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## Formal definition of inverse functions (optional)

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# Introduction to the inverse of a function

## Video transcript

Let's say we have some function,
f, and it's a mapping from the set X to Y. So if I were to draw the set X
right there, that's my set X. If I were to draw the set
Y, just like that. We know, and I've done this
several videos ago, that a function just associates any
member of our set X-- so I have some member of my set
X there-- if I apply the function to it, or if we're
dealing with vectors, we can imagine instead of using the
word function, we would use the word transformation. But it's the same thing. We would associate with this
element, or this member of X, a member of Y. So that's why we call
it a mapping. When I apply this function--
I'll do it in a different color-- this little member of
X is associated with this member of Y. If this is right here,
this is a capital X. Let's say we call this a,
and let's call that b. We would say that the function
where a is a member of X and b is a member of Y, we would say
that f of a is equal to b. This is all a review of
everything that we've learned already about functions. Now I'm going to
define a couple of interesting functions. The first one-- I guess it's
really just one function, I said it's a couple-- but I'll
call it the identity function. This is a function. I'll just call it
a big capital I. This identity function
operates on some set. So let's say this is the
identity function on set X, and it's a mapping
from X to X. What's interesting about the
identity function is that if you give it some a that is a
member of X-- so lets say you give it that a-- the identity
function applied to that member of X, the identity
function of a, is going to be equal to a. So it literally just maps
things back to itself. So the identity function, if I
were draw it on this diagram right here, would
look like this. It would look like-- we pick a
nice suitable color-- it would look like this. It would just kind
of be a circle. It just points back
at the point that you started off with. It associates all points
with themselves. That's the identity function on
X, especially as it applies to the point a. If you apply it to some other
point in X, it would just refer back to itself. That's the identity
function on X. You could also have an identity
function on Y. So let's say that b
is a member of Y. So I drew b right there. Then the Y identity function--
so this would be that identity function on Y applied
to b-- would just refer back to itself. So that would be equal to b. This is the identity
function on Y. So you might say, hey
Sal, these are kind of silly functions. But we'll use them. They're actually at least a
useful notation to use as we progress through our explorations of linear algebra. But I'm going to make
a new definition. I'm going to say that a
function-- let me pick a nice color, pink-- I'm going to say
that a function, let me say f since we already established
it right over here. I'm going to say that f is
invertible, introducing some new terminology. f is invertible if and only
if the following is true. I could either write it with
these two-way arrows like that, or I could write it
as iff with two f's. That means that if this is true,
then this is true, and only if this is true. So this implies that, and
that implies this. So f is invertible-- I'm kind
of making a definition right here-- if and only if there
exists a function, I'll call it nothing just yet. I'll call it something
in a second. I'll write it as this f
with this negative 1 superscript on it. So f is invertible if and only
if there exists a function f inverse-- well I guess I just
called it something. Let me do that in purple. Remember f is just a mapping
from X to Y. So this function, f inverse,
is going to be a mapping from Y to X. So I'm saying that f is
invertible if there exists a function, f inverse, that's a
mapping from Y to X such that if I take the composition of f
inverse with f, this is equal to the identity function
over X. So let's think about
what's happening. This is just part
of it actually. Let me just complete the
whole definition. This is true, this has to be
true, and the composition of f with the inverse function has
to be equal to the identity function over Y. So let's just think about
what's this saying. There's some function-- I'll
call it right now, this called the inverse of f-- and it's
a mapping from Y to X. Let me draw it up here. So f is a mapping from X to Y. We showed that. This is the mapping
of f right there. It goes in that direction. We're saying there has be some
other function, f inverse, that's a mapping from Y to X. So let's write it here. So f inverse is a mapping from Y
to X. f inverse, if you give me some value in set
Y, I go to set X. So this guy's domain is this
guy's codomain, and this guy's codomain is this guy's domain. Fair enough. But let's see what
it's saying. It's saying that the composition
of f inverse with f, has to be equal to
the identity matrix. So essentially it's saying if I
apply f to some value in X-- right, if you think about
what's this composition doing-- this guy's going
from X to Y. This guy goes from Y to X. So let's think about what's
happening here. f is going from X to Y. Then f inverse is going
from Y to X. So this composition is going to
be a mapping from X to X, which the identity function
needs to do. It needs to go from X to X. They're saying this equals
the identity function. So that means when you apply f
on some value in our domain, so you go here, and then you
apply f inverse to that point over there, you go back to
this original point. So another way of saying this
is that f-- let me do it in another color-- the composition
of f inverse with f of some member of the set
X is equal to the identity function applied on that item. These two statements
are equivalent. So by definition, this
thing is going to be your original thing. Or another way of writing this
is that f inverse applied to f of a is going to
be equal to a. That's what this first
statement tells us. If you think of it visually,
it's saying you start with an a, you apply f to it, and you
get this value right here. That is f of a. I'm saying it equals b, or I
said it equalled b earlier on. Then if you apply this f
inverse-- and it doesn't always exist-- but if you apply
that f inverse to this function, it needs to
go back to this. By definition it needs to go
back to your original a. It has to be equivalent to just
doing this little closed loop right when I introduce you
to the identity function. Now that's what this statement
is telling us right here. The second statement is saying
look, if I apply f to f inverse, I'm getting the
identity function on Y. So if I start at some point in
Y right there, and I apply f inverse first, maybe
I go right here. Let's call that lowercase y So
this would be f inverse of lowercase y. Then if I were to apply f to
that-- I know this chart is getting very confusing-- if I
apply f to this right here, I need to go right back
