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HERBERT GROSS: Hi, our lecture
today is entitled inverse
functions, and it's almost what
you could call a natural
follow-up to our lecture of
last time when we talked
briefly about 1:1 and
onto functions.
Inverse functions have a
tremendous application as we
progress through calculus, but
of even more exciting impact
is the fact that inverse
functions are valuable in
their own right.
They are a pre-calculus topic.
In fact, they appear as early
in the curriculum as
approximately the first grade.
See roughly speaking, inverse
functions in plain English,
mean that all we've done is
made a switch in emphasis.
Let's take a look at that.
Let's go back roughly to our
first grade curriculum when
one learns that 2 plus
3 equals 5, or that 5
minus 3 equals 2.
Both of these statements say
the same thing, but with a
change in emphasis.
It's as if 2 is being emphasized
here while 5 is
being emphasized here.
This is rather interesting
you see.
For example, in the new
mathematics, one talks about
subtraction being the
inverse of addition.
And this is the same inverse
that we want to talk about
today as it applies to
mathematics in general.
What do we that subtraction is
the inverse of addition?
It may sound fancy, but all it
means is that if you know how
to add, if you define
subtraction properly, you
automatically know
how to subtract.
And this, of course, is what's
prevalent in the old
change-making technique of going
into a store, making a
purchase, paying for the
purchase, and when you receive
your change, the clerk rarely,
if ever, performs subtraction.
You may recall that what he
does is he adds onto the
amount of the purchase the
amount necessary to make up
the denomination of the bill
with which you paid him.
In other words, what we're
saying here is, for example,
that one may think of 5
minus 3 as being what?
That number, which must be
added onto 3 to give 5.
You see, in this sense,
subtraction is
the inverse of addition.
Once we know how to add, we
automatically know how to subtract.
Now you see, this idea
goes with us.
We learned that multiplication
and division are inverses of
one another.
And as we go on through higher
mathematics, even on the
pre-calculus level, we find
additional examples of this.
For example, when one knows
how to use exponents, one
automatically knows how
to study logarithms.
Namely, if y equals the log of
x to the base b, this is a
synonym for saying that
b to the y equals x.
And what is the basic difference
between these two
statements?
They are paraphrases
of one another.
In one case it seems that the
number y is being emphasized
and in the other case it's
the number x that's being
emphasized.
And as in the case of most
examples of paraphrasing,
which of the two forms we use
depends on what problem it is
that we're trying to solve.
In other words, if we know
one of these two, then we
automatically can study the
other in terms of the one with
which we feel more familiar.
This, of course, continues when
one gets to trigonometry
and studies the so-called
inverse
trigonometric functions.
If y equals the inverse sine of
x that's the same thing as
saying that x equals sine y.
Again, what is the
basic difference?
In one case, it seems that
y is being emphasized.
In the other case, it's x
that's being emphasized.
In terms of the usual calculus
jargon of independent variable
versus dependent variable
it appears that what?
In one case, x is the
independent variable, y the
dependent variable.
In the other case, y is the
independent variable, x the
dependent variable.
To generalize this result what
we're saying is simply this.
If y equals f of x, you see
where y is being emphasized,
the dependent variable.
If we wish to switch the
emphasis, then we write x
equals f inverse of y.
This is read f inverse of y and
it's the function which in
a sense, inverts the
roles of x and y.
Let's see what this means more
explicitly in terms of a
particular example.
Let's suppose that we're
given the equation y
equals 2x minus 7.
Or to represent this somewhat
more abstractly, y equals f of
x where f of x is 2x minus 7.
Now you see, without mentioning
the word inverse
function, it turns out that
early in our high school
career we were finding inverse
functions as soon as, for
example, someone were to give us
this problem and say solve
for x in terms of y.
Solve for x in terms of y.
You see, if we solve for x in
terms of y, we now have what?
x is being emphasized.
These two statements tell
us the same thing.
But now we write what? x
equals f inverse of y.
And f inverse of y is just
y plus 7 over 2.
Notice again the connection
between f inverse and f.
How one undoes the other.
