15.

OK, so our first point to plot
is minus one and 10 which will

give us something there zero and

five. There's a point here. One

and four. It gives the points

here. Two and Seven.

Should be around here and
three and 14 which will be

about here so we can see the
kind of shape that this

function is starting to take
here. And we can draw in the

graph.

What we want is to say that
every input gives us only one

single output, so we can get our
ruler again and just quickly

check by going along and we can
see that as we go along. Our

rule across is the curve once

and once only. Which means that
this function is valid.

However, an interesting point to
note is this point here. The

minimum point which actually
occurs when X is North .75.

So with X value of North .75
are outputs can take a minimum

value of 3.875.

So this means when we look at
our domain and range, we need to

make no restrictions on the
domain because our function was

valid. However. Our range has
a minimum of 3.875, so we write

this. As F of X equals.

Two X squared minus three X +5.

And the range F of X has
always been more than or equal

to 3.875. So for
the next example.

What would happen if we had a
function F defined by?

F of X equals
one over X.

Well, that's always the first
stage is to substitute in some

values for the arguments.

So for F of one.

The argument is one is 1 / 1
just gives US1.

For Port F of two.

We just get one half.

F of three gives us 1/3.

And F4 will
give us 1/4.

And as before, we can also look
at some negative arguments.

So if I look at F of minus one.

Skip 1 divided by minus one,
which is just minus one.

F of minus 2.

Is 1 divided by minus two, which
just gives us minus 1/2?

F of minus three. Same thing
will give us minus 1/3.

And F of minus 4.

Will give us minus 1/4?

Now if we look at F of 0.

We have 1 /

0. Which is obviously a problem

for us. Because of this
problem, we have to restrict

our domain so that it does not
include the arguments X equals

0. So let's have a look at
what the graph of this

function actually looks like.

So let's be 4F of X and are
vertical axis for the Outputs.

An argument sax on
the horizontal axis.

OK, so we've gone
over here 1234.

And minus 1  2 
3  4 over here.

As we go.

All the web to one down to minus

one. Has it as well? So
we've got one and one.

We've got 2

1/2. 3

1/3. 4

one quarter.
We've got minus 1  1.

Minus 2 

1/2. Minus
3  1/3.

A minus four and minus 1/4.

OK, and obviously we've excluded
0 from our domain. As we said

before. So if we join these

points up. And a smooth curve.

Get something that
looks like this.

Now, obviously we've excluded X
equals 0 from our domain, but

it's also worth noticing here.
Thought there's nothing at the

output of F of X equals 0, so
that also ends up being excluded

from the range.

So we actually end up with F of
X equals one over X.

And we've got X not equal to 0
from the domain.

And also. In the range F of X
never equals 0 either.

But what's actually happening at

this point? X equals 0 when the
arguments is zero. What is going

on? Well, let's have a look and
will start off by having a look

what happens as we get closer
and closer to 0.

Now.

If we start off with a value of
1/2 of one and remember F of X

was just equal to one over X.

Half of 1 just gives US1, so if
I get closer to 0 again, let's

look at half of 1/2.

At 1 / 1/2.

This one over 1/2 which just
gives us 2.

So about F of

110th. It just gives us 1 / 1
tenth, which gives us 10.

Half of one over 1000 will
just give us 1 / 1

over 1000. Which gives

us 1000. What about
one over 1,000,000, so F of

one over a million?

It's actually just in the same
way as before, just going to

give us a million.

So we can see.

The US we get closer and closer
to zero from the right hand side

as we saw on our graph before.

We're getting closer and closer
to positive Infinity to the

graph goes off to positive

Infinity that. What happens when
we approach zero from the left

hand side? Well, let's have a

look. This is minus one, just
gives us 1 divided by minus one,

which is minus one.

F of minus 1/2.

It's going to be just one
divided by minus 1/2.

Just give his minus 2.

Half of minus

110th. Is 1 divided
by minus 110th?

