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HERBERT GROSS: Hi, our lecture
today is entitled inverse
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functions, and it's almost what
you could call a natural
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follow-up to our lecture of
last time when we talked
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00:00:41,280 --> 00:00:44,220
briefly about 1:1 and
onto functions.
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Inverse functions have a
tremendous application as we
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progress through calculus, but
of even more exciting impact
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00:00:52,430 --> 00:00:55,320
is the fact that inverse
functions are valuable in
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00:00:55,320 --> 00:00:56,340
their own right.
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00:00:56,340 --> 00:00:58,760
They are a pre-calculus topic.
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00:00:58,760 --> 00:01:02,110
In fact, they appear as early
in the curriculum as
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00:01:02,110 --> 00:01:04,560
approximately the first grade.
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See roughly speaking, inverse
functions in plain English,
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mean that all we've done is
made a switch in emphasis.
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00:01:12,680 --> 00:01:14,600
Let's take a look at that.
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00:01:14,600 --> 00:01:17,980
Let's go back roughly to our
first grade curriculum when
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00:01:17,980 --> 00:01:21,760
one learns that 2 plus
3 equals 5, or that 5
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00:01:21,760 --> 00:01:23,990
minus 3 equals 2.
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00:01:23,990 --> 00:01:27,820
Both of these statements say
the same thing, but with a
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00:01:27,820 --> 00:01:29,340
change in emphasis.
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00:01:29,340 --> 00:01:35,020
It's as if 2 is being emphasized
here while 5 is
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00:01:35,020 --> 00:01:36,770
being emphasized here.
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00:01:36,770 --> 00:01:38,220
This is rather interesting
you see.
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00:01:38,220 --> 00:01:41,740
For example, in the new
mathematics, one talks about
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00:01:41,740 --> 00:01:44,530
subtraction being the
inverse of addition.
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00:01:44,530 --> 00:01:47,470
And this is the same inverse
that we want to talk about
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00:01:47,470 --> 00:01:50,540
today as it applies to
mathematics in general.
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00:01:50,540 --> 00:01:53,410
What do we that subtraction is
the inverse of addition?
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00:01:53,410 --> 00:01:57,790
It may sound fancy, but all it
means is that if you know how
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00:01:57,790 --> 00:02:01,650
to add, if you define
subtraction properly, you
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00:02:01,650 --> 00:02:03,920
automatically know
how to subtract.
40
00:02:03,920 --> 00:02:06,910
And this, of course, is what's
prevalent in the old
41
00:02:06,910 --> 00:02:09,940
change-making technique of going
into a store, making a
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00:02:09,940 --> 00:02:13,100
purchase, paying for the
purchase, and when you receive
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00:02:13,100 --> 00:02:17,420
your change, the clerk rarely,
if ever, performs subtraction.
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00:02:17,420 --> 00:02:20,280
You may recall that what he
does is he adds onto the
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00:02:20,280 --> 00:02:23,590
amount of the purchase the
amount necessary to make up
46
00:02:23,590 --> 00:02:26,450
the denomination of the bill
with which you paid him.
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00:02:26,450 --> 00:02:30,120
In other words, what we're
saying here is, for example,
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00:02:30,120 --> 00:02:33,910
that one may think of 5
minus 3 as being what?
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00:02:33,910 --> 00:02:40,070
That number, which must be
added onto 3 to give 5.
50
00:02:40,070 --> 00:02:42,420
You see, in this sense,
subtraction is
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00:02:42,420 --> 00:02:44,130
the inverse of addition.
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00:02:44,130 --> 00:02:45,770
Once we know how to add, we
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00:02:45,770 --> 00:02:47,860
automatically know how to subtract.
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00:02:47,860 --> 00:02:50,110
Now you see, this idea
goes with us.
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00:02:50,110 --> 00:02:53,450
We learned that multiplication
and division are inverses of
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00:02:53,450 --> 00:02:54,160
one another.
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00:02:54,160 --> 00:02:57,200
And as we go on through higher
mathematics, even on the
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00:02:57,200 --> 00:03:00,570
pre-calculus level, we find
additional examples of this.
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00:03:00,570 --> 00:03:05,340
For example, when one knows
how to use exponents, one
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00:03:05,340 --> 00:03:08,230
automatically knows how
to study logarithms.
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00:03:08,230 --> 00:03:12,380
Namely, if y equals the log of
x to the base b, this is a
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00:03:12,380 --> 00:03:16,330
synonym for saying that
b to the y equals x.
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00:03:16,330 --> 00:03:18,360
And what is the basic difference
between these two
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statements?
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00:03:19,270 --> 00:03:22,170
They are paraphrases
of one another.
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In one case it seems that the
number y is being emphasized
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and in the other case it's
the number x that's being
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00:03:28,630 --> 00:03:30,610
emphasized.
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00:03:30,610 --> 00:03:34,460
And as in the case of most
examples of paraphrasing,
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00:03:34,460 --> 00:03:38,970
which of the two forms we use
depends on what problem it is
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00:03:38,970 --> 00:03:40,570
that we're trying to solve.
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00:03:40,570 --> 00:03:43,960
In other words, if we know
one of these two, then we
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00:03:43,960 --> 00:03:47,260
automatically can study the
other in terms of the one with
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which we feel more familiar.
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00:03:48,900 --> 00:03:52,420
This, of course, continues when
one gets to trigonometry
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and studies the so-called
inverse
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trigonometric functions.
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If y equals the inverse sine of
x that's the same thing as
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00:03:59,280 --> 00:04:01,840
saying that x equals sine y.
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00:04:01,840 --> 00:04:03,930
Again, what is the
basic difference?
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00:04:03,930 --> 00:04:07,580
In one case, it seems that
y is being emphasized.
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00:04:07,580 --> 00:04:10,930
In the other case, it's x
that's being emphasized.
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00:04:10,930 --> 00:04:15,680
In terms of the usual calculus
jargon of independent variable
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00:04:15,680 --> 00:04:18,370
versus dependent variable
it appears that what?
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00:04:18,370 --> 00:04:21,839
In one case, x is the
independent variable, y the
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00:04:21,839 --> 00:04:23,010
dependent variable.
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00:04:23,010 --> 00:04:26,890
In the other case, y is the
independent variable, x the
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00:04:26,890 --> 00:04:28,350
dependent variable.
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00:04:28,350 --> 00:04:32,850
To generalize this result what
we're saying is simply this.
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00:04:32,850 --> 00:04:37,740
If y equals f of x, you see
where y is being emphasized,
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00:04:37,740 --> 00:04:39,170
the dependent variable.
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00:04:39,170 --> 00:04:43,780
If we wish to switch the
emphasis, then we write x
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00:04:43,780 --> 00:04:47,290
equals f inverse of y.
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00:04:47,290 --> 00:04:50,850
This is read f inverse of y and
it's the function which in
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00:04:50,850 --> 00:04:53,750
a sense, inverts the
roles of x and y.
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00:04:53,750 --> 00:04:56,540
Let's see what this means more
explicitly in terms of a
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00:04:56,540 --> 00:04:58,660
particular example.
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00:04:58,660 --> 00:05:01,220
Let's suppose that we're
given the equation y
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00:05:01,220 --> 00:05:03,040
equals 2x minus 7.
100
00:05:03,040 --> 00:05:07,230
Or to represent this somewhat
more abstractly, y equals f of
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00:05:07,230 --> 00:05:10,800
x where f of x is 2x minus 7.
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00:05:10,800 --> 00:05:12,870
Now you see, without mentioning
the word inverse
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00:05:12,870 --> 00:05:15,970
function, it turns out that
early in our high school
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00:05:15,970 --> 00:05:19,640
career we were finding inverse
functions as soon as, for
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00:05:19,640 --> 00:05:23,750
example, someone were to give us
this problem and say solve
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00:05:23,750 --> 00:05:26,160
for x in terms of y.
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00:05:26,160 --> 00:05:28,640
Solve for x in terms of y.
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00:05:28,640 --> 00:05:34,820
You see, if we solve for x in
terms of y, we now have what?
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00:05:34,820 --> 00:05:36,490
x is being emphasized.
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00:05:36,490 --> 00:05:38,940
These two statements tell
us the same thing.
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00:05:38,940 --> 00:05:42,640
But now we write what? x
equals f inverse of y.
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00:05:42,640 --> 00:05:46,920
And f inverse of y is just
y plus 7 over 2.