In terms of our function machine
idea, what we're
saying is we may visualize the f
machine whereby the input is
x and the output will
be twice x minus 7.
In other words, the output
will always be twice
the input minus 7.
Now the question that comes up
is, suppose we reverse the
roles of the output
and the input.
In other words, suppose now we
let the input be y, what will
the output be?
If we reverse the terminal so
to speak, what we have shown
is that now the f inverse
machine would be what?
The input is y, the output
is y plus 7 over 2.
And by the way, a question that
we shall come back to
very shortly that plays a rather
important role here and
which I'll emphasize from
another point of view is that
if you get into the idea of
always wanting to call the
input x and the output y, which
is how we get geared to
do things in terms
of calculus.
x is always the horizontal axis,
y the vertical axis, and
we always agree to plot
the independent
variable along the x-axis.
In other words, the input along
the x-axis, the output
along the y-axis.
Then the question is, could we
have called this x and called
this x plus 7 over 2?
And we'll go with this in more
detail in a little while.
But obviously, what we call the
name of the input should
not affect how the
machine behaves.
By the way, as a little aside,
I thought it might be
interesting to show why we use
such notation as f inverse.
f to the minus 1.
It's rather interesting here.
Let's suppose we let a number
go into the f machine.
Call that number c.
Notice that any number that goes
into the f machine has as
its output twice that
number minus 7.
Suppose we now let that number
be the input of the f inverse
machine, what does the
f inverse machine do?
It adds 7 onto any input and
then divides that result by 2.
In other words, if we now run
2c minus 7 through the f
inverse machine, we have
2c minus 7 plus 7
over 2 equals c.
In other words, notice how
the f inverse machine
undoes the f machine.
If we wanted to use the language
of last time in terms
of composition of functions,
what we do is what?
What we're saying is that if
you combine f followed by f
inverse, f inverse following f,
that that gives you what we
can call the identity
function.
The identity function.
Namely, if the input is c, the
output will again be c.
In other words, f inverse of f
of c is just c back again.
In a similar way, notice that
we can reverse these roles.
We saw last time that
composition of functions
depends on which order you
combine the functions.
But notice that if you run d
through the f inverse machine,
the output will be
d plus 7 over 2.
If this becomes the input of the
f machine, remember what
the f machine does.
It doubles the input
and subtracts 7.
In other words, again,
f of f inverse of d
gives me d back again.
In other words, in terms of
composition of functions, f
followed by f inverse or f
inverse followed by f is what
we call the identity function.
That one is truly the inverse
of the other from that
particular point of view.
However, let's correlate what
we're talking about now with
the circle diagrams that we
used in our last lecture.
You see, first of all, let's
recall that unless our
function is both 1:1
and onto, we do not
have an inverse function.
Namely, for example, if our
function had not been onto,
then when we--
see, here's the idea again.
Let me make sure
this is clear.
To get an inverse function,
essentially all we do is this.
If f is a function from A to
B, the inverse function is
defined by reversing the
input and the output.
Which means in terms of this
diagram, we reverse the sense
of our arrows.
We reverse which end the
arrowhead goes on.
And what we're saying is, if we
had a function from A to B,
which was not onto, then you
see when we reverse the
arrowheads, f is not defined
on all of b.
In other words, the domain of
f, the domain of the inverse
function, would not exist
because it would not be
defined on all of B.
Secondly, if two different
elements of A went into the
same element of B when we
reversed the arrowheads, the
resulting function would
not be single-valued.
And hence, in terms of modern
mathematics, it would not be a
well defined function.
So in other words, for the
inverse to exist it must be
that the original function
is both 1:1 and onto.
And as an example of that, this
is what this diagram here
represents.
And to make sure that we can
read this all right, I have
singled out a typical element
of capital A, a typical
element of capital B. Remember
what our notation is.
The notation is that
f of a equals b.
That the image of
a under f is b.
And to use that in terms of
the inverse language, if I
called g the function that
I get when I reversed the
arrowheads, g of b equals a.
And g is what I'm calling
f inverse.
