Such that this inverse function, the composition of the inverse with the function, is equal to the identity function.

And the inverse and the function in the composition of the function, with the inverse function, should be the identity on y.

The function is its own inverse.

And this is the inverse function.

So any function has a unique inverse.

Only one inverse function which is true.

The inverse function, if you take f inverse of 4, f inverse of 4 is equal to 0.

What does the inverse function look like, as a function of x?

Because this is the only possible inverse function.

And so we now have our inverse function as a function of x.

So given that f is invertible, we know that there is this f inverse function, and I can apply that f inverse function.

So let's apply our f inverse function to this.

For the inverse function, 0 gets mapped to 4.

In the first inverse function video, I talked about how a function and their inverse they are the reflection over the line y equals x.

And that the composition of the function with the inverse function is equal to the identity function on y.

Or the inverse function is mapping us from 4 to 0.

Then the composition of the function with the inverse has to be the identity function on Y.

That if f is invertible, it only has one unique inverse function.

We could say f inverse as a function of y so we can have 10 or 8 so now the range is now the domain for f inverse. f inverse as a function of y is equal to 1 2y minus 2.

We use the definition of invertibility that there exists this inverse function right there.

So this function, f inverse, is going to be a mapping from Y to X.

Solve for x in terms of y, and that's essentially your inverse function as a function of y, but then you can rename it as a function of x.

The composition of the inverse with the function has to become the identity matrix on x.

The definition of this inverse function is that when you take the composition with f, you're going to end up with the identity function.

We're saying there has be some other function, f inverse, that's a mapping from Y to X.

Is there some other function, we can call that the inverse of f, that'll take us back?

So, if I take the inverse function on both sides of this equation, where some element over here in y, and I'm taking the inverse function to get to some element in x, what's this going to be equal to?

So this is the inverse function right here, and we've written it as a function of y, but we can just rename the y as x so it's a function of x.

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.

I'll do it in the same color this is equal to f inverse as a function of y.

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.

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.

That the function takes you from 2 to 8, the inverse will take us back from 8 to 2.

And one thing I want to point out is what happens when you graph the function and the inverse.

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.

The second statement is saying look, if I apply f to f inverse, I'm getting the identity function on Y.

And that unique solution, if you really care about it, is going to be the inverse function applied to y.

Let's think about what functions really do, and then we'll think about the idea of an inverse of a function.

And C inverse is obviously its inverse.

That if a function's Laplace Transform, if I take a function against the Laplace Transform, and then if I were take the inverse Laplace Transform, the only function whose Laplace Transform that that is, is that original function.

And you could see, you have the function and its inverse, they're reflected about the line y is equal to x.

Inverse

Inverse

Inverse

A is also the inverse of A inverse.