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?