By way of further review, the
domain of f is equal to the
image of f inverse.
And that's A. The image
of f is the same as
the domain of f inverse.
And that's B.
And now, what the question
is, is this.
Notice that as long as you want
to use the same diagram,
all we have to do to express f
inverse in terms of f is sort
of to reverse the arrowheads.
The question that comes up is,
suppose you insist that the
domain be listed first.
In other words, when we're going
to talk about g or f
inverse in this case, that's a
function from B to A. So why
don't we list B first
And you see again,
we can do this.
Here are the elements of B, here
are the elements of A.
And all we have to do is see
what happens over here.
For example, if we come back to
here notice that the first
element listed in B comes from
the third element listed in A.
So when I make up the inverse
function, I just capitalize on
this by writing the
same thing.
The only problem is--
and this is going to become
a crucial one--
is the fact that if somehow or
other you couldn't see these
labels, if you couldn't see
these labels and all you knew
was that the first set was
called the domain and the
second set was called
the image.
If you now looked at these two
functions, you see they
wouldn't look anything
at all alike.
In other words, f and f inverse,
while not independent
of one another, do look
quite different.
For example, notice that f
inverse causes the first two
elements in here to sort of
crisscross as they have
images, and the second two
elements of here crisscross.
Notice though in terms of f,
it's the second and third that
crisscross and the first and
fourth that don't intersect at
all this way.
In other words, if you look at
this curve or this diagram and
compare it with this diagram,
notice that there is a
difference in what seems
to be going on.
Well again, this is
quite abstract.
Let's try to relate this as
much as possible to the
language of calculus and our
coordinate geometry graphing
techniques.
To begin with, let's suppose
that we have a function f
whose domain is the closed
interval from a to b and whose
range is the closed interval
from to c to d.
And the question that we'd like
to raise is, under what
conditions will f possess
an inverse function?
What does onto mean?
What does 1:1 mean and
things of this type?
Well, the first thing I'd like
to point out is that if the
graph y equals f of x looks
something like this.
See, notice that the domain
is from a to b.
The image is from c to d.
Notice the fact that we have a
break in the curve over here,
tells us that our function
is not onto.
Namely, given this number p,
which is in our image between
c and d, there is no element
of the domain
that maps into p.
So, in other words, if there is
a break in the curve, the
function is not onto
and hence, it
will not have an inverse.
Now suppose there is no
break in the curve.
Let's suppose now that the
curve doubles back.
It comes up and doubles back.
Now what my claim is is that the
function will not be 1:1.
Well, how can we see that?
Pick any part where the curve
doubles back, pick a point
like this in that range.
Call that point q.
Noticed that q is in the
proper range of f now.
y equals f of x.
Now the question is, given the
y value of q, are there any
x-values that map
into q under y?
And the answer is
yes, there are.
In fact, there are
more than one.
Namely, notice that both f of
x1 and f of x2 equal q.
In other words, in this case,
f of x1 equals f of x2, even
though x1 is unequal to x2.
That means that this function
is not 1:1.
And because it's not 1:1, it
doesn't have a well defined
inverse function.
Well, putting these two cases
together, what it means for
the function to be onto, what
it means for the function to
be 1:1, it turns out that if
our curve is unbroken, then
the only way our function can
have an inverse function is
that the curve must either
always be rising or always be
falling, and it can't
have a break in it.
And by the way, as an aside,
let me point out here the
difference between a continuous
variable--
meaning one that's defined
on a whole interval--
and a discrete variable--
meaning where you get isolated
pieces of data.
Notice, for example, if you plot
y versus x the way we do
in a lab experiment where for
a particular value of x, you
measure a value of y.
Notice that the data can
double back without the
function being multi-valued.
In other words, notice for
example, that even though the
curve doubles back here--
I can't call it a curve.
The data doubles back.
Notice, for example, that no two
different pieces of data
have the same y-coordinate.
In other words, given this point
here as being q, there
is only one piece of
data that has its
y-coordinate equal to q.
However, of course, keep in
mind it is possible that
another piece of data will
have the same coordinate.