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00:05:46,920 --> 00:05:50,490
Notice again the connection
between f inverse and f.
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00:05:50,490 --> 00:05:52,460
How one undoes the other.
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00:05:52,460 --> 00:05:55,240
In terms of our function machine
idea, what we're
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00:05:55,240 --> 00:06:00,370
saying is we may visualize the f
machine whereby the input is
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00:06:00,370 --> 00:06:04,220
x and the output will
be twice x minus 7.
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00:06:04,220 --> 00:06:06,420
In other words, the output
will always be twice
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00:06:06,420 --> 00:06:08,460
the input minus 7.
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00:06:08,460 --> 00:06:11,990
Now the question that comes up
is, suppose we reverse the
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00:06:11,990 --> 00:06:14,600
roles of the output
and the input.
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00:06:14,600 --> 00:06:20,560
In other words, suppose now we
let the input be y, what will
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00:06:20,560 --> 00:06:22,160
the output be?
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00:06:22,160 --> 00:06:25,440
If we reverse the terminal so
to speak, what we have shown
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00:06:25,440 --> 00:06:29,050
is that now the f inverse
machine would be what?
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00:06:29,050 --> 00:06:33,400
The input is y, the output
is y plus 7 over 2.
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00:06:33,400 --> 00:06:35,670
And by the way, a question that
we shall come back to
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00:06:35,670 --> 00:06:39,080
very shortly that plays a rather
important role here and
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00:06:39,080 --> 00:06:41,740
which I'll emphasize from
another point of view is that
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00:06:41,740 --> 00:06:46,440
if you get into the idea of
always wanting to call the
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00:06:46,440 --> 00:06:50,110
input x and the output y, which
is how we get geared to
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00:06:50,110 --> 00:06:53,130
do things in terms
of calculus.
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00:06:53,130 --> 00:06:57,430
x is always the horizontal axis,
y the vertical axis, and
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00:06:57,430 --> 00:07:00,120
we always agree to plot
the independent
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00:07:00,120 --> 00:07:01,560
variable along the x-axis.
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00:07:01,560 --> 00:07:04,240
In other words, the input along
the x-axis, the output
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00:07:04,240 --> 00:07:05,520
along the y-axis.
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00:07:05,520 --> 00:07:09,360
Then the question is, could we
have called this x and called
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00:07:09,360 --> 00:07:11,810
this x plus 7 over 2?
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00:07:11,810 --> 00:07:14,480
And we'll go with this in more
detail in a little while.
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00:07:14,480 --> 00:07:18,640
But obviously, what we call the
name of the input should
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00:07:18,640 --> 00:07:21,380
not affect how the
machine behaves.
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00:07:21,380 --> 00:07:24,870
By the way, as a little aside,
I thought it might be
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00:07:24,870 --> 00:07:29,100
interesting to show why we use
such notation as f inverse.
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00:07:29,100 --> 00:07:30,600
f to the minus 1.
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00:07:30,600 --> 00:07:32,170
It's rather interesting here.
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00:07:32,170 --> 00:07:37,320
Let's suppose we let a number
go into the f machine.
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00:07:37,320 --> 00:07:38,810
Call that number c.
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00:07:38,810 --> 00:07:42,120
Notice that any number that goes
into the f machine has as
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00:07:42,120 --> 00:07:46,470
its output twice that
number minus 7.
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00:07:46,470 --> 00:07:51,470
Suppose we now let that number
be the input of the f inverse
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00:07:51,470 --> 00:07:54,950
machine, what does the
f inverse machine do?
153
00:07:54,950 --> 00:08:03,030
It adds 7 onto any input and
then divides that result by 2.
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00:08:03,030 --> 00:08:07,420
In other words, if we now run
2c minus 7 through the f
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00:08:07,420 --> 00:08:13,360
inverse machine, we have
2c minus 7 plus 7
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00:08:13,360 --> 00:08:16,480
over 2 equals c.
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00:08:16,480 --> 00:08:20,260
In other words, notice how
the f inverse machine
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00:08:20,260 --> 00:08:22,420
undoes the f machine.
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00:08:22,420 --> 00:08:26,370
If we wanted to use the language
of last time in terms
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00:08:26,370 --> 00:08:30,250
of composition of functions,
what we do is what?
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00:08:30,250 --> 00:08:36,980
What we're saying is that if
you combine f followed by f
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00:08:36,980 --> 00:08:42,980
inverse, f inverse following f,
that that gives you what we
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00:08:42,980 --> 00:08:47,000
can call the identity
function.
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00:08:47,000 --> 00:08:48,400
The identity function.
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00:08:48,400 --> 00:08:53,310
Namely, if the input is c, the
output will again be c.
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00:08:53,310 --> 00:09:00,610
In other words, f inverse of f
of c is just c back again.
167
00:09:00,610 --> 00:09:04,090
In a similar way, notice that
we can reverse these roles.
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00:09:04,090 --> 00:09:07,290
We saw last time that
composition of functions
169
00:09:07,290 --> 00:09:09,620
depends on which order you
combine the functions.
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00:09:09,620 --> 00:09:13,500
But notice that if you run d
through the f inverse machine,
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00:09:13,500 --> 00:09:16,470
the output will be
d plus 7 over 2.
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00:09:16,470 --> 00:09:19,900
If this becomes the input of the
f machine, remember what
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00:09:19,900 --> 00:09:21,290
the f machine does.
174
00:09:21,290 --> 00:09:23,740
It doubles the input
and subtracts 7.
175
00:09:23,740 --> 00:09:26,480
176
00:09:26,480 --> 00:09:32,190
In other words, again,
f of f inverse of d
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00:09:32,190 --> 00:09:34,250
gives me d back again.
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00:09:34,250 --> 00:09:37,600
In other words, in terms of
composition of functions, f
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00:09:37,600 --> 00:09:41,200
followed by f inverse or f
inverse followed by f is what
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00:09:41,200 --> 00:09:43,520
we call the identity function.
181
00:09:43,520 --> 00:09:46,220
That one is truly the inverse
of the other from that
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00:09:46,220 --> 00:09:47,750
particular point of view.
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00:09:47,750 --> 00:09:51,120
However, let's correlate what
we're talking about now with
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00:09:51,120 --> 00:09:53,960
the circle diagrams that we
used in our last lecture.
185
00:09:53,960 --> 00:09:57,680
186
00:09:57,680 --> 00:10:01,200
You see, first of all, let's
recall that unless our
187
00:10:01,200 --> 00:10:05,270
function is both 1:1
and onto, we do not
188
00:10:05,270 --> 00:10:07,130
have an inverse function.
189
00:10:07,130 --> 00:10:12,490
Namely, for example, if our
function had not been onto,
190
00:10:12,490 --> 00:10:13,800
then when we--
191
00:10:13,800 --> 00:10:14,900
see, here's the idea again.
192
00:10:14,900 --> 00:10:16,400
Let me make sure
this is clear.
193
00:10:16,400 --> 00:10:20,270
To get an inverse function,
essentially all we do is this.
194
00:10:20,270 --> 00:10:23,680
If f is a function from A to
B, the inverse function is
195
00:10:23,680 --> 00:10:27,530
defined by reversing the
input and the output.
196
00:10:27,530 --> 00:10:31,330
Which means in terms of this
diagram, we reverse the sense
197
00:10:31,330 --> 00:10:32,340
of our arrows.
198
00:10:32,340 --> 00:10:35,430
We reverse which end the
arrowhead goes on.
199
00:10:35,430 --> 00:10:38,490
And what we're saying is, if we
had a function from A to B,
200
00:10:38,490 --> 00:10:41,860
which was not onto, then you
see when we reverse the
201
00:10:41,860 --> 00:10:45,870
arrowheads, f is not defined
on all of b.
202
00:10:45,870 --> 00:10:49,860
In other words, the domain of
f, the domain of the inverse
203
00:10:49,860 --> 00:10:52,880
function, would not exist
because it would not be
204
00:10:52,880 --> 00:10:54,490
defined on all of B.
205
00:10:54,490 --> 00:10:59,380
Secondly, if two different
elements of A went into the
206
00:10:59,380 --> 00:11:02,840
same element of B when we
reversed the arrowheads, the
207
00:11:02,840 --> 00:11:05,570
resulting function would
not be single-valued.
208
00:11:05,570 --> 00:11:08,890
And hence, in terms of modern
mathematics, it would not be a
209
00:11:08,890 --> 00:11:11,680
well defined function.