to my original Y. So when I apply f to f inverse
of Y this has to be equivalent of just doing the identity
function on y. So that's what the second
statement is saying. Or another way to write it is
that f of f inverse of y, where y is a member of the
set capital Y, it has to be equal to Y. You've been exposed to the idea
of an inverse before. We're just doing it a little
bit more precisely because we're going to start dealing
with these notions with transformations and matrices
in the very near future. So it's good to be exposed to it
in this more precise form. Now the first thing you might
ask is let's say that I have a function f, and there does exist
a function f inverse that satisfies these
two requirements. So f is invertible. The obvious question, or maybe
it's not an obvious question is, is f inverse unique? Actually probably the obvious
question is how do you know when something's invertible. We're going to talk a
lot about that in the very near future. But let's say we know that
f is invertible. How do we know, or do we know
whether f inverse is unique? To answer that question, let's
assume it's not unique. So if it's not unique, let's say
that there's two functions that satisfy our two constraints
that can act as inverse functions of f. Let's say that g
is one of them. So let's say g is a mapping. Remember f is a mapping
from X to Y. Let's say that g is a mapping
from Y to X such that if I apply f to something and then
apply g to it-- so this gets me from X to Y. Then when I do the composition
with g, that gets me back into X. This is equivalent to the
identity function. This was part of the definition
of what it means to be an inverse. I'm assuming that g is
an inverse of f. This assumption implies
these two things. Now let's say that h is
another inverse of f. By definition, by what I just
called an inverse, h has to satisfy two requirements. It has to be a mapping
from Y to X. Then if I take the composition
of h with f, I have to get the identity matrix on the set X. Now that wasn't just part
of the definition. It implies even more
than that. If something is an inverse, it
has to satisfy both of these. The composition of the inverse
with the function has to become the identity
matrix on x. Then the composition of the
function with the inverse has to be the identity
function on Y. Let's write that. So g is an inverse of f. It implies this. It also implies-- I'll do
it in yellow-- that the composition of f with
g is equal to the identity function on y. Then if we do it with h, the
fact that h is an inverse of f implies that the composition
of f with h is equal to the identity function
on y as well. Let me redraw what I drew in the
beginning just so we know what we're doing. So if this is a set X right
here-- let me do it in a different color-- let's say this
right here is the set Y. We know that f is a
mapping X to Y. What we're trying to determine
is is f's inverse unique. So any inverse, so we're saying
that g is a situation that if you take the composition
of g with f, you get the identity matrix. So f does that. If you could take g
you're going to go back to the same point. So it's equivalent. So taking the composition of g
with f-- that means doing f first then g-- this is the
equivalent of just taking the identity function in X, so just
taking an X and going back to an X. It's equivalent to that. So this is g right here. The same thing is true with h. h should also be. If I start with some element in
X and go into Y, and then apply h, it should also be
equivalent to the identity transformation. That's what this statement and
this statement are saying. Now this statement is saying
that if I start with some entry in Y here and I apply g,
which is the inverse of f, I'm going to go here. So g will take me there. When I apply f then to that, I'm
going to go back to that same element of Y. That's equivalent to
just doing the identity function on Y. That's the same thing as the
identity function of Y. I could do the same
thing here with h. I just take a point here, apply
h, then apply f back. I should just go back
to that point. That's all of what
this is saying. So let's go back to
the question of whether g is unique. Can we have two different
inverse functions g and h? So let's start with g. Remember g is just a mapping
from Y to X. So this is going to be equal to,
this is the same thing as the composition of the identity function over x with g. To show you why that's the case,
remember g just goes from-- these diagrams get me
confused very quickly-- so let's say this is
x and this is y. Remember g is a mapping
from y to x. So g will take us there. There's a mapping from y to x. I'm saying that this g is
equivalent to the identity mapping, or the identity
function in composition with this. Because all this is saying is
you apply g, and then you apply the identity
mapping on x. So obviously you're going to get
to the exact same mapping or the exact same point. So these are equivalent. But what is another
way of writing the identity mapping on x? What's another way
of writing that? Well by definition, if
h is another inverse of f, this is true. So I can replace this in this
expression with a composition of h with f. So this is going to be equal to
the composition of h with f, and the composition
of that with g. You might want to put
parentheses here. I'll do it very lightly. You might want to put
parentheses there. But I showed you a couple
of videos ago that the composition of functions, or
of transformations, is associative. It doesn't matter if you put
the parentheses there or if you put the parentheses there. Actually I'll do that. I'll put the parentheses there
at first just so you can understand that this
is the same thing as that right there. But we know that composition
is associative. So this is equal to the
composition of h with the composition of f and g. Now what is this equal to, the
composition of f and g? Well it's equal to, by
definition, it's equal to the identity transformation
over y. So this is equal to h composed
with, or the composition of h with, the identity function over
y with this right here. Now what is this going to be? Remember h is a mapping
from y to x. Let me redraw it. So that's my x and
that is my y. h could take some element
in y and gives me some element in x. If I take the composition of the
identity in y-- so that's essentially I take some element,
let me do it this way-- I take some element in
y, I apply the identity function, which essentially
just gives me that element again, and then I
apply h to that. That's the same thing as just
applying h to the function to begin with. So just going through this
little exercise we've shown, even though we started off
saying I have these two different inverses, we've
just shown that g must be equal to h. So any function has
a unique inverse. You can't set up two
different inverses. If you do you'll find that
they're always going to be equal to each other. So far we know what
an inverse is. We don't know what causes
someone to be able to have an inverse or not, but we know if
they have an inverse, how to think about it. We also know that that
inverse is unique.