All I'm saying is that the
idea of whether the curve
always has to be rising or
following certainly depends on
whether you have a continuous
curve or not.
Well, again, let's continue on
with what inverse functions
are all about.
You see, this comes up with our
whole idea of why do we
make fun or why do we minimize
single-valued
functions in calculus?
And the answer is that
single-valued--
I'm sorry.
Why do we always stick to
single-valued functions and do
away with multi-valued
functions?
And the answer is if you have a
smooth curve, we can always
break down a multi-valued
function into a union of
single-valued functions.
For example, if we take the
curve c here as being y equals
f of x, which plots like this.
Notice if I take the points at
which I have vertical tangents
and break the curve up at
those particular points.
In this case, I'll
get what curves?
c1, c2, and c3.
Notice that c is the union of
c1, c2, and c3, but that each
of the curves c1, c2, and c3
are either always rising or
always falling.
In a similar way, when we have
a function which doubles
back-- and by the way, notice
what the connection is between
multi-valued and not 1:1.
You see, notice that in terms
of a function versus its
inverse function idea, that if a
function is multi-valued the
inverse function
cannot be 1:1.
In other words, the idea being
that when you interchange the
domain and the range, sort of
the curve flips over idea, all
I want you to see here
is that what?
If you're given a function which
is not single-valued, if
we take the points at which
horizontal tangents occur and
break down the curve like this,
we can break the curve
down into a union of
1:1 functions.
The hardship being of course,
that when you start with a
point like this, analytically
speaking, it's rather
difficult unless you invent some
scheme to know which of
the points here you want
to single out.
In terms of our previous
experience, it's sort of like
saying to a person, I am
thinking of the angle whose
sine is 1/2.
There are, you see, infinitely
many functions whose sine is
equal to 1/2.
Of course if we say to the
person, I am thinking of the
angle whose sine is 1/2 and the
angle is between minus 90
degrees and plus 90 degrees,
then the only possible answer
is the angle must
be 30 degrees.
But notice that when you have
a function which is not
single-valued, the
inverse will be a
multi-valued function.
And we'll talk more about
that in a little while.
Again, I just want to keep this
shotgun approach going on
just what an inverse function
is in relationship to the
function itself.
Again, let's look at this
more abstractly.
Here I have drawn
a curve which is
continuous and always rising.
So I can talk about the
inverse function.
If the equation is y
equals f of x, the
inverse is written what?
x equals f inverse y.
And if this seems a little bit
too abstract for you, think of
a concrete representation.
Suppose the curve happened
to represent y
equals 10 to the x.
Then another way of saying
the same thing would be
x equals log y.
The convention of course here
being that you don't usually
write base 10.
But we won't worry about that.
You see, this is what?
Two different ways of expressing
the same curve.
Whether I write y equals f of
x or x equals f inverse y, I
have the same curve this way.
In terms of our arrows, you see
what I'm saying is, if I
start with x1, by going this
way, my function determines
the output y1.
Inversely, if I start with y1
and reverse the arrows, I wind
up with x1.
Again, the basic difference
being as to which of the two
variables is being emphasized.
What the real problem is, is
that most people say look it.
I'm not used to studying
curves this way.
I'm not used to looking at the
input being along the vertical
axis and the output along the
horizontal axis according to
the way I've been trained
when we're
studying the inverse function.
In other words, when y is the
input, aren't we used to
having y over here and
then plotting the
output along this axis?
In other words, the question
is given this graph, how do
you arrive at this one?
You see, somehow or other, let's
observe that if all you
did was switch your orientation
and say let me
switch this by 90 degrees,
notice that we would be in a
little bit of trouble.
In other words, if we start with
this kind of a set up and
we say, let's rotate through
a positive 90 degrees.
Notice now what we would
wind up with is what?
Our x-axis would be the way we
want it, but the y-axis would
now have the opposite sense of
what we usually want our input
axis to look like.
So after we rotate through 90
degrees, it would seem that
the next step is to do what?
Flip with respect
to the x-axis.
That means fold this
thing over.
