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HERBERT GROSS: Hi, our lesson
today concerns functions.
And in the certain manner of
speaking, this is perhaps
where a course in calculus
should usually begin.
After all, what will be studying
is a relationship
between certain variables.
This is what we mean by
a function in general.
Or at least, in particular,
with our emphasis on real
variables and graphs that
we were talking about.
And what we would like to do now
is to motivate the concept
of function from a more general
point of view in terms
of the language of sets that we
have talked about and read
about in our supplementary
notes.
Let's then begin with a basic
general definition of what a
function is, after which we will
specialize and talk about
functions of a real
variable for the
remainder of the lecture.
We begin with a definition
written this way.
A function f from A to B--
see, it's written this way.
It means f is a function from A
to B means that f is a rule,
which assigns to each element in
set A, an element in set B.
Now of course, this may
sound like just an
empty bunch of words.
But if we come here and look a
little bit at a picture, we
get the idea as to
what's going on.
Namely, we may visualize
our sets as circles.
And what does the function do?
The function assigns to each
element of A, an element of B.
In fact, notice the geometric
wording again.
This element is mapped into
this element by f.
This element is mapped into
this element by f.
And this element is mapped
into this one.
And while we're looking at this,
perhaps it would be a
good chance to emphasize not to
read more into a definition
than what's already there.
You see when we say that f
assigns to each element of A,
what we mean is we will
not allow something
like this to happen.
We will not say, for example,
let f send this element into
this one and this one.
This becomes non well-defined.
It becomes ambiguous.
In other words, we do not want
to have to make the value
judgment as to which of two
things we are going to look
back at as being what
A maps into.
And why that's the case
we'll mention
as our course proceeds.
Also, notice that when we say
that each element of A is
assigned to an element of B, two
things are implied here.
First of all, notice that we
do not insist that all
of B be used up.
You see, in other words, here's
some surplus B's over
here, which are not used up
by A's with respect to f.
And secondly, whereas we
prohibit the same A from
having two different images in
B, we do not prohibit one B
from being the image of two
different elements in A. In
other words, notice that even
though both of these elements
here are assigned to the same
element in B with respect to
f, there certainly is no
violation of our definition
that each element here was
assigned to an element here.
By the way, this leads in the
literature to three different
terms that we should
define right now.
One of these is the set A itself
and that's called the
domain of f.
You see this is what
f is defined on.
The domain is the
set A over here.
Often abbreviated--
well, there are many different
abbreviations.
Sometimes one just writes D-O-M
with an f or D-O-M with
a subscript.
Or a capital D with
a subscript f.
But we will adapt to these
notations as we need them.
But for our purposes all the
domain of f is, it's the set
A. It's the set on which f
operates to assign values, ok.
And then the companion to that
is the set B. And B, almost
again in graphic terms, is
called the range of f.
And by the way, as you look at
the range of f, you may get
the feeling that somehow or
other, this little element
over here is kind
of out of place.
That maybe somehow or other
let me just circle this.
Let me call this the set C.
Somehow or other, you get the
feeling that C describes much
better what f does to A than
B. Because you see, each of the
elements in C is used up
as being the image of at least
one element in A. Somehow or
other you see, we could have
deleted this element B without
too much loss of continuity.
And to get around this we
interject still a third
definition, C. It's called
the image of f.
You see, image verses range.
Image is the part that's
actually used up
by f with the mapping.
And sometimes this is even
abbreviated as follows.
You use the same f as one uses
for the function and then a
capital A in parentheses
to indicate what?
This is a set of all things of
the form f of x where x is in
A. In other words, again, the
standard notation if we want
to look at it this way.
If we call this element here a
and this element here b, to
indicate that a was mapped into
b by the function f or
the rule f, we often write
that as f of a equals b.
Now in many places, including
our text, there is no
distinction made between the
image and the range.
And the reason for this is that
in many situations, the
image and the range turn out
to be quite the same thing.
In fact, if they do turn out to
be the same thing, we call
that a rather special
type of function.