210
00:11:11,680 --> 00:11:15,290
So in other words, for the
inverse to exist it must be
211
00:11:15,290 --> 00:11:19,590
that the original function
is both 1:1 and onto.
212
00:11:19,590 --> 00:11:22,790
And as an example of that, this
is what this diagram here
213
00:11:22,790 --> 00:11:23,780
represents.
214
00:11:23,780 --> 00:11:26,140
And to make sure that we can
read this all right, I have
215
00:11:26,140 --> 00:11:29,440
singled out a typical element
of capital A, a typical
216
00:11:29,440 --> 00:11:32,740
element of capital B. Remember
what our notation is.
217
00:11:32,740 --> 00:11:36,390
The notation is that
f of a equals b.
218
00:11:36,390 --> 00:11:39,110
That the image of
a under f is b.
219
00:11:39,110 --> 00:11:42,700
And to use that in terms of
the inverse language, if I
220
00:11:42,700 --> 00:11:45,340
called g the function that
I get when I reversed the
221
00:11:45,340 --> 00:11:48,140
arrowheads, g of b equals a.
222
00:11:48,140 --> 00:11:50,870
And g is what I'm calling
f inverse.
223
00:11:50,870 --> 00:11:55,200
By way of further review, the
domain of f is equal to the
224
00:11:55,200 --> 00:11:56,780
image of f inverse.
225
00:11:56,780 --> 00:11:59,750
And that's A. The image
of f is the same as
226
00:11:59,750 --> 00:12:01,820
the domain of f inverse.
227
00:12:01,820 --> 00:12:03,180
And that's B.
228
00:12:03,180 --> 00:12:05,300
And now, what the question
is, is this.
229
00:12:05,300 --> 00:12:08,850
Notice that as long as you want
to use the same diagram,
230
00:12:08,850 --> 00:12:13,180
all we have to do to express f
inverse in terms of f is sort
231
00:12:13,180 --> 00:12:15,350
of to reverse the arrowheads.
232
00:12:15,350 --> 00:12:19,150
The question that comes up is,
suppose you insist that the
233
00:12:19,150 --> 00:12:21,550
domain be listed first.
234
00:12:21,550 --> 00:12:25,030
In other words, when we're going
to talk about g or f
235
00:12:25,030 --> 00:12:29,890
inverse in this case, that's a
function from B to A. So why
236
00:12:29,890 --> 00:12:31,720
don't we list B first
237
00:12:31,720 --> 00:12:33,640
And you see again,
we can do this.
238
00:12:33,640 --> 00:12:38,050
Here are the elements of B, here
are the elements of A.
239
00:12:38,050 --> 00:12:42,250
And all we have to do is see
what happens over here.
240
00:12:42,250 --> 00:12:45,510
For example, if we come back to
here notice that the first
241
00:12:45,510 --> 00:12:50,830
element listed in B comes from
the third element listed in A.
242
00:12:50,830 --> 00:12:55,120
So when I make up the inverse
function, I just capitalize on
243
00:12:55,120 --> 00:12:57,740
this by writing the
same thing.
244
00:12:57,740 --> 00:12:59,290
The only problem is--
245
00:12:59,290 --> 00:13:01,430
and this is going to become
a crucial one--
246
00:13:01,430 --> 00:13:04,880
is the fact that if somehow or
other you couldn't see these
247
00:13:04,880 --> 00:13:08,640
labels, if you couldn't see
these labels and all you knew
248
00:13:08,640 --> 00:13:11,170
was that the first set was
called the domain and the
249
00:13:11,170 --> 00:13:13,090
second set was called
the image.
250
00:13:13,090 --> 00:13:17,760
If you now looked at these two
functions, you see they
251
00:13:17,760 --> 00:13:20,470
wouldn't look anything
at all alike.
252
00:13:20,470 --> 00:13:23,910
In other words, f and f inverse,
while not independent
253
00:13:23,910 --> 00:13:26,330
of one another, do look
quite different.
254
00:13:26,330 --> 00:13:30,370
For example, notice that f
inverse causes the first two
255
00:13:30,370 --> 00:13:33,320
elements in here to sort of
crisscross as they have
256
00:13:33,320 --> 00:13:37,070
images, and the second two
elements of here crisscross.
257
00:13:37,070 --> 00:13:40,870
Notice though in terms of f,
it's the second and third that
258
00:13:40,870 --> 00:13:44,570
crisscross and the first and
fourth that don't intersect at
259
00:13:44,570 --> 00:13:45,060
all this way.
260
00:13:45,060 --> 00:13:48,570
In other words, if you look at
this curve or this diagram and
261
00:13:48,570 --> 00:13:51,870
compare it with this diagram,
notice that there is a
262
00:13:51,870 --> 00:13:54,800
difference in what seems
to be going on.
263
00:13:54,800 --> 00:13:57,340
Well again, this is
quite abstract.
264
00:13:57,340 --> 00:14:01,020
Let's try to relate this as
much as possible to the
265
00:14:01,020 --> 00:14:04,780
language of calculus and our
coordinate geometry graphing
266
00:14:04,780 --> 00:14:05,790
techniques.
267
00:14:05,790 --> 00:14:10,080
To begin with, let's suppose
that we have a function f
268
00:14:10,080 --> 00:14:13,690
whose domain is the closed
interval from a to b and whose
269
00:14:13,690 --> 00:14:17,390
range is the closed interval
from to c to d.
270
00:14:17,390 --> 00:14:20,300
And the question that we'd like
to raise is, under what
271
00:14:20,300 --> 00:14:23,580
conditions will f possess
an inverse function?
272
00:14:23,580 --> 00:14:24,930
What does onto mean?
273
00:14:24,930 --> 00:14:27,860
What does 1:1 mean and
things of this type?
274
00:14:27,860 --> 00:14:30,720
Well, the first thing I'd like
to point out is that if the
275
00:14:30,720 --> 00:14:34,780
graph y equals f of x looks
something like this.
276
00:14:34,780 --> 00:14:37,690
See, notice that the domain
is from a to b.
277
00:14:37,690 --> 00:14:40,410
The image is from c to d.
278
00:14:40,410 --> 00:14:43,600
Notice the fact that we have a
break in the curve over here,
279
00:14:43,600 --> 00:14:46,470
tells us that our function
is not onto.
280
00:14:46,470 --> 00:14:51,730
Namely, given this number p,
which is in our image between
281
00:14:51,730 --> 00:14:55,000
c and d, there is no element
of the domain
282
00:14:55,000 --> 00:14:56,720
that maps into p.
283
00:14:56,720 --> 00:15:00,090
So, in other words, if there is
a break in the curve, the
284
00:15:00,090 --> 00:15:01,950
function is not onto
and hence, it
285
00:15:01,950 --> 00:15:04,800
will not have an inverse.
286
00:15:04,800 --> 00:15:07,530
Now suppose there is no
break in the curve.
287
00:15:07,530 --> 00:15:10,870
Let's suppose now that the
curve doubles back.
288
00:15:10,870 --> 00:15:13,230
It comes up and doubles back.
289
00:15:13,230 --> 00:15:17,030
Now what my claim is is that the
function will not be 1:1.
290
00:15:17,030 --> 00:15:18,770
Well, how can we see that?
291
00:15:18,770 --> 00:15:22,100
Pick any part where the curve
doubles back, pick a point
292
00:15:22,100 --> 00:15:25,100
like this in that range.
293
00:15:25,100 --> 00:15:26,760
Call that point q.
294
00:15:26,760 --> 00:15:31,290
Noticed that q is in the
proper range of f now.
295
00:15:31,290 --> 00:15:33,590
y equals f of x.
296
00:15:33,590 --> 00:15:38,190
Now the question is, given the
y value of q, are there any
297
00:15:38,190 --> 00:15:41,410
x-values that map
into q under y?
298
00:15:41,410 --> 00:15:43,630
And the answer is
yes, there are.
299
00:15:43,630 --> 00:15:46,560
In fact, there are
more than one.
300
00:15:46,560 --> 00:15:51,830
Namely, notice that both f of
x1 and f of x2 equal q.
301
00:15:51,830 --> 00:15:56,950
In other words, in this case,
f of x1 equals f of x2, even
302
00:15:56,950 --> 00:16:01,080
though x1 is unequal to x2.