In other words, a 90 degree
rotation followed by a folding
over gives me the orientation if
I insist that the input has
to be along the horizontal axis
and the output along the
vertical axis.
What I want you to also notice
though, is that if we don't
insist on this, there is no
reason why we have to use two
separate diagrams.
Again notice, these are
two different ways of
giving the same --
two different equations for
giving the same curve.
It's only when we want to switch
the role and make sure
that the input is along the
horizontal axis that we have
to go through this kind
of a process.
Let's look at this a little
bit more concretely.
What I call a semi-concrete
illustration.
What I'm saying now is let's
suppose this is the curve I've
drawn in here, y
equals f of x.
Another way of saying that is
x equals f inverse of y.
And the question is, suppose I
now want to plot this same
curve, same equation, but now
with the y-axis as my
horizontal axis.
You see again, in terms of
geometry, how I shift my axes
will not change this equation.
But what the picture of this
equation looks like will
certainly depend on how
I orient my axes.
So the idea is what?
I simply fold this, rotate this
thing, through 90 degrees.
And if I do that, the resulting
picture looks like this.
And once the picture looks
like this, the
next step is what?
Flip this with respect
to the x-axis.
And now my picture
looks like this.
In other words, x equals f
inverse of y here and x equals
f inverse of y here are
the same equation.
The reason that the picture
looks different is because I
didn't allow myself
to use this as the
axis of inputs here.
In other words, again, as soon
as I wanted to make this axis
orient so it would be the
horizontal axis, this is what
I had to go through over here.
Now you see, the next refinement
is that a person
says look it, I'm not used to
calling this the y-axis.
What's in a name?
Why don't we always agree to
call the horizontal axis the
x-axis and the vertical
axis the y-axis?
And if I agree to do that,
notice what happens just by
changing the names
of the variables.
All that happens is, is that
now this becomes y equals f
inverse of x.
This is an important thing
to notice then.
In other words, if you insist
that the horizontal axis in
both cases will be called the
x-axis and the vertical axis
the y-axis, then this would be
the curve y equals f of x and
this would be the curve y
equals f inverse of x.
But again, the whole thing
comes about only when you
insist on how you want
your axes oriented.
Let's go back to our problem of
y equals 2x minus 7 and see
what this thing means
in terms of a graph.
As we saw previously, if y
equals 2x minus 7, x is equal
to y plus 7 over 2.
And the idea is what?
Let's see what this thing
really means.
If I plot the straight line y
equals 2x minus 7, this is the
line that I get.
Notice that as long as I'm
going to use the same
orientation of axes here, it
makes no difference whether I
call this line y equals 2x minus
7 or whether I call it x
equals y plus 7 over 2.
They are two different names
for the same line.
The problem occurs when I insist
that the independent
variable always be plotted
along the x-axis, the
horizontal axis, and the
dependent variable along the
vertical axis.
Again, going through what we did
before, I first take this
thing and I rotate it through
a positive 90 degrees.
That takes this picture and
transforms it into this one.
I now take this and I flip it
with respect to the x-axis,
and that gives me this
picture here.
Now what is this line here?
It's x equals y plus 7 over 2.
Again, this is the same
equation as this one.
The reason that the pictures
look differently is the fact
that we have changed the
orientation of the axis.
Again, if we now say OK, let's
rename this the x-axis, let's
rename this the y-axis, then
this becomes what? y equals x
plus 7 over 2.
It's in this sense that we
call this curve of this
equation and this equation here,
that these two equations
are inverses of one another.
Again, in terms of what we said
before, if you pick a
particular value of x and
compute y this way, then you
apply this recipe to that.
In other words, if you now take
twice y, twice the output
here, and subtract 7, your
original input returns.
In other words, this
works exactly the
same as we did before.
But again, the whole basic
difference is what?
How you want to orient
your axis.
That the curves look different
because your coordinate system
is different.
Of course, the interesting
question now is, if we
compared these two curves, since
the-- see, granted that
the function and its inverse are
different functions, they
are somewhat related.
They're not random.
How are these two
graphs related?
That might be the next natural
question to ask.