Namely, the function from A to
B is said to be onto if its
image equals its range.
Now again, in terms of words,
that doesn't say very much.
The image equals the range.
What does that mean, image
equals the range?
Perhaps the best way
to see this is
by means of an example.
The example I have
in mind is this.
And this happens many, many
times in mathematics.
One begins with a set.
In this case, I picked A to be
the set consisting of the
numbers 1, 2, and 3.
Frequently, one defines a
function explicitly on A
without any regard to
a second set B.
For example, in this
illustration I've said, let f
of a be 4a for each a in set
capital A. Well, what are the
elements of capital A?
They are 1, 2, and 3.
So according to this recipe,
what do we have?
Well, f of 1 is 4.
f of 2, see, it's what?
4 times 2 is 8.
And f of 3.
That's what?
4 times 3 is 12.
Let me now invent a new set B.
And the elements of B, as you
could probably guess, are going
to be 4, 8, and 12.
And now I look at my function
f from A to B.
Notice that in this case the
function from A to B uses up
all of B. In fact, in terms
of a diagram, you see
here's A. Has what?
1, 2, and 3.
And here's B, which is made
up of 4, 8, and 12.
Now, what does have f do?
It maps 1 into 4.
It maps 2 into 8.
And it maps 3 into 12.
What's happened here?
A has not only mapped
into B, but all of B
is used up in this.
In other words, notice then that
the range of B and the
image of B in this particular
case happen to be the same.
This will happen in every single
case where we start
with a set A and define a
function on A. Namely, we see
what f of x is for each x in A,
take the collection of all
those images, call that set B,
and then you see by default so
to speak, B will be both the
range and the image of f.
And it's in this sense that in
most textbook examples that we
deal with, we need not make any
distinction between the
range of the function and the
image of the function.
But roughly speaking then, just
to keep things straight,
a function is called onto if the
entire range is used up in
the mapping.
But if there are elements of the
range which are not used
up as images, then the function
is simply called
into, or not onto.
But at any rate, this
is the concept of
what we mean by onto.
Now, a second feature that one
talks about with functions
which in no way is connected
with onto, this, but which is
a very important independent
feature is something which is
called a 1:1 function.
Let's look at that
for a moment too.
For example, let's suppose I
have a function f defined on
my set A. The question is, f
maps a1 into a particular
element of B and it maps a2 into
a particular element of
B.
Now, there are two possibilities
that can happen.
One is that f of a1 and
f of a2 will be
different elements of b.
In other words, what will
happen is, is that two
distinct elements of A will
have distinct images in B.
On the other hand, it's
possible that the two
different elements of A have the
same element of the same
image in B.
Now, by and large, whereas
nothing is wrong when this
happens, it does cut down our
operating speed to some extent
when it does.
Because you see, frequently to
study a particular function,
we may want to look at the image
rather than the domain.
And somehow or other, you see if
two different elements can
map into the same element in the
image, then you see when
we look at the image we have
no way of knowing which of
these two elements we're
talking about.
So in other words then, if it
should turn out that no two
different elements in A can have
the same image in B, in
other words, notice what
this thing here says.
You see, if what?
This is the image of a1.
This is the image of a2.
It says what?
If a1 and a2 have the same
image, then they must be the
same element.
Now if that happens, and again
as I show you over here, it
doesn't have to happen.
If that happens, the function
is called 1:1.
For example, here is a picture
I've drawn in which
a function is 1:1.
By the way, I've drawn this
picture so that my function f
is both 1:1.
Meaning what?
That no two elements in A have
the same image in B. And
secondly, it also happened
to be onto here.
Namely, no element of
B was left out by f.
Now that wasn't crucial.
For example, if I do this notice
now that the function
from A to B is still 1:1.
No two different elements in A
have the same image in B. But
now you see the function is no
longer onto because there
happened to be elements in B
which are not mapped into
under F by elements of A.
Now, what is nice of course is
that if a function happens to
be both 1:1 and onto, notice
that we can induce a new
function, which I'll call g
from B to A by essentially
reversing the arrowheads here.