303
00:16:01,080 --> 00:16:04,090
That means that this function
is not 1:1.
304
00:16:04,090 --> 00:16:07,550
And because it's not 1:1, it
doesn't have a well defined
305
00:16:07,550 --> 00:16:09,220
inverse function.
306
00:16:09,220 --> 00:16:12,630
Well, putting these two cases
together, what it means for
307
00:16:12,630 --> 00:16:15,270
the function to be onto, what
it means for the function to
308
00:16:15,270 --> 00:16:20,870
be 1:1, it turns out that if
our curve is unbroken, then
309
00:16:20,870 --> 00:16:25,310
the only way our function can
have an inverse function is
310
00:16:25,310 --> 00:16:29,350
that the curve must either
always be rising or always be
311
00:16:29,350 --> 00:16:32,480
falling, and it can't
have a break in it.
312
00:16:32,480 --> 00:16:36,020
And by the way, as an aside,
let me point out here the
313
00:16:36,020 --> 00:16:38,750
difference between a continuous
variable--
314
00:16:38,750 --> 00:16:42,320
meaning one that's defined
on a whole interval--
315
00:16:42,320 --> 00:16:43,600
and a discrete variable--
316
00:16:43,600 --> 00:16:46,030
meaning where you get isolated
pieces of data.
317
00:16:46,030 --> 00:16:50,350
Notice, for example, if you plot
y versus x the way we do
318
00:16:50,350 --> 00:16:53,480
in a lab experiment where for
a particular value of x, you
319
00:16:53,480 --> 00:16:55,170
measure a value of y.
320
00:16:55,170 --> 00:17:00,670
Notice that the data can
double back without the
321
00:17:00,670 --> 00:17:03,850
function being multi-valued.
322
00:17:03,850 --> 00:17:06,550
In other words, notice for
example, that even though the
323
00:17:06,550 --> 00:17:08,109
curve doubles back here--
324
00:17:08,109 --> 00:17:09,560
I can't call it a curve.
325
00:17:09,560 --> 00:17:11,060
The data doubles back.
326
00:17:11,060 --> 00:17:14,869
Notice, for example, that no two
different pieces of data
327
00:17:14,869 --> 00:17:16,990
have the same y-coordinate.
328
00:17:16,990 --> 00:17:20,760
In other words, given this point
here as being q, there
329
00:17:20,760 --> 00:17:23,280
is only one piece of
data that has its
330
00:17:23,280 --> 00:17:25,319
y-coordinate equal to q.
331
00:17:25,319 --> 00:17:29,570
However, of course, keep in
mind it is possible that
332
00:17:29,570 --> 00:17:32,330
another piece of data will
have the same coordinate.
333
00:17:32,330 --> 00:17:35,440
All I'm saying is that the
idea of whether the curve
334
00:17:35,440 --> 00:17:39,130
always has to be rising or
following certainly depends on
335
00:17:39,130 --> 00:17:41,120
whether you have a continuous
curve or not.
336
00:17:41,120 --> 00:17:45,820
Well, again, let's continue on
with what inverse functions
337
00:17:45,820 --> 00:17:47,030
are all about.
338
00:17:47,030 --> 00:17:50,970
You see, this comes up with our
whole idea of why do we
339
00:17:50,970 --> 00:17:54,670
make fun or why do we minimize
single-valued
340
00:17:54,670 --> 00:17:56,770
functions in calculus?
341
00:17:56,770 --> 00:18:00,380
And the answer is that
single-valued--
342
00:18:00,380 --> 00:18:00,850
I'm sorry.
343
00:18:00,850 --> 00:18:04,490
Why do we always stick to
single-valued functions and do
344
00:18:04,490 --> 00:18:06,500
away with multi-valued
functions?
345
00:18:06,500 --> 00:18:09,410
And the answer is if you have a
smooth curve, we can always
346
00:18:09,410 --> 00:18:12,900
break down a multi-valued
function into a union of
347
00:18:12,900 --> 00:18:14,370
single-valued functions.
348
00:18:14,370 --> 00:18:18,240
For example, if we take the
curve c here as being y equals
349
00:18:18,240 --> 00:18:20,680
f of x, which plots like this.
350
00:18:20,680 --> 00:18:25,640
Notice if I take the points at
which I have vertical tangents
351
00:18:25,640 --> 00:18:27,970
and break the curve up at
those particular points.
352
00:18:27,970 --> 00:18:29,500
In this case, I'll
get what curves?
353
00:18:29,500 --> 00:18:31,910
c1, c2, and c3.
354
00:18:31,910 --> 00:18:37,070
Notice that c is the union of
c1, c2, and c3, but that each
355
00:18:37,070 --> 00:18:43,320
of the curves c1, c2, and c3
are either always rising or
356
00:18:43,320 --> 00:18:44,940
always falling.
357
00:18:44,940 --> 00:18:50,720
In a similar way, when we have
a function which doubles
358
00:18:50,720 --> 00:18:54,660
back-- and by the way, notice
what the connection is between
359
00:18:54,660 --> 00:18:57,870
multi-valued and not 1:1.
360
00:18:57,870 --> 00:19:01,110
You see, notice that in terms
of a function versus its
361
00:19:01,110 --> 00:19:07,040
inverse function idea, that if a
function is multi-valued the
362
00:19:07,040 --> 00:19:11,140
inverse function
cannot be 1:1.
363
00:19:11,140 --> 00:19:13,960
In other words, the idea being
that when you interchange the
364
00:19:13,960 --> 00:19:18,540
domain and the range, sort of
the curve flips over idea, all
365
00:19:18,540 --> 00:19:20,660
I want you to see here
is that what?
366
00:19:20,660 --> 00:19:24,720
If you're given a function which
is not single-valued, if
367
00:19:24,720 --> 00:19:27,960
we take the points at which
horizontal tangents occur and
368
00:19:27,960 --> 00:19:34,980
break down the curve like this,
we can break the curve
369
00:19:34,980 --> 00:19:38,840
down into a union of
1:1 functions.
370
00:19:38,840 --> 00:19:41,940
The hardship being of course,
that when you start with a
371
00:19:41,940 --> 00:19:44,900
point like this, analytically
speaking, it's rather
372
00:19:44,900 --> 00:19:48,170
difficult unless you invent some
scheme to know which of
373
00:19:48,170 --> 00:19:52,340
the points here you want
to single out.
374
00:19:52,340 --> 00:19:55,090
In terms of our previous
experience, it's sort of like
375
00:19:55,090 --> 00:19:58,320
saying to a person, I am
thinking of the angle whose
376
00:19:58,320 --> 00:19:59,820
sine is 1/2.
377
00:19:59,820 --> 00:20:03,010
There are, you see, infinitely
many functions whose sine is
378
00:20:03,010 --> 00:20:04,760
equal to 1/2.
379
00:20:04,760 --> 00:20:08,210
Of course if we say to the
person, I am thinking of the
380
00:20:08,210 --> 00:20:12,200
angle whose sine is 1/2 and the
angle is between minus 90
381
00:20:12,200 --> 00:20:16,690
degrees and plus 90 degrees,
then the only possible answer
382
00:20:16,690 --> 00:20:19,050
is the angle must
be 30 degrees.
383
00:20:19,050 --> 00:20:21,740
But notice that when you have
a function which is not
384
00:20:21,740 --> 00:20:24,420
single-valued, the
inverse will be a
385
00:20:24,420 --> 00:20:25,690
multi-valued function.
386
00:20:25,690 --> 00:20:28,180
And we'll talk more about
that in a little while.
387
00:20:28,180 --> 00:20:31,340
Again, I just want to keep this
shotgun approach going on
388
00:20:31,340 --> 00:20:35,280
just what an inverse function
is in relationship to the
389
00:20:35,280 --> 00:20:36,560
function itself.
390
00:20:36,560 --> 00:20:39,780
Again, let's look at this
more abstractly.
391
00:20:39,780 --> 00:20:42,710
Here I have drawn
a curve which is
392
00:20:42,710 --> 00:20:44,950
continuous and always rising.
393
00:20:44,950 --> 00:20:47,890
So I can talk about the
inverse function.
394
00:20:47,890 --> 00:20:50,710
If the equation is y
equals f of x, the
395
00:20:50,710 --> 00:20:52,110
inverse is written what?
396
00:20:52,110 --> 00:20:54,610
x equals f inverse y.