If we do that, the idea
is simply this.
Let's suppose we have y equals
f of x as one of our curves.
The curve happens to be
invertible, meaning that f is
always rising and it's
unbroken, et cetera.
The question is, if we now try
to plot y equals f inverse of
x in the same diagram.
See notice now what
I'm saying.
In other words, I am not
saying x equals f
inverse of y here.
I'm saying suppose I have the
curve y equals f of x and also
the curve y equals
f inverse of x.
How do these two curves look
with respect to an x- and
y-coordinate system?
See, let me do that part
more slowly again.
Let me come over here
for a moment.
Notice that y equals 2x minus
7 was my original curve in
what I dealt with before.
If I want to keep the same
orientation of axis, the
inverse function we saw was
y equals x plus 7 over 2.
The question that we're asking
quite in general, not in this
specific case, is how are these
two curves related?
And the solution goes
something like this.
Let's suppose that the
point x1, y1 belongs
to the curve c1.
By definition of c1, that
says that y1 is f of x1.
By definition of inverse
function, see if y1 is f of
x1, that means if you
interchange the input and the
output, that's another
way of saying what?
That x1 is f inverse of y1.
In other words again, if f maps
x1 into y1, f inverse
maps y1 into x1,
by definition.
Now you see, compare this
with the curve c2.
See, this says what?
That the input y1 maps
into the output x1.
In other words, notice that if
you look at the f inverse
situation here, when the input
is y1, the output is x1.
That's another way of saying
that y1 comma x1 belongs to
the curve c2.
In other words, if x1, y1
belongs to y1 equals f of x1,
then y1, x1 belongs to y
equals f inverse of x.
Now, what is the relationship
between the point x1 comma y1
and the point y1 comma x1?
If we draw this little diagram,
we observe that we
have a couple of congruent
triangles here.
This length equals
this length.
This angle equals this angle.
And this gives me a hint.
This makes triangle
OPQ isosceles.
I draw the angle bisector of
angle O. The angle bisector of
an isosceles triangle
is the perpendicular
bisector of the base.
And angle bisector of the
vertex angle is the
perpendicular bisector
of the base.
Well, you see that makes this
angle equal to this angle.
That makes this a
45 degree angle.
In other words, the line
that I've drawn is
indeed, y equals x.
And notice that P and Q are
symmetrically located with
respect to the line
y equals x.
In other words, going back to
our original problem here, the
curve c1 and c2 are related by
the fact that they are mirror
images of one another
with respect to the
line y equals x.
That's exactly what I've
drawn over here.
In other words, going back to
the problem of how are the
curves y equal 2x minus 7 and
y equal x plus 7 over 2
related, the answer is they
are mirror images of one
another with respect to
the line y equals x.
They are symmetric with
respect to that line.
Now you see, let's talk about
this from another point of
view also, and show what
the tough thing is.
You see, so far my whole
discussion seems to have
hinged on the fact that we
have a function, which is
invertible.
What if you have a function
which is non-invertible?
Going back to something more
familiar, why do we, talk
about -- when y equals the
square root of x, why do we
have this convention that we
take the positive square root?
After all, doesn't the square
root of x and minus the square
root of x have the property that
when you square them you
get the same result?
Plus or minus squared
is always plus.
And the answer is that if you
square both sides here and
think of this as being the curve
y squared equals x, what
happens is you get a
multi-valued function.
One value of x yields
two values of y.
And the way we get around that
is we break this curve down
into two pieces, c1 and c2,
where c1 is always rising.
c2 is always falling here.
In other words, we broke this
thing off at the point of
vertical tangency.
And we can now think of this
curve as being the union of
two curves.
One of which is y equals the
positive square root of x and
the other is y equals the
negative square root of x.
Now the question is, what
happens when you have a
function which is not
single-valued.
In other words, let's just
invert this one.
Let's suppose we started with
the curve y equals x squared.
You see, now for a given value
of y, I'm in trouble.
Because if y1 is positive, there
are two different values
of x which yield this
particular result.
In other words, both of these
have the property that when
you square them you get y1.