You see, if the function is
both 1:1 and onto, by
reversing the arrowheads,
instead of getting a function
from A to B, I do get a function
from B to A. This
function is called the inverse
function and will play a very
important role in much of our
course which follows.
The important point to notice,
however, is that if the
function is not both 1:1 and
onto, you cannot reverse the
arrowheads, believe it or not.
Well, you say, I can reverse
them, can't I?
Why can't I reverse
them over here?
And the answer is
well, look it.
If we include these being in
here, suppose we reverse the
arrowheads now.
Look at B.
What is the domain of g?
Well, for B to be the domain,
every element of B has to be
assigned to something
in A by g.
But look at these two elements
over here, g doesn't act on
those at all.
In other words, if the original
function is not onto,
then when you reverse the
arrowheads you haven't defined
the new function on your
whole domain here.
In another sense, if the
function was not
1:1 when you started.
In other words, suppose
this happened.
So f was not 1:1.
Now you see when you try to
reverse your arrowheads,
notice that the element here
in B is assigned to two
different elements in A. And
we agreed that we wouldn't
allow that to happen.
OK, so far so good.
Notice that that particular part
of our course has nothing
to do with real variables
and the like.
Meaning when we're talking about
sets they can be sets of
arbitrary numbers.
Now what I'd like to do is
zero in, on our specific
calculus of a single
variable course.
And let's go back to our old
friend who somehow or other
has made an appearance
in every lecture that
we've had so far.
Let's go back to s equals
16t squared.
Only now, we're not going to
repeat the same old stuff that
we did before with it.
We're now going to get slightly
more sophisticated.
Namely, when we talk about s
equals 16t squared, what
problem was being done here?
You are assuming that there
is no air resistance.
An object is being held
above the ground.
You release the object and the
distance s that the object
falls in feet after
t seconds is given
by s equal 16t squared.
Now if we think about that for
a while, we realize that that
does not tell the
whole picture.
Obviously, the s equals 16t
squared applies only to the
time in which that object
is falling.
Perhaps what we should have said
was this, that until you
release the object it doesn't
fall any distance at all.
Then from the instance you
release it, it starts to fall
a distance s given
by 16t squared.
Not forever, but until
it hits the ground.
Let's call t sub g the
time at which this
thing hits the ground.
You see this recipe that we
called s equals 16t squared is
not in effect forever.
It's in effect only when t
is between 0 and t sub g.
And by the way, hopefully once
the object hits the ground it
won't fall any further.
In other words, for any time
after t sub g, the distance
that it's fallen is
16t sub g squared.
Meaning this is the distance
that its fallen when it hits
the ground and it stays there.
If we wanted to graph this, you
see, and notice how we are
refining our previous result.
The graph is not this, you
see, the graph is what?
The distance is 0 until
t equals 0.
Then the distance that it falls
increases up till the
time the object hits
the ground.
And then it levels
off like this.
And by the way, in terms of
making a few asides, notice
that this curve here does
represent a 1:1 function.
Namely, if you pick two
different times in this strip,
you have two different
distances.
Two different times cannot
yield the same distance.
As opposed to the fact, let's
call this t1 and t2.
As opposed to the fact that
once the thing hits the
ground, our function
is no longer 1:1.
In fact, the any two values of
t once the thing has hit the
ground, we have the
same s value.
In other words, what we're
saying is what?
That once the object hits the
ground, it really makes no
difference what t is,
s is still going to
be 16t sub g squared.
Now that was just an aside.
The reason I mentioned this is
to motivate a very important
type of domain that takes
place when we deal with
functions of real numbers.
In most cases, when we do a
physical experiment it's over
some time interval.
We put something into effect
and say, let's
measure it for one hour.
Or let's measure if from now
until 3 o'clock tomorrow.
In other words, in general,
whereas a domain of a function
can be anything we want it
to be, in most real life
laboratory situations, our
domain happens to be a
connected interval, whatever
that means intuitively.
In fact, let's try to talk about
that in more detail.