397
00:20:54,610 --> 00:20:57,920
And if this seems a little bit
too abstract for you, think of
398
00:20:57,920 --> 00:21:00,260
a concrete representation.
399
00:21:00,260 --> 00:21:02,600
Suppose the curve happened
to represent y
400
00:21:02,600 --> 00:21:04,570
equals 10 to the x.
401
00:21:04,570 --> 00:21:07,180
Then another way of saying
the same thing would be
402
00:21:07,180 --> 00:21:09,300
x equals log y.
403
00:21:09,300 --> 00:21:11,600
The convention of course here
being that you don't usually
404
00:21:11,600 --> 00:21:13,000
write base 10.
405
00:21:13,000 --> 00:21:14,850
But we won't worry about that.
406
00:21:14,850 --> 00:21:16,080
You see, this is what?
407
00:21:16,080 --> 00:21:21,500
Two different ways of expressing
the same curve.
408
00:21:21,500 --> 00:21:26,210
Whether I write y equals f of
x or x equals f inverse y, I
409
00:21:26,210 --> 00:21:29,010
have the same curve this way.
410
00:21:29,010 --> 00:21:32,380
In terms of our arrows, you see
what I'm saying is, if I
411
00:21:32,380 --> 00:21:36,760
start with x1, by going this
way, my function determines
412
00:21:36,760 --> 00:21:38,690
the output y1.
413
00:21:38,690 --> 00:21:43,440
Inversely, if I start with y1
and reverse the arrows, I wind
414
00:21:43,440 --> 00:21:44,830
up with x1.
415
00:21:44,830 --> 00:21:48,140
Again, the basic difference
being as to which of the two
416
00:21:48,140 --> 00:21:50,300
variables is being emphasized.
417
00:21:50,300 --> 00:21:53,910
What the real problem is, is
that most people say look it.
418
00:21:53,910 --> 00:21:56,300
I'm not used to studying
curves this way.
419
00:21:56,300 --> 00:22:00,000
I'm not used to looking at the
input being along the vertical
420
00:22:00,000 --> 00:22:03,820
axis and the output along the
horizontal axis according to
421
00:22:03,820 --> 00:22:05,860
the way I've been trained
when we're
422
00:22:05,860 --> 00:22:07,750
studying the inverse function.
423
00:22:07,750 --> 00:22:11,640
In other words, when y is the
input, aren't we used to
424
00:22:11,640 --> 00:22:16,140
having y over here and
then plotting the
425
00:22:16,140 --> 00:22:18,450
output along this axis?
426
00:22:18,450 --> 00:22:21,540
In other words, the question
is given this graph, how do
427
00:22:21,540 --> 00:22:24,240
you arrive at this one?
428
00:22:24,240 --> 00:22:27,450
You see, somehow or other, let's
observe that if all you
429
00:22:27,450 --> 00:22:30,990
did was switch your orientation
and say let me
430
00:22:30,990 --> 00:22:34,890
switch this by 90 degrees,
notice that we would be in a
431
00:22:34,890 --> 00:22:36,080
little bit of trouble.
432
00:22:36,080 --> 00:22:39,230
In other words, if we start with
this kind of a set up and
433
00:22:39,230 --> 00:22:43,790
we say, let's rotate through
a positive 90 degrees.
434
00:22:43,790 --> 00:22:46,930
Notice now what we would
wind up with is what?
435
00:22:46,930 --> 00:22:51,170
Our x-axis would be the way we
want it, but the y-axis would
436
00:22:51,170 --> 00:22:54,620
now have the opposite sense of
what we usually want our input
437
00:22:54,620 --> 00:22:55,900
axis to look like.
438
00:22:55,900 --> 00:22:59,840
So after we rotate through 90
degrees, it would seem that
439
00:22:59,840 --> 00:23:01,560
the next step is to do what?
440
00:23:01,560 --> 00:23:04,200
441
00:23:04,200 --> 00:23:10,190
Flip with respect
to the x-axis.
442
00:23:10,190 --> 00:23:12,130
That means fold this
thing over.
443
00:23:12,130 --> 00:23:16,660
In other words, a 90 degree
rotation followed by a folding
444
00:23:16,660 --> 00:23:22,840
over gives me the orientation if
I insist that the input has
445
00:23:22,840 --> 00:23:29,420
to be along the horizontal axis
and the output along the
446
00:23:29,420 --> 00:23:30,380
vertical axis.
447
00:23:30,380 --> 00:23:33,180
What I want you to also notice
though, is that if we don't
448
00:23:33,180 --> 00:23:37,390
insist on this, there is no
reason why we have to use two
449
00:23:37,390 --> 00:23:38,710
separate diagrams.
450
00:23:38,710 --> 00:23:41,230
Again notice, these are
two different ways of
451
00:23:41,230 --> 00:23:43,070
giving the same --
452
00:23:43,070 --> 00:23:46,040
two different equations for
giving the same curve.
453
00:23:46,040 --> 00:23:49,810
It's only when we want to switch
the role and make sure
454
00:23:49,810 --> 00:23:54,930
that the input is along the
horizontal axis that we have
455
00:23:54,930 --> 00:23:58,210
to go through this kind
of a process.
456
00:23:58,210 --> 00:24:00,410
Let's look at this a little
bit more concretely.
457
00:24:00,410 --> 00:24:03,500
458
00:24:03,500 --> 00:24:06,170
What I call a semi-concrete
illustration.
459
00:24:06,170 --> 00:24:08,960
What I'm saying now is let's
suppose this is the curve I've
460
00:24:08,960 --> 00:24:11,040
drawn in here, y
equals f of x.
461
00:24:11,040 --> 00:24:15,170
Another way of saying that is
x equals f inverse of y.
462
00:24:15,170 --> 00:24:18,450
And the question is, suppose I
now want to plot this same
463
00:24:18,450 --> 00:24:22,780
curve, same equation, but now
with the y-axis as my
464
00:24:22,780 --> 00:24:24,220
horizontal axis.
465
00:24:24,220 --> 00:24:28,330
You see again, in terms of
geometry, how I shift my axes
466
00:24:28,330 --> 00:24:30,710
will not change this equation.
467
00:24:30,710 --> 00:24:33,560
But what the picture of this
equation looks like will
468
00:24:33,560 --> 00:24:36,770
certainly depend on how
I orient my axes.
469
00:24:36,770 --> 00:24:38,170
So the idea is what?
470
00:24:38,170 --> 00:24:40,600
I simply fold this, rotate this
471
00:24:40,600 --> 00:24:42,770
thing, through 90 degrees.
472
00:24:42,770 --> 00:24:44,450
And if I do that, the resulting
473
00:24:44,450 --> 00:24:46,110
picture looks like this.
474
00:24:46,110 --> 00:24:48,170
And once the picture looks
like this, the
475
00:24:48,170 --> 00:24:49,460
next step is what?
476
00:24:49,460 --> 00:24:51,650
Flip this with respect
to the x-axis.
477
00:24:51,650 --> 00:24:54,360
478
00:24:54,360 --> 00:24:56,510
And now my picture
looks like this.
479
00:24:56,510 --> 00:25:01,750
In other words, x equals f
inverse of y here and x equals
480
00:25:01,750 --> 00:25:05,340
f inverse of y here are
the same equation.
481
00:25:05,340 --> 00:25:08,410
The reason that the picture
looks different is because I
482
00:25:08,410 --> 00:25:11,080
didn't allow myself
to use this as the
483
00:25:11,080 --> 00:25:12,340
axis of inputs here.
484
00:25:12,340 --> 00:25:18,490
In other words, again, as soon
as I wanted to make this axis
485
00:25:18,490 --> 00:25:21,280
orient so it would be the
horizontal axis, this is what
486
00:25:21,280 --> 00:25:23,300
I had to go through over here.
487
00:25:23,300 --> 00:25:26,060
Now you see, the next refinement
is that a person
488
00:25:26,060 --> 00:25:29,640
says look it, I'm not used to
calling this the y-axis.
489
00:25:29,640 --> 00:25:30,640
What's in a name?
490
00:25:30,640 --> 00:25:33,790
Why don't we always agree to
call the horizontal axis the
491
00:25:33,790 --> 00:25:37,370
x-axis and the vertical
axis the y-axis?
492
00:25:37,370 --> 00:25:40,520
And if I agree to do that,
notice what happens just by
493
00:25:40,520 --> 00:25:43,260
changing the names
of the variables.
