WEBVTT

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

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

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of X equals Y.

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

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

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

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that F acts on.

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Here's a simple
example of how

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to workout an
inverse function.

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Let's have F of

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X. Equals 4X.

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Now F of X takes
number and multiplies it before.

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

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

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So F inverse of X.

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Equals 1/4 of
X.

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

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

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

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though the notation makes it
look like they should be.

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Because whenever up
of X is one over 4X.

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

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denominator, and there the X is
on the numerator.

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Let's workout
another inverse

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function.

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This time will have F of X.

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Equals. 3X.
+2.

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

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We multiplied by three.

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To get 3X.

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Then we add onto.

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

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give us back X.

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To workout if inverse we
need to undo every

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operation that ever did.

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So if we started with X.

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

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take away too. That gives
us X minus 2.

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

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divide by three. And that
gives us X minus

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two, all divided by

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

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divided by three.

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

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

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Being 7 minus X cubed.

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

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Would love F of X written as
minus X cubed?

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

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were doing 2X. We start with X.

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We cubit.

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That gives us X cubed.

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

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And then
finally we

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add on 7.

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So again to workout
F inverse we start

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with X. And we undo all the

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

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

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

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And we started off by cubing.

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So this time we take cube roots.

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That gives us the cube
root of 7 minus X.

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So this time F inverse of X.

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Is cube roots?

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7 minus X.

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Now we can also
use algebraic manipulation to

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workout in versus.

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

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was. F of X.

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Equals 3X.

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

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it back to X.

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

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We want to F inverse.

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To take Y and give us back X.

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

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

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I can rearrange this.

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And I get why minus 2?

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

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divided by three.

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

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and divided by three?

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This means that if inverse of Y.

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Equals Y minus two, all divided

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

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X. Is X minus two all divided by

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three? So this is how we use
algebraic manipulation to

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workout in versus.

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Now we can use this
method to workout some slightly

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more difficult in versus 2.

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

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

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

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for X greater than one.

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

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

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

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

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times X minus one.

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

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

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get YX minus Y equals X.

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

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us. Wax. Minus
X equals Y.

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

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

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

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Of Why? Is Y
over Y minus one?

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

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

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to be exactly the same as F.

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This example shows how useful
algebraic manipulation is

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because it would have been
really, really difficult to try

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

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Not all
functions have

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

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

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can. Now
when

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we
look

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for
an

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inverse

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

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And send it back to X.

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But in this case.

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

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

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also X squared.

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Now we can't define an inverse

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for a function. If there
are two things we could

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define each value to be.

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

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function we look at.

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

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do is we cut down the graph.

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We say F of

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X. Equals X squared.

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

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And that gives us a graph
like this.

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Now, since we've cut out all the

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values this side.
Each value of FX.

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Only comes
from One X.

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So in this case we can
define F inverse.

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And F inverse of X.

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Is plus square root of X.

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

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

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

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

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graph like this.

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

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There's only one value.

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

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of X defines.

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And it's equal to minus root X.

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

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be able to define an inverse.

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Would have F of X.

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Equals sign X.

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And this graph.

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Looks a bit like this. I'll try

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

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thing and carries on forever.

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Then repeats itself down this
way as well forever.

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

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

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

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

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to define an inverse.

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

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than or equal to minus 90
degrees and less than or

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equal to plus 90 degrees.

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And if I block out the rest of

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the function. Hopefully.

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You'll be able to see that in

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

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F of X. So for this.

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We restrict the domain.

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2X is greater than or equal to

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minus 90. Less than or equal
2 + 90.

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And the inverse.

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Of Cynex Is called arc
sine X.

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Now this notation can
be particularly confusing

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

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over sine X. But
you'll often see

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things like sine
squared of X.

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Which means cynex all squared.

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Just remember that sign.

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Minus one of X is

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not. One over sine X.

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

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

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restricted for us to define

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inverses. But we cover
these more fully the trig

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functions video.

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There are some functions that
counts have inverses even if

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we do restrict the domains.

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

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something like say F of X equals

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4. The growth of this
looks like this.

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

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

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

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

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

00:15:57.606 --> 00:16:03.393
an inverse function for the
graph of a function.

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

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I can think of.

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So we have X here.

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That's Why.

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

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so something like that.

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Notice that.

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A point on this graph.

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Has coordinates X.

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F of X.

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

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We want the coordinates on
the graph of F inverse to

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

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Now for turnover this
transparency so that my old X

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

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

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X here. And X here.

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So these points.

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Must form the graph of F
inverse.

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

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down that line.

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The sort of bottom left, top
right diagonal reflecting

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down here to get from one
graph to the other.

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And this line is precisely the
line Y equals X.

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So that must mean.

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But the graph of F inverse.

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