In other words, a very special
type of domain that one uses
when one talks about functions
of a real variable.
They are called intervals.
Written as sets, if a is less
than b, we talk about what?
The set of all x which greater
than a and less than b.
By the way, that's called the
open interval from a to b.
It's written this way
with parentheses.
The set of all elements from
less than b and greater than a
inclusively is called the
closed set or the closed
interval from a to b.
And it's written this way.
And pictorially, you can't
tell these apart.
Namely, if this is a and this
is b, both of these
pictorially are what?
An interval as we think
of it intuitively.
Namely, it's this stretch.
But in one case, the endpoints
a and b are included.
And in the other case, the
endpoints are excluded.
They're included in the
closed interval.
They're excluded in
the open interval.
And again, notice that since a
point has no thickness, we
have no way of telling just by
looking at the figure which of
these two is meant unless we
draw in the appropriate
diagram this way.
By the way, notice also that an
interval can be half open
and have closed.
I mean, for example, one could
talk about how about including
the left endpoint but excluding
the right endpoint.
See, why couldn't we talk about
something like this?
In which case we would have
written the half open half
closed interval this
particular way.
Now again, this is
all notation.
It's things that you
can memorize.
Things that are emphasized
in the text.
But the thing that I wanted to
try to have you see from the
lecture is why we concentrate
so heavily on the things
called intervals.
It's because in most situations
when we deal with
functions of a real variable,
our so-called input, is
usually defined on some
continuous interval.
All times from such and
such to such and such.
Now, by the way again, notice
that the picture-- just as
we've been talking
about before.
The picture comes in handy.
Namely, 1/2 being in the open
interval from 0 to 1 does not
need a picture to
interpret it.
Namely since 1/2 is greater
than 0 but less than 1, by
definition 1/2 is in
this interval.
On the other hand, by use of a
picture, I think it becomes
rather easy to visualize what
it is that we're saying when
1/2 is in this particular
interval.
Again, notice when somebody
says does 0
belong to this interval?
Notice that subtlety about
open and closed,
point versus dot.
Namely, 0 does not belong to the
open interval from 0 to 1,
but it does belong, for
example, to the closed
interval from 0 to 1.
Because what is the basic
difference between these two?
In this one, the endpoints
are not included.
In this one, the endpoints
are included.
Now, a companion to interval is
a very important building
block of this course.
It's something called
a neighborhood.
Now, in terms of a definition,
a neighborhood isn't a very
exciting thing.
A neighborhood of a point c, a
neighborhood of x equal c is
simply an interval which
contains c inside.
You want c to be inside
the interval.
Now what does that
mean intuitively?
Well, what it means is pick any
interval which has c inside.
Maybe we can go from this
point to this point.
This would be called a
neighborhood of c.
By the way, you may
notice I've drawn
this as an open interval.
The idea is that we
really want c to
be inside the interval.
We do not want the
situation where c
is one of the endpoints.
And whereas we'll talk about
this in more detail later, the
important point is that in many
of our investigations in
calculus we will want to study
what's happening just before
we get to a certain point
and just after
we leave that point.
And somehow or other, if we let
that point be at the very
end of our interval, we
have no information.
For example, if that point is
the left endpoint, we don't
know what's happening before
we get to the point.
If it's the right endpoint, we
don't know what's happening
afterwards.
And that's why you'll find
in the textbook that a
neighborhood is defined to be
an open interval, which
contains c.
In other words, we want to make
sure that c is in the
interior here.
By the way, in many cases it
turns out algebraically to be
easier if this happens to be
what we call a symmetric
neighborhood.
In other words, if c
is in the middle.
We won't go into that right now,
but if c happens to be in
the middle that's called a
symmetric neighborhood.
In fact, another way of writing
that is to say what?
Pick some definite distance h
and what do you write down?
You write down c minus h to c
plus h and that puts c right
in the middle of this
particular interval.
And you see, the idea here is
that when you're looking at
what's happening to a function,
you may lose symmetry.
