the function f of x and it's equal to two x plus five,

over four minus three x.

And what we wanna do is figure out what

is the inverse of our function.

Pause this video and try to figure

that out before we work on that together.

All right, now let's work on it together.

Just as a reminder of what a function

and an inverse even does,

if this is the domain of a function

and that's the set of all values that you could input

into the function for x and get a valid output.

And so let's say you have an x here,

it's a member of the domain.

And if I were to apply the function to it,

or if I were to input that x into that function.

Then the function is going to output a value

in the range of the function and we call that value f of x.

Now an inverse, that goes the other way.

If you were to input the f of x value into the function

that's going to take us back to x.

So that's exactly what f inverse does.

Now how do we actually figure out the inverse

of a function especially a function

that's defined with a rational expression like this.

Well the way that I think about it is,

let's say that y is equal to our function of x

or y is a function of x so we could say

that y is equal to two x plus five,

over four minus three x.

For our inverse the relationship

between x and y is going to be swapped.

And so in our inverse it's going

to be true that x is going to be equal to two y plus five,

over four minus three y.

And then to be able to express this

as a function of x, to say that what is y as a function

of x for our inverse we now have to solve for y.

So it's just a little bit of algebra here.

So let's see if we can do that.

So the first thing that I would do

is multiply both sides of this equation by four minus 3 y.

If we do that, on the left hand side we are going

to get x times each of these terms.

So we're going to get four x minus three yx

and then that's going to be equal to

on the right hand side, since we multiplied

by the denominator here we're just going

to be left with the numerator.

It's going to be equal to two y plus five.

And this could be a little bit intimidating

'cause we're seeing xs and ys, what are we trying to do,

remember we're trying to solve for y.

So let's gather all the y terms on one side

and all the non-y terms on the other side.

So let's get rid of this two y here.

Actually, well I could go either way.

Let's get rid of this two y here,

so let's subtract two y from both sides.

And let's get rid of this four x from the left hand side,

so let's subtract four x from both sides.

And then what're we going to be left with.

On the left hand side we're left with minus

or negative five, or actually it would be this way,

it would be negative three yx minus two y.

And you might say hey where is this going,

but I'll show you in a second,

is equal to, those cancel out

and we're gonna have five minus four x.

Now once again we are trying to solve for y.

So let's factor out a y here,

and then we are going to have y,

times negative three x minus two

is equal to five minus four x.

And now this is the homestretch.

We can just divide both sides of this equation

by negative three x minus two

and we're going to get y is equal

to five minus four x, over negative three x minus two.

Now another way that you could express this

is you could multiply both the numerator

and the denominator by negative one,

that won't change the value.

And then you would get, you would get

in the numerator four x minus five,

and in the denominator you would get a three x plus two.

So there you have it.

Our f inverse as a function of x,

which we could say is equal to this y

is equal to this right over there.