Determining if a function is invertible | Mathematics III | High School Math | Khan Academy

- [Voiceover] So if x is equal to a

then, so if we input a into our function

then we output -6.

f of a is -6.

We input b we get three,
we input c we get -6,

we input d we get two,
we input e we get -6.

Alright, so let's see
what's going on over here.

Let me scroll down a little bit more.

So in this purple oval, this
is representing the domain

of our function f and this is the range.

So the function is going to,

if you give it a member of the domain

it's going to map from
that member of domain

to a member of the range.

So, for example, you
input a into the function

it goes to -6.

So a goes to -6, so I drag
that right over there.

b goes to three,

b goes to three.

c goes to -6, so it's already interesting

that we have multiple
values that point to -6.

So this is okay for f to be a function

but we'll see it might
make it a little bit tricky

for f to be invertible.

So let's see, d is points
to two, or maps to two.

So you input d into our
function you're gonna output two

and then finally e maps to -6 as well.

e maps to -6 as well.

So, that's a visualization
of how this function f maps

from a through e to members of the range

but also ask ourselves is
this function invertible?

And I already hinted at it a little bit.

Well in order for it to
be invertible you need a,

you need a function that could take

go from each of these points to,

they can do the inverse mapping.

But it has to be a function.

So, if you input three
into this inverse function

it should give you b.

If you input two into
this inverse function

it should output d.

If you input -6 into
this inverse function,

well this hypothetical inverse function.

what should it do?

Well you can't have a function
that if you input one,

if you input a number it could
have three possible values,

a, c, or e, you can only map to one value.

So there isn't, you actually can't set up

an inverse function that does this

'cause it wouldn't be a function.

You can't go from input -6
into that inverse function

and get three different values.

So this is not invertible.

Let's do another example.

So here, so this is the same drill.

We have our members of our
domain, members of our range.

We can build our mapping diagram.

a maps to -36,

b maps to nine,

c maps to -4,

d maps to 49,

and then finally e maps to 25.

e maps to 25.

Now is this function invertible?

Well let's think about it.

The inverse, woops, the,
was it d maps to 49.

So, let's think about what the inverse,

this hypothetical inverse
function would have to do.

It would have to take each
of these members of the range

and do the inverse mapping.

So if you input 49 into
our inverse function

it should give you d.

Input 25 it should give you e.

Input nine it gives you b.

You input -4 it inputs c.

You input -36 it gives you a.

So you could easily construct
an inverse function here.

So this is very much, this
is very much invertible.

One way to think about it is these are a,

this is a one to one mapping.

Each of the members of the domain

correspond to a unique
member of the range.

You don't have two members of the domain

pointing to the same member of the range.

Anyway, hopefully you
found that interesting.