For example, in this particular
graph that I've
drawn, notice that at this
particular point I've marked
off equal intervals on
both sides of l.
But notice that when I come down
here, they do not project
onto equal intervals on
either side of c.
In other words, if this had
been a straight line.
Frequently what we do in
a case like this is we
say well look it.
If we're interested in seeing
what happens near c,
why don't we just--
this is non-symmetric.
Why don't we just take the
smaller of these two widths
and see what happens in
the symmetric part?
In other words, if the
neighborhood is not symmetric,
we can always make
it symmetric.
And so there really isn't that
much to worry about in that
particular respect.
But why are we interested
in neighborhoods
in the first place?
And the answer is that in many
cases what we're going to be
doing is studying what's
happening near a particular
point c and want to know what's
happened just before
and what's happening
just after.
The next important concept
that's connected with
neighborhoods is the idea of
a deleted neighborhood.
And that in turn, is very
strongly connected with 0/0.
For example, consider the
function f of x which is x
squared minus 9 over
x minus 3.
If we let x equal 3, if the
input is 3, notice that the
output becomes 9 minus 9
over 3 minus 3, or 0/0.
On the other hand, if x is any
number whatsoever except x
equals 3, no harm is done
with this as an input.
Consequently, what one is
talking about now is the only
time you get that 0/0
form is when x is 3.
What happens if you're in a
neighborhood of 3, but not
equal to 3 itself?
You see what I'm driving at here
is pick any number x in
this interval other
than 3 itself.
And notice that f of x can
be written this way.
As long as x is not equal to
3, we can cancel x minus 3
from numerator and
denominator.
Remember, we can't
divide by 0.
And now we see what?
That as long as x is not
equal to 3, f of x is
perfectly well defined.
And consequently, this is what
motivates the concept of a
deleted neighborhood.
Namely, everything is fine in
this whole neighborhood except
for 3 itself.
So to avoid that unpleasantry,
let's just delete that point.
And that's called a deleted
neighborhood.
And you see what we do when the
neighborhood is deleted,
we're still going to
talk about what?
How close you are
to that point.
And by the way, this brings us
to another very fascinating
aspect of what's going on
between our geometry and our
arithmetic.
Do you really talk about the
distance between numbers?
I mean, is 7 near 3?
And the guys says, well, what
do you mean, is 7 near 3?
Well, I would say here that
7 is very near to 3.
When you say that 7 is near 3,
you certainly don't mean close
to in the geometric sense.
You mean the difference
between them is small.
In other words, the next thing
that we have to talk about is
how when we talk about being
close to a point which is a
geometric term, how do we talk
about that algebraically?
You see geometrically, how do
you talk about the distance
between x1 and x2?
Well, if you're going this
way, it's just what?
x2 minus x1.
If you're going the other way,
the direct distance this way,
it's x1 minus x2.
In any event, the distance
between these two points is
just the magnitude
of the difference
of these two numbers.
And that leads, you see, to the
concept that's hit quite
heavily in our text,
and that is the
concept of absolute value.
Perhaps one of the most critical
analytical geometric
topics that we tackle in our
early part of our course.
Analytically, we define the
absolute value written with
vertical bars here,
x1 minus x2.
The absolute value of x1 minus
x2 to be the positive square
root of x1 minus x2 squared.
And in plain English all
this says is what?
See when you square and then
take the positive square root,
you haven't undone what
you've done before.
All you've done is what?
If it's positive here you
haven't changed anything.
But if x1 minus x2 are negative,
when you square it
and extract the positive square
root, all you've done
is changed the sign just like
you're supposed to.
Let me give you an example.
Suppose you're faced with the
absolute value of x minus 3 is
less than 2.
What does this say
geometrically?
Geometrically what it says
is that x is within
two units of 3.
in terms of a picture, all you
have to do now is draw in 3,
mark off two units on either
side, and for x to be within
two units of 3, all you
know is is that x
has to be in here.
In other words, look at how
easily you can solve this
particular problem.
On the other hand, you can
always go back to the basic
definition and say,
wait a second.