494
00:25:43,260 --> 00:25:48,180
All that happens is, is that
now this becomes y equals f
495
00:25:48,180 --> 00:25:49,440
inverse of x.
496
00:25:49,440 --> 00:25:52,770
This is an important thing
to notice then.
497
00:25:52,770 --> 00:25:55,420
In other words, if you insist
that the horizontal axis in
498
00:25:55,420 --> 00:25:59,220
both cases will be called the
x-axis and the vertical axis
499
00:25:59,220 --> 00:26:04,250
the y-axis, then this would be
the curve y equals f of x and
500
00:26:04,250 --> 00:26:08,130
this would be the curve y
equals f inverse of x.
501
00:26:08,130 --> 00:26:11,110
But again, the whole thing
comes about only when you
502
00:26:11,110 --> 00:26:14,750
insist on how you want
your axes oriented.
503
00:26:14,750 --> 00:26:18,830
Let's go back to our problem of
y equals 2x minus 7 and see
504
00:26:18,830 --> 00:26:21,140
what this thing means
in terms of a graph.
505
00:26:21,140 --> 00:26:26,550
As we saw previously, if y
equals 2x minus 7, x is equal
506
00:26:26,550 --> 00:26:30,000
to y plus 7 over 2.
507
00:26:30,000 --> 00:26:31,250
And the idea is what?
508
00:26:31,250 --> 00:26:33,160
Let's see what this thing
really means.
509
00:26:33,160 --> 00:26:36,950
If I plot the straight line y
equals 2x minus 7, this is the
510
00:26:36,950 --> 00:26:38,880
line that I get.
511
00:26:38,880 --> 00:26:41,650
Notice that as long as I'm
going to use the same
512
00:26:41,650 --> 00:26:45,270
orientation of axes here, it
makes no difference whether I
513
00:26:45,270 --> 00:26:49,560
call this line y equals 2x minus
7 or whether I call it x
514
00:26:49,560 --> 00:26:51,650
equals y plus 7 over 2.
515
00:26:51,650 --> 00:26:54,450
They are two different names
for the same line.
516
00:26:54,450 --> 00:26:59,440
The problem occurs when I insist
that the independent
517
00:26:59,440 --> 00:27:03,890
variable always be plotted
along the x-axis, the
518
00:27:03,890 --> 00:27:07,040
horizontal axis, and the
dependent variable along the
519
00:27:07,040 --> 00:27:08,070
vertical axis.
520
00:27:08,070 --> 00:27:11,170
Again, going through what we did
before, I first take this
521
00:27:11,170 --> 00:27:15,790
thing and I rotate it through
a positive 90 degrees.
522
00:27:15,790 --> 00:27:20,360
That takes this picture and
transforms it into this one.
523
00:27:20,360 --> 00:27:24,600
I now take this and I flip it
with respect to the x-axis,
524
00:27:24,600 --> 00:27:28,020
and that gives me this
picture here.
525
00:27:28,020 --> 00:27:30,380
Now what is this line here?
526
00:27:30,380 --> 00:27:33,440
It's x equals y plus 7 over 2.
527
00:27:33,440 --> 00:27:36,880
Again, this is the same
equation as this one.
528
00:27:36,880 --> 00:27:39,850
The reason that the pictures
look differently is the fact
529
00:27:39,850 --> 00:27:43,190
that we have changed the
orientation of the axis.
530
00:27:43,190 --> 00:27:47,810
Again, if we now say OK, let's
rename this the x-axis, let's
531
00:27:47,810 --> 00:27:53,020
rename this the y-axis, then
this becomes what? y equals x
532
00:27:53,020 --> 00:27:54,940
plus 7 over 2.
533
00:27:54,940 --> 00:27:58,840
It's in this sense that we
call this curve of this
534
00:27:58,840 --> 00:28:02,760
equation and this equation here,
that these two equations
535
00:28:02,760 --> 00:28:05,580
are inverses of one another.
536
00:28:05,580 --> 00:28:08,970
Again, in terms of what we said
before, if you pick a
537
00:28:08,970 --> 00:28:12,780
particular value of x and
compute y this way, then you
538
00:28:12,780 --> 00:28:15,530
apply this recipe to that.
539
00:28:15,530 --> 00:28:21,540
In other words, if you now take
twice y, twice the output
540
00:28:21,540 --> 00:28:26,620
here, and subtract 7, your
original input returns.
541
00:28:26,620 --> 00:28:28,120
In other words, this
works exactly the
542
00:28:28,120 --> 00:28:29,830
same as we did before.
543
00:28:29,830 --> 00:28:33,230
But again, the whole basic
difference is what?
544
00:28:33,230 --> 00:28:37,150
How you want to orient
your axis.
545
00:28:37,150 --> 00:28:40,680
That the curves look different
because your coordinate system
546
00:28:40,680 --> 00:28:42,660
is different.
547
00:28:42,660 --> 00:28:44,880
Of course, the interesting
question now is, if we
548
00:28:44,880 --> 00:28:48,610
compared these two curves, since
the-- see, granted that
549
00:28:48,610 --> 00:28:51,480
the function and its inverse are
different functions, they
550
00:28:51,480 --> 00:28:52,740
are somewhat related.
551
00:28:52,740 --> 00:28:54,490
They're not random.
552
00:28:54,490 --> 00:28:56,370
How are these two
graphs related?
553
00:28:56,370 --> 00:28:59,280
That might be the next natural
question to ask.
554
00:28:59,280 --> 00:29:02,060
If we do that, the idea
is simply this.
555
00:29:02,060 --> 00:29:06,060
Let's suppose we have y equals
f of x as one of our curves.
556
00:29:06,060 --> 00:29:09,370
The curve happens to be
invertible, meaning that f is
557
00:29:09,370 --> 00:29:12,820
always rising and it's
unbroken, et cetera.
558
00:29:12,820 --> 00:29:17,700
The question is, if we now try
to plot y equals f inverse of
559
00:29:17,700 --> 00:29:20,320
x in the same diagram.
560
00:29:20,320 --> 00:29:21,570
See notice now what
I'm saying.
561
00:29:21,570 --> 00:29:24,070
In other words, I am not
saying x equals f
562
00:29:24,070 --> 00:29:25,150
inverse of y here.
563
00:29:25,150 --> 00:29:29,060
I'm saying suppose I have the
curve y equals f of x and also
564
00:29:29,060 --> 00:29:31,620
the curve y equals
f inverse of x.
565
00:29:31,620 --> 00:29:35,780
How do these two curves look
with respect to an x- and
566
00:29:35,780 --> 00:29:38,090
y-coordinate system?
567
00:29:38,090 --> 00:29:40,450
See, let me do that part
more slowly again.
568
00:29:40,450 --> 00:29:42,700
Let me come over here
for a moment.
569
00:29:42,700 --> 00:29:47,580
Notice that y equals 2x minus
7 was my original curve in
570
00:29:47,580 --> 00:29:49,010
what I dealt with before.
571
00:29:49,010 --> 00:29:52,200
If I want to keep the same
orientation of axis, the
572
00:29:52,200 --> 00:29:56,370
inverse function we saw was
y equals x plus 7 over 2.
573
00:29:56,370 --> 00:29:58,410
The question that we're asking
quite in general, not in this
574
00:29:58,410 --> 00:30:02,040
specific case, is how are these
two curves related?
575
00:30:02,040 --> 00:30:04,370
And the solution goes
something like this.
576
00:30:04,370 --> 00:30:08,240
Let's suppose that the
point x1, y1 belongs
577
00:30:08,240 --> 00:30:10,050
to the curve c1.
578
00:30:10,050 --> 00:30:15,690
By definition of c1, that
says that y1 is f of x1.
579
00:30:15,690 --> 00:30:19,500
By definition of inverse
function, see if y1 is f of
580
00:30:19,500 --> 00:30:22,510
x1, that means if you
interchange the input and the
581
00:30:22,510 --> 00:30:24,620
output, that's another
way of saying what?
582
00:30:24,620 --> 00:30:28,920
That x1 is f inverse of y1.
583
00:30:28,920 --> 00:30:36,810
In other words again, if f maps
x1 into y1, f inverse
584
00:30:36,810 --> 00:30:40,230
maps y1 into x1,
by definition.
585
00:30:40,230 --> 00:30:45,970
Now you see, compare this
with the curve c2.
586
00:30:45,970 --> 00:30:48,530
See, this says what?
