www.mathcentre.ac.uk/.../Inverse%20functions.mp4

In this video, we'll define the
inverse, the function F.

Let's suppose we have a function
F that sends XY. So I mean F

of X equals Y.

The inverse of F is denoted F to
the minus one.

And it's the function that sends
why back to X?

For this to be an inverse,
it needs to work for every X

that F acts on.

Here's a simple
example of how

to workout an
inverse function.

Let's have F of

X. Equals 4X.

Now F of X takes
number and multiplies it before.

And we want F inverse to take 4X
back to X.

Now, this must mean that F
inverse of X must divide by 4.

So F inverse of X.

Equals 1/4 of
X.

Now we can see from this that
since F inverse of X is 1/4

X and one over F of X
is one over 4X.

That F inverse and one over F of
X and not the same thing, even

though the notation makes it
look like they should be.

Because whenever up
of X is one over 4X.

But that's not equal to 1/4 X,
because here The X is on the

denominator, and there the X is
on the numerator.

Let's workout
another inverse

function.

This time will have F of X.

Equals. 3X.
+2.

Now what does F of X do we
start with X?

We multiplied by three.

To get 3X.

Then we add onto.

Now, since we want F inverse
to take F of X and

give us back X.

To workout if inverse we
need to undo every

operation that ever did.

So if we started with X.

First, we undo the last
operation that F did, so we

take away too. That gives
us X minus 2.

And before that, we'd multiplied
by three. So to undo that, we

divide by three. And that
gives us X minus

two, all divided by

three. So F inverse of X in
this case is X minus two, all

divided by three.

Here's one more example of how
we can undo the operations of F

to workout its inverse. Let's
have F of X.

Being 7 minus X cubed.

And I'll rewrite the slightly
to make it easier to workout.

Would love F of X written as
minus X cubed?

Plus Seven, so now it's a bit
easier to see which operations

were doing 2X. We start with X.

We cubit.

That gives us X cubed.

Then we send X cubed minus X
Cube, so we times Y minus one.

And then
finally we

add on 7.

So again to workout
F inverse we start

with X. And we undo all the

operations of ETH. So we start
by taking off 7 to undo the plus

7 bit. And then we
multiplied by minus one. So now

we divide by minus one. That
gives us 7 minus X.

And we started off by cubing.

So this time we take cube roots.

That gives us the cube
root of 7 minus X.

So this time F inverse of X.

Is cube roots?

7 minus X.

Now we can also
use algebraic manipulation to

workout in versus.

I'll show you how to do this
with our second example, which

was. F of X.

Equals 3X.

+2. Now remember that
we want to take F of X and send

it back to X.

So if I set F of X
equals Y equals 3 X +2.

We want to F inverse.

To take Y and give us back X.

So we need to workout how to get
2X from Y?

So if I write down again, Y
equals 3 X +2.

I can rearrange this.

And I get why minus 2?

Equals 3X. So X
equals Y minus two, all

divided by three.

So to get from Y and go to X,
you need to take why take off 2

and divided by three?

This means that if inverse of Y.

Equals Y minus two, all divided

by three. But that's just the
same as saying that F inverse of

X. Is X minus two all divided by

three? So this is how we use
algebraic manipulation to

workout in versus.

Now we can use this
method to workout some slightly

more difficult in versus 2.

This time will have F of X
being X over X minus one.

And we have to set X is greater
than one because otherwise

you'll get a zero denominators.
So we only look at this function

for X greater than one.

That work at the inverse again
will have Y equals X over X

minus one. Remember again we
need to get to X from Y, so we

need to write X in terms of Y,
so will rearrange this.

Or multiply both sides by X
minus one and that gives us why

times X minus one.

Equals X. And we want and
X equals out of this with all

the access on one side, so I'll
multiply it. These brackets you

get YX minus Y equals X.

Then I'll take all the ex is
over to one side, so that gives

us. Wax. Minus
X equals Y.

Factor out the X you get X
times Y minus one equals Y

and that gives you X
equals Y over Y minus one.

So here we have X in terms of Y
again so F inverse.

Of Why? Is Y
over Y minus one?