And all we're saying is that in
a problem such as this, we
can study this curve as
two separate pieces.
Call one of these curves k1.
That will be the curve y equals
x squared, where x is
non-negative.
So this will be the curve k1.
And call the other one k2,
where k2 will be what?
The same curve y equals x
squared, but its domain is the
negative values of x.
In other words, k2 will
be this one over here.
And now the point is, if we deal
with either of these two
pieces separately, we can talk
about inverse functions.
Now the point is, which of these
two halves do we use?
And this is where the word
principal values comes in.
And you see what I'd like you
to keep in mind is this, a
little cliche I've written
down here.
It's called misinterpretation
versus non-comprehension.
If you don't understand what
something means, there's no
danger you're going to
misinterpret it.
The danger is when you think
that you know what something
means and you have the
thing twisted around.
You see, the idea is this.
Let's go back to our old
friend y equals sine x.
Let's pick the value of
y equal to 1/2 say.
And now we say to the person the
same problem as we asked
before, find the angle
whose sine is 1/2.
Well, the point is to
find that angle.
If I draw this particular line,
I can find all sorts of
candidates.
The point is that what we tried
to do instead is to say,
OK, well now restrict
the function.
We'll break this down to be
a union of several curves.
In other words, it'll be this
curve union this one.
Union this one.
Union this one, et cetera.
What do all of these separate
pieces have in common?
What they have in common
is that what?
They are onto the range from
minus 1 to 1 and on that range
they are also 1:1.
1:1 and onto.
Every value of y between minus 1
and 1 is taken on along each
of these pieces.
And no value occurs more than
once on any of these pieces.
The point is it's not so crucial
whether you take this
particular one or whether you
take this particular piece.
That's something that's
sort of arbitrary.
What we must do to avoid
misinterpretation is unless
otherwise specified we say look
it, unless you hear from
me to the contrary, let's always
agree that this is the
little piece of the curve that
we're talking about, or this
is the piece that we're
talking about.
But the idea being what?
Unless you make such a
restriction, we cannot talk
about inverse functions.
The idea being that for an
inverse function to exist, we
must be able to back map.
We must be able to go from the
value in the image to the
value in the domain without
any danger of
misinterpretation.
We can conclude our example with
returning to our y equals
x squared idea again.
You see the idea is given the
curve y equals x squared, we
can think of it in terms
of our pieces
k2 and k1 as before.
The accented piece
being k2, the
non-accented piece being k1.
And what we're saying is the
inverse of k1 is this curve
here, which I'll call
k1 inverse.
The inverse of k2 is this
curve here, which
I'll call k2 inverse.
In other words, the important
thing is I can find the
inverse of either this
curve or this curve.
And in fact, how do I do that?
Again, with respect to the 45
degree line, the line y equals
x, notice that k1 and k1 inverse
are symmetric with
respect to this 45
degree line.
And similarly, so are
k2 and k2 inverse.
The thing I must be very careful
about and this is
where problems occur, is
I must not confuse--
for example, what I can't do is
take, for example, k2 and
k1 inverse.
Notice the built-in idea here.
These two curves together are
not symmetric with respect to
the 45 degree line.
You see what we're saying
here is, is what?
That for this particular
curve, x
and y are both positive.
So obviously, anything that
matches it must have x and y
both positive.
And that doesn't happen
over here.
What we're saying is you
can't do these things
completely at random.
However, what you can do is
either take k1 and match that
with k1 inverse, k2 and match
that with k2 inverse.
It's not important which of the
two ways you do this, as
long as you understand that
there is a danger of getting
mixed up once the curve
itself is not 1:1.
In other words, when we break
the curve down into 1:1
pieces, we have to make
sure that we match
these things up properly.
Now, this is all we're going to
say about inverse functions
for the time being.
The rest will be taken care of
in the exercises in this unit.
However, we will return to this
point very, very strongly
later in our discussion
of calculus.
The important point to remember
is that a function
and its inverse function give
us two different ways of
expressing the same
information.
And that we can use whichever
one happens
to be to our advantage.
Well, 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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