587
00:30:48,530 --> 00:30:54,690
That the input y1 maps
into the output x1.
588
00:30:54,690 --> 00:30:59,380
In other words, notice that if
you look at the f inverse
589
00:30:59,380 --> 00:31:04,740
situation here, when the input
is y1, the output is x1.
590
00:31:04,740 --> 00:31:09,330
That's another way of saying
that y1 comma x1 belongs to
591
00:31:09,330 --> 00:31:10,870
the curve c2.
592
00:31:10,870 --> 00:31:17,070
In other words, if x1, y1
belongs to y1 equals f of x1,
593
00:31:17,070 --> 00:31:22,430
then y1, x1 belongs to y
equals f inverse of x.
594
00:31:22,430 --> 00:31:27,170
Now, what is the relationship
between the point x1 comma y1
595
00:31:27,170 --> 00:31:30,080
and the point y1 comma x1?
596
00:31:30,080 --> 00:31:33,110
If we draw this little diagram,
we observe that we
597
00:31:33,110 --> 00:31:36,010
have a couple of congruent
triangles here.
598
00:31:36,010 --> 00:31:38,280
This length equals
this length.
599
00:31:38,280 --> 00:31:41,650
This angle equals this angle.
600
00:31:41,650 --> 00:31:43,180
And this gives me a hint.
601
00:31:43,180 --> 00:31:48,020
This makes triangle
OPQ isosceles.
602
00:31:48,020 --> 00:31:53,260
I draw the angle bisector of
angle O. The angle bisector of
603
00:31:53,260 --> 00:31:56,200
an isosceles triangle
is the perpendicular
604
00:31:56,200 --> 00:31:59,080
bisector of the base.
605
00:31:59,080 --> 00:32:01,350
And angle bisector of the
vertex angle is the
606
00:32:01,350 --> 00:32:03,350
perpendicular bisector
of the base.
607
00:32:03,350 --> 00:32:06,810
Well, you see that makes this
angle equal to this angle.
608
00:32:06,810 --> 00:32:09,720
That makes this a
45 degree angle.
609
00:32:09,720 --> 00:32:11,860
In other words, the line
that I've drawn is
610
00:32:11,860 --> 00:32:13,790
indeed, y equals x.
611
00:32:13,790 --> 00:32:18,100
And notice that P and Q are
symmetrically located with
612
00:32:18,100 --> 00:32:20,780
respect to the line
y equals x.
613
00:32:20,780 --> 00:32:25,070
In other words, going back to
our original problem here, the
614
00:32:25,070 --> 00:32:31,140
curve c1 and c2 are related by
the fact that they are mirror
615
00:32:31,140 --> 00:32:33,600
images of one another
with respect to the
616
00:32:33,600 --> 00:32:35,560
line y equals x.
617
00:32:35,560 --> 00:32:38,120
That's exactly what I've
drawn over here.
618
00:32:38,120 --> 00:32:40,760
In other words, going back to
the problem of how are the
619
00:32:40,760 --> 00:32:45,540
curves y equal 2x minus 7 and
y equal x plus 7 over 2
620
00:32:45,540 --> 00:32:49,250
related, the answer is they
are mirror images of one
621
00:32:49,250 --> 00:32:52,090
another with respect to
the line y equals x.
622
00:32:52,090 --> 00:32:55,210
They are symmetric with
respect to that line.
623
00:32:55,210 --> 00:32:58,990
Now you see, let's talk about
this from another point of
624
00:32:58,990 --> 00:33:01,440
view also, and show what
the tough thing is.
625
00:33:01,440 --> 00:33:04,190
You see, so far my whole
discussion seems to have
626
00:33:04,190 --> 00:33:07,810
hinged on the fact that we
have a function, which is
627
00:33:07,810 --> 00:33:08,700
invertible.
628
00:33:08,700 --> 00:33:12,090
What if you have a function
which is non-invertible?
629
00:33:12,090 --> 00:33:16,540
Going back to something more
familiar, why do we, talk
630
00:33:16,540 --> 00:33:18,700
about -- when y equals the
square root of x, why do we
631
00:33:18,700 --> 00:33:21,930
have this convention that we
take the positive square root?
632
00:33:21,930 --> 00:33:25,520
After all, doesn't the square
root of x and minus the square
633
00:33:25,520 --> 00:33:28,150
root of x have the property that
when you square them you
634
00:33:28,150 --> 00:33:29,290
get the same result?
635
00:33:29,290 --> 00:33:32,250
Plus or minus squared
is always plus.
636
00:33:32,250 --> 00:33:35,860
And the answer is that if you
square both sides here and
637
00:33:35,860 --> 00:33:39,210
think of this as being the curve
y squared equals x, what
638
00:33:39,210 --> 00:33:43,990
happens is you get a
multi-valued function.
639
00:33:43,990 --> 00:33:47,730
One value of x yields
two values of y.
640
00:33:47,730 --> 00:33:50,650
And the way we get around that
is we break this curve down
641
00:33:50,650 --> 00:33:56,050
into two pieces, c1 and c2,
where c1 is always rising.
642
00:33:56,050 --> 00:33:57,970
c2 is always falling here.
643
00:33:57,970 --> 00:34:00,560
In other words, we broke this
thing off at the point of
644
00:34:00,560 --> 00:34:01,850
vertical tangency.
645
00:34:01,850 --> 00:34:04,990
And we can now think of this
curve as being the union of
646
00:34:04,990 --> 00:34:05,850
two curves.
647
00:34:05,850 --> 00:34:09,080
One of which is y equals the
positive square root of x and
648
00:34:09,080 --> 00:34:12,480
the other is y equals the
negative square root of x.
649
00:34:12,480 --> 00:34:14,880
Now the question is, what
happens when you have a
650
00:34:14,880 --> 00:34:16,820
function which is not
single-valued.
651
00:34:16,820 --> 00:34:19,090
In other words, let's just
invert this one.
652
00:34:19,090 --> 00:34:23,800
Let's suppose we started with
the curve y equals x squared.
653
00:34:23,800 --> 00:34:28,050
You see, now for a given value
of y, I'm in trouble.
654
00:34:28,050 --> 00:34:32,520
Because if y1 is positive, there
are two different values
655
00:34:32,520 --> 00:34:36,300
of x which yield this
particular result.
656
00:34:36,300 --> 00:34:40,139
In other words, both of these
have the property that when
657
00:34:40,139 --> 00:34:42,460
you square them you get y1.
658
00:34:42,460 --> 00:34:46,440
And all we're saying is that in
a problem such as this, we
659
00:34:46,440 --> 00:34:50,250
can study this curve as
two separate pieces.
660
00:34:50,250 --> 00:34:52,440
Call one of these curves k1.
661
00:34:52,440 --> 00:34:55,560
That will be the curve y equals
x squared, where x is
662
00:34:55,560 --> 00:34:56,659
non-negative.
663
00:34:56,659 --> 00:34:58,730
So this will be the curve k1.
664
00:34:58,730 --> 00:35:02,890
And call the other one k2,
where k2 will be what?
665
00:35:02,890 --> 00:35:06,390
The same curve y equals x
squared, but its domain is the
666
00:35:06,390 --> 00:35:07,900
negative values of x.
667
00:35:07,900 --> 00:35:10,680
In other words, k2 will
be this one over here.
668
00:35:10,680 --> 00:35:13,620
And now the point is, if we deal
with either of these two
669
00:35:13,620 --> 00:35:18,870
pieces separately, we can talk
about inverse functions.
670
00:35:18,870 --> 00:35:22,690
Now the point is, which of these
two halves do we use?
671
00:35:22,690 --> 00:35:25,150
And this is where the word
principal values comes in.
672
00:35:25,150 --> 00:35:28,790
And you see what I'd like you
to keep in mind is this, a
673
00:35:28,790 --> 00:35:30,770
little cliche I've written
down here.
674
00:35:30,770 --> 00:35:35,220
It's called misinterpretation
versus non-comprehension.
675
00:35:35,220 --> 00:35:38,060
If you don't understand what
something means, there's no
676
00:35:38,060 --> 00:35:40,390
danger you're going to
misinterpret it.
677
00:35:40,390 --> 00:35:42,500
The danger is when you think
that you know what something
678
00:35:42,500 --> 00:35:45,240
means and you have the
thing twisted around.
679
00:35:45,240 --> 00:35:46,370
You see, the idea is this.