And that's just the same as
saying F inverse of X equals X

over X minus one. So in this
case the inverse of F turns out

to be exactly the same as F.

This example shows how useful
algebraic manipulation is

because it would have been
really, really difficult to try

and get this inverse by just
reversing the operations of F.

Not all
functions have

straightforward inverses.
Let's look at F of X.

Equals X squared and I'll just
quickly sketch you a graph if I

can. Now
when

we
look

for
an

inverse

function. We want to
take the value of F of X.

And send it back to X.

But in this case.

There are two X is we could send
F of X back to.

That's because F of X is X
squared, but F of minus X is

also X squared.

Now we can't define an inverse

for a function. If there
are two things we could

define each value to be.

To get around this problem, we
restrict how much of the

function we look at.

So instead of defining F of X
like this for every X, what we

do is we cut down the graph.

We say F of

X. Equals X squared.

But only look at X greater than
or equal to 0.

And that gives us a graph
like this.

Now, since we've cut out all the

values this side.
Each value of FX.

Only comes
from One X.

So in this case we can
define F inverse.

And F inverse of X.

Is plus square root of X.

Now we didn't have to define F
for X greater than or equal to

0. We could have cut out the
other half of the graph instead,

so I could have defined F of X
equals X squared.

For X less than or equal to 0
and that would have given us a

graph like this.

Now again, if a pack of value of
F of X.

There's only one value.

Of ex. That gives us the F
of X, so in this case F inverse

of X defines.

And it's equal to minus root X.

Here's another function where we
need to restrict the domain to

be able to define an inverse.

Would have F of X.

Equals sign X.

And this graph.

Looks a bit like this. I'll try

and sketch it. So for X greater
than zero, does this kind of

thing and carries on forever.

Then repeats itself down this
way as well forever.

Now, if a pick a value of
F of X here.

You can see there's certainly
more than One X giving this F of

X. In fact, there will
be an infinite number.

So we'll certainly need to cut
down the domain of this function

to define an inverse.

What we do in this case is
we look at X for X greater

than or equal to minus 90
degrees and less than or

equal to plus 90 degrees.

And if I block out the rest of

the function. Hopefully.

You'll be able to see that in

this case. For every F of X
there's only one ex giving that

F of X. So for this.

We restrict the domain.

2X is greater than or equal to

minus 90. Less than or equal
2 + 90.

And the inverse.

Of Cynex Is called arc
sine X.

Now this notation can
be particularly confusing

because. F inverse of X. Like I
said before, is not equal to one

over sine X. But
you'll often see

things like sine
squared of X.

Which means cynex all squared.

Just remember that sign.

Minus one of X is

not. One over sine X.

But the inverse function sign X,
which is also called AC sine X.

The functions, calls, and turn
also need the domains to be

restricted for us to define

inverses. But we cover
these more fully the trig

functions video.

There are some functions that
counts have inverses even if

we do restrict the domains.

A good example of this is a
constant function and that is

something like say F of X equals

4. The growth of this
looks like this.

Now you can see here that the
only way we could get One X for

each F of X here is to cut down
the domain to a single point,

and this isn't a very useful
thing to do.

So in this case, we say that
this function has no inverse.

Now there is a noise easy
way of getting the graph of

an inverse function for the
graph of a function.

If I just get you the graph
of some random function that

I can think of.

So we have X here.

That's Why.

And I'll just draw some
function that's appropriate,

so something like that.

Notice that.

A point on this graph.

Has coordinates X.

F of X.

Now since. F inverse sends F
of X 2X.

We want the coordinates on
the graph of F inverse to

be the coordinates F of XX.
So these two interchanged.

Now for turnover this
transparency so that my old X

axis was, well, my Y axis was
and vice versa. Do that for you.

You can see by doing this I've
got precisely that I've got F of

X here. And X here.

So these points.

Must form the graph of F
inverse.

So notice that in swapping these
axes all I did was I reflected

down that line.

The sort of bottom left, top
right diagonal reflecting

down here to get from one
graph to the other.

And this line is precisely the
line Y equals X.

So that must mean.

But the graph of F inverse.

Is the graph of F reflected in
the line Y equals X?