This means the positive square
root of x minus 3 squared is
less than 2.
So I will square both sides.
If you do that you get x minus
3 squared is less than 4.
If you now collect terms and
expand, you get x squared
minus 6x plus 9 minus 4.
That's what?
Plus 5 is less than 0.
This factors into x minus
1 times x minus 5
is less than 0.
The only way the product of two
numbers can be negative is
if the factors have
different signs.
Since this is x minus 5 is less
than x minus 1, this must
be the smaller of the two.
This must be the larger
of the two.
To say that x minus 1 is greater
than 0 is the same as
saying that x is
greater than 1.
To say that x minus 5 is less
than 0 is the same as saying
that x is less than 5.
You put that all together and
notice that even though it
wasn't quite as comfortable, we
can obtain the same answer
algebraically as we can
obtain geometrically.
In other words, our relationship
between algebra
and geometry remains the same.
Again, when you can draw
the picture, it's
worth a thousand words.
If you can't draw the picture or
you're suspicious about the
picture, especially when it
involves point versus dot,
then what you do is resort to
the analytic definition.
These do not replace
one another, they
work hand in hand.
Finally, what we must talk
about now is the
arithmetic of functions.
Can we combine functions
to form functions?
And the answer is yes.
First of all, in talking about
the arithmetic of functions,
what must we do?
We must first, at least, define
what it means for two
functions to be equal.
Well, for two functions to be
equal, all we insist on is
that first of all, they're
defined on the same domain.
And secondly, that for each
input in the domain, each
function gives you
the same output.
For example, suppose a is
a set whose elements
consists of 0 and 1.
And suppose b is also the
set whose elements
consist of 0 and 1.
One such function would be f.
It maps 0 into 0.
It maps 1 into 1.
Another function, which
I'll call g--
see, f does what?
It maps 0 into 0 and 1 into 1.
What does g do?
g maps 0 into 1 and 1 into 0.
Notice that f and
g are different.
They both have the
same domain.
They both have the same image.
But notice that element for
element, they're not the same.
Namely, f and g do different
things to 0.
f sends 0 into 0.
g sends 0 into 1.
So I can tell f and g apart.
And because I can tell them
apart, they're not equal.
All right, so equality means
I can't tell f from g.
Now the next kind of question
is, how do you do arithmetic
with f and g?
Can I add two functions?
Can I multiply two functions?
Can I subtract two functions?
And the answer again,
turns out to be yes.
And not only yes, but yes
in a rather simple way.
Let's again do this by
means of examples.
Suppose we defined f of x to be
2x for all x in A. Namely,
if A is 1, 2, 3, f of 1 will
be 2, f of 2 will be 4, f
of 3 will be 6.
Let's define another function
on A, let's call it g.
g of x will be x plus 1 for each
x in A. In other words, g
of 1 will be 2, g of 2 will
be 3, g of 3 will be 4.
Now the point is, can I
add f of x and g of x?
Well, sure. f of x is 2x.
g of x is x plus 1.
So if I add these I get
h of x is 3x plus 1.
In a similar way, could I have
multiplied these two?
Well, sure.
Again, f of x is 2x,
g of x is x plus 1.
If I multiply these together,
I get what?
2x times x plus 1, which
is the same as 2x
squared plus 2x.
Now of course this probably
doesn't look too smooth
because there's no
pictures here.
All we're saying is this.
Here's A and all you're saying
is that if you add f and g,
what do you get?
If A is 1, 3x plus 1 is 4.
2 times 3 plus 1 is 7.
3 times 3 plus 1 is 10.
In other words, in this case,
our image, if I want to call
it b, would look like this.
This would be the sum of the
two functions f and g.
And similarly, for the product
I could do the
same kind of a thing.
In other words, I can just
arithmetically, since both the
output of f and the g machines
are real numbers, and the sum
of two real numbers is a real
number, I can add and multiply
functions to form functions.
But there's one other important
way of combining
functions in calculus.