680
00:35:46,370 --> 00:35:49,450
Let's go back to our old
friend y equals sine x.
681
00:35:49,450 --> 00:35:53,290
Let's pick the value of
y equal to 1/2 say.
682
00:35:53,290 --> 00:35:55,650
And now we say to the person the
same problem as we asked
683
00:35:55,650 --> 00:35:59,470
before, find the angle
whose sine is 1/2.
684
00:35:59,470 --> 00:36:02,200
Well, the point is to
find that angle.
685
00:36:02,200 --> 00:36:05,350
If I draw this particular line,
I can find all sorts of
686
00:36:05,350 --> 00:36:07,240
candidates.
687
00:36:07,240 --> 00:36:10,370
The point is that what we tried
to do instead is to say,
688
00:36:10,370 --> 00:36:12,700
OK, well now restrict
the function.
689
00:36:12,700 --> 00:36:17,280
We'll break this down to be
a union of several curves.
690
00:36:17,280 --> 00:36:20,900
In other words, it'll be this
curve union this one.
691
00:36:20,900 --> 00:36:22,090
Union this one.
692
00:36:22,090 --> 00:36:23,310
Union this one, et cetera.
693
00:36:23,310 --> 00:36:26,210
What do all of these separate
pieces have in common?
694
00:36:26,210 --> 00:36:29,600
What they have in common
is that what?
695
00:36:29,600 --> 00:36:35,740
They are onto the range from
minus 1 to 1 and on that range
696
00:36:35,740 --> 00:36:38,610
they are also 1:1.
697
00:36:38,610 --> 00:36:39,490
1:1 and onto.
698
00:36:39,490 --> 00:36:44,280
Every value of y between minus 1
and 1 is taken on along each
699
00:36:44,280 --> 00:36:45,590
of these pieces.
700
00:36:45,590 --> 00:36:50,930
And no value occurs more than
once on any of these pieces.
701
00:36:50,930 --> 00:36:54,360
The point is it's not so crucial
whether you take this
702
00:36:54,360 --> 00:36:59,520
particular one or whether you
take this particular piece.
703
00:36:59,520 --> 00:37:01,580
That's something that's
sort of arbitrary.
704
00:37:01,580 --> 00:37:05,110
What we must do to avoid
misinterpretation is unless
705
00:37:05,110 --> 00:37:08,280
otherwise specified we say look
it, unless you hear from
706
00:37:08,280 --> 00:37:11,700
me to the contrary, let's always
agree that this is the
707
00:37:11,700 --> 00:37:14,310
little piece of the curve that
we're talking about, or this
708
00:37:14,310 --> 00:37:17,030
is the piece that we're
talking about.
709
00:37:17,030 --> 00:37:18,530
But the idea being what?
710
00:37:18,530 --> 00:37:21,380
Unless you make such a
restriction, we cannot talk
711
00:37:21,380 --> 00:37:22,700
about inverse functions.
712
00:37:22,700 --> 00:37:26,260
The idea being that for an
inverse function to exist, we
713
00:37:26,260 --> 00:37:28,070
must be able to back map.
714
00:37:28,070 --> 00:37:32,090
We must be able to go from the
value in the image to the
715
00:37:32,090 --> 00:37:35,540
value in the domain without
any danger of
716
00:37:35,540 --> 00:37:37,890
misinterpretation.
717
00:37:37,890 --> 00:37:42,760
We can conclude our example with
returning to our y equals
718
00:37:42,760 --> 00:37:44,420
x squared idea again.
719
00:37:44,420 --> 00:37:48,270
You see the idea is given the
curve y equals x squared, we
720
00:37:48,270 --> 00:37:50,590
can think of it in terms
of our pieces
721
00:37:50,590 --> 00:37:53,040
k2 and k1 as before.
722
00:37:53,040 --> 00:37:55,530
The accented piece
being k2, the
723
00:37:55,530 --> 00:37:58,260
non-accented piece being k1.
724
00:37:58,260 --> 00:38:03,290
And what we're saying is the
inverse of k1 is this curve
725
00:38:03,290 --> 00:38:06,480
here, which I'll call
k1 inverse.
726
00:38:06,480 --> 00:38:11,240
The inverse of k2 is this
curve here, which
727
00:38:11,240 --> 00:38:13,530
I'll call k2 inverse.
728
00:38:13,530 --> 00:38:15,980
In other words, the important
thing is I can find the
729
00:38:15,980 --> 00:38:21,240
inverse of either this
curve or this curve.
730
00:38:21,240 --> 00:38:22,800
And in fact, how do I do that?
731
00:38:22,800 --> 00:38:26,040
Again, with respect to the 45
degree line, the line y equals
732
00:38:26,040 --> 00:38:31,630
x, notice that k1 and k1 inverse
are symmetric with
733
00:38:31,630 --> 00:38:33,880
respect to this 45
degree line.
734
00:38:33,880 --> 00:38:38,870
And similarly, so are
k2 and k2 inverse.
735
00:38:38,870 --> 00:38:41,840
The thing I must be very careful
about and this is
736
00:38:41,840 --> 00:38:45,320
where problems occur, is
I must not confuse--
737
00:38:45,320 --> 00:38:52,040
for example, what I can't do is
take, for example, k2 and
738
00:38:52,040 --> 00:38:54,380
k1 inverse.
739
00:38:54,380 --> 00:38:56,960
Notice the built-in idea here.
740
00:38:56,960 --> 00:39:00,260
These two curves together are
not symmetric with respect to
741
00:39:00,260 --> 00:39:01,990
the 45 degree line.
742
00:39:01,990 --> 00:39:04,590
You see what we're saying
here is, is what?
743
00:39:04,590 --> 00:39:06,660
That for this particular
curve, x
744
00:39:06,660 --> 00:39:08,790
and y are both positive.
745
00:39:08,790 --> 00:39:11,950
So obviously, anything that
matches it must have x and y
746
00:39:11,950 --> 00:39:13,070
both positive.
747
00:39:13,070 --> 00:39:14,970
And that doesn't happen
over here.
748
00:39:14,970 --> 00:39:16,860
What we're saying is you
can't do these things
749
00:39:16,860 --> 00:39:18,200
completely at random.
750
00:39:18,200 --> 00:39:22,150
However, what you can do is
either take k1 and match that
751
00:39:22,150 --> 00:39:26,520
with k1 inverse, k2 and match
that with k2 inverse.
752
00:39:26,520 --> 00:39:30,470
It's not important which of the
two ways you do this, as
753
00:39:30,470 --> 00:39:34,010
long as you understand that
there is a danger of getting
754
00:39:34,010 --> 00:39:37,680
mixed up once the curve
itself is not 1:1.
755
00:39:37,680 --> 00:39:40,430
In other words, when we break
the curve down into 1:1
756
00:39:40,430 --> 00:39:42,930
pieces, we have to make
sure that we match
757
00:39:42,930 --> 00:39:44,540
these things up properly.
758
00:39:44,540 --> 00:39:47,630
Now, this is all we're going to
say about inverse functions
759
00:39:47,630 --> 00:39:49,290
for the time being.
760
00:39:49,290 --> 00:39:52,690
The rest will be taken care of
in the exercises in this unit.
761
00:39:52,690 --> 00:39:56,540
However, we will return to this
point very, very strongly
762
00:39:56,540 --> 00:39:59,500
later in our discussion
of calculus.
763
00:39:59,500 --> 00:40:02,250
The important point to remember
is that a function
764
00:40:02,250 --> 00:40:05,820
and its inverse function give
us two different ways of
765
00:40:05,820 --> 00:40:08,260
expressing the same
information.
766
00:40:08,260 --> 00:40:10,840
And that we can use whichever
one happens
767
00:40:10,840 --> 00:40:13,060
to be to our advantage.
768
00:40:13,060 --> 00:40:15,200
Well, until next
time, goodbye.
769
00:40:15,200 --> 00:40:18,020
770
00:40:18,020 --> 00:40:20,550
ANNOUNCER: Funding for the
publication of this video was
771
00:40:20,550 --> 00:40:25,260
provided by the Gabriella and
Paul Rosenbaum Foundation.
772
00:40:25,260 --> 00:40:29,440
Help OCW continue to provide
free and open access to MIT
773
00:40:29,440 --> 00:40:33,640
courses by making a donation
at ocw.mit.edu/donate.
774
00:40:33,640 --> 00:40:38,378