It just gives us minus 10 and we
can kind of see a pattern here.

OK so F of minus one over 1000.

Will actually give us minus

1000. And F of
minus one over 1,000,000.

Actually gives
us minus

1,000,000. So you can see that
as we approach zero from the

left or outputs approaches

negative Infinity. And as we
approach zero from the right

hand side are output approaches
positive Infinity, and these are

very different things. OK, for
the last example.

I just like to look at the
function F defined by.

F of X equals one over
X minus two all squared.

So as always with the examples
we've done, it's worthwhile

started off by looking at some
different values for the

arguments. So we start off with
an argument of minus two, so

half of minus two gives us one
over minus 2  2 squared.

Which gives us one over minus 4
squared, which is one over 16.

F of minus one.

Will give us one over minus 1
 2 all squared, which gives us

one over minus 3 squared which
is one over 9.

Now, FO arguments of zero will
give us one over 0  2

all squared, which is one over
minus 2 squared, which works out

as one quarter.

And F of one will give
us one over 1  2

squared. Which is just one over
minus one squared, which gives

us just one.

OK. Half
of two gives us one over.

2  2 or squared, which gives us
one over 0 which presents us

with exactly the same problem we
had in the previous example when

we had one over X and so we have
to exclude X equals 2 from the

domain. Half of three gives us
one over 3  2 all squared.

Which is one over 1 squared is
just gives US1 again.

After four gives us one over 4
 2 or squared, which gives us

one over 4.

After 5.
Gives us one over 5  2 or

squared which gives us one over
3 squared which is 1 ninth and

finally F of six gives us one
over 6  2 all squared which is

one over 4 squared which works
out as one over 16.

Now if we want to plot the graph
of this function will probably

need to put this into a table

first. So as usual, do our
table of X&F of X.

OK, and we went from minus 2
 1 zero all the way.

Up to and arguments of sex.

And the values we got for the

Outputs. One over

16. One 9th,
one quarter and

1:01. One quarter,
1 ninth and

one over 16.

So we plot that onto.

The graph as before.

So we have arguments going along
the horizontal axis.

And Outputs going along
the vertical axis.

We've gone from minus
1  2 over

there 123456 along this

way. And then going off, we've

gone too. One up here, so put in
a few of the marks 1/2.

It's put in 1/4 that.

Put in 3/4. OK, so we've got
minus two and 116th, which is

going to come in down here.

Minus one and one 9th.

For coming over here zero
and one quarter.

Over here. 1 on one.

Right, the way up here?

To an. Obviously this was the
divide by zero, so we couldn't

do anything with that. We've
excluded, uh, from our domain.

Three and one.

Pay up. Four and one quarter.

I'm here. Five and one,
9th and six and 116th.

Because we've excluded X equals
2 from our domain.

Put dotted line there,
so that's an asymptotes.

And we can draw our curve.

Up through the points.

On this side.

And we can see differently to
the other example where F of X

is one over X this time.

As we get approach to from both
the left and from the right,

both of the outputs are heading
towards positive Infinity, so a

little bit different, and also
because we've excluded X equals

2 from the domain of function is

now valid. But most also notice
that our range is never zero,

and it's also never negative.

So to write this out properly.

Our function F of X equals one
over X minus two all squared.

And we said.

They are domain is restricted so

it doesn't include two. And our
range is always more than 0.

OK, so let's just recap on what
we've done in this unit.

So firstly, the definition of a

function. And that was that. A
function is a rule that Maps are

unique number X to another
unique number F of X.

Secondly, was the idea that
an argument is exactly the

same as an input.

Thirdly, we looked at the idea
of independent and dependent

variables. And we said that
the input axe was the

independent variable and the
output was the dependent

variable.

4th, we looked at the domain and
we said that the domain was the

set of possible inputs.

And finally we looked at the
range and we said that the range

was the set of possible outputs.

So now you know how to define a

function. And how to find
the outputs of a function

for a given argument?