A way which is very, very
important and one which we may
not have seen too
much of before.
And so let me close our lecture
for today with an
emphasis on that particular
topic.
It's called composition
of functions.
And to see what composition of
functions means think of a
particular example where maybe
the f machine f of x is 2x.
In other words, think
of it this way.
We run x through the
f machine, the
output will be 2x.
Now we run the output of the f
machine into the g machine.
Now what does the
g machine do?
If x is the input, x plus 1
means 1 more than the input.
The g machine always adds
on 1 to the input to
give you the output.
Well, if the input is 2x, the
output will be 2x plus 1.
Notice that these two together
can be thought of as being one
function machine, which I'll
call the q machine.
In other words, what happens
for the q machine is what?
x runs into the f machine, the
output of the f machine
becomes the input of the g
machine, and the output of the
g machine is then the output
of the q machine.
The q machine is sort of build
with component parts here.
And the reason that this is
very, very important is that
this comes up in calculus all
the time, where the first
variable is related to the
second variable, the second
variable has a definite
relationship to the third
variable, and we now want to
relate the first variable to
the third variable.
And the way we write
that-- and I guess
this is hard to see.
This is not an O over here, it's
a little circle, like a
dot, and it's called the
composition of g and f.
It's not gof.
Its g circle f.
And the q machine is what?
You write it this way and
maybe if you look at the
picture you can see exactly
what's happening here.
You apply f to x and then
apply g to the result.
In other words, just looking
at this picture it becomes
rather apparent that q of
x is just 2x plus 1.
Notice you see, that the domain
of the q machine is the
same as the domain of f.
The input of the q machine is
what goes into the f machine.
The output, the image of
the q machine, is the
image of the g machine.
In other words, just this
particular thing.
Now this type of function
combination called
composition, is a very
intricate thing.
It depends on the order in which
you do these things.
This is a rather interesting
point.
For example, when you add two
numbers, a and b, it makes no
difference in which order
you add them.
On the other hand, when you
divide two numbers, a and b,
the quotient does depend
on the order in
which you divided them.
Well, the same thing
is true here.
Let's call p the function
which starts with the g
machine followed by
the f machine.
And as this lecture wears on, I
think maybe there's a reason
for making this circle look
like an O. Maybe this is
starting to look a little bit
like fog at this time.
It's not.
All I want you to see is that
what we do now is we start--
see, what are we going
to do here now?
We're going to start
with the g machine
first, then the f machine.
In other words, what
happens now?
If the input is x, the output
of the g machine is one more
than the input.
That would make the input x
plus 1 to the f machine.
What does f do?
Remember, f doubles.
f doubles the input.
So the output here
would be what?
Twice x plus 1.
In other words, what would
p of x look like?
If x goes into the p machine,
what comes out is twice x plus
1, or 2x plus 2.
On the other hand, when we put
the f and the g machine in the
other order and formed q of
x, what was the output?
Let's go back here and look.
The output was 2x plus 1.
In other words, do you build a
different function machine by
interchanging the f and the g?
And the answer is yes.
In other words, what I'd like
you to see for concluding this
part is that whereas everything
was pretty
straightforward up until now,
the most important new
concept, one which was not so
intuitive is the one that's
called the composition
of functions.
It's the one that occurs
all the time in
related rates problems.
We'll be using it over and over
again in this course.
And all I want you to see is
that first of all when you use
the composition of functions,
what you get depends on the
order in which you
combine them.
And that secondly, and most
importantly, that neither of
these is the same as this.
That combining two functions is
not the same as multiplying
the outputs of two different
functions.
Well, I think that's
sort of enough of a
mouthful for one sitting.
Our main aim today was to
introduce functions, the
language that we're going to be
using the rest of the way.
Because after all if we don't
have the vocabulary
established it will not be
second nature to talk about
the concepts.
Starting next time and beyond
we will deal more with
specific calculus contexts.
But until next time, goodbye.
ANNOUNCER: Funding for the
publication of this video was
provided by the Gabriella and
Paul Rosenbaum Foundation.
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