Let's look at Inverse Functions. [SOUND] For example, let's find the inverse of

this 1 to 1 funtion f and then, we'll state it's domain and range.

The first thing we do, when we're finding an inverse of a function is, in the

definition of that function, we replace f(x) by y.

In other words, we have y = x + 3 / x - 4.

And then, we interchange the rules of x and y.

In other words, wherever we see a y we put an x and wherever we see an x we put

a y, which gives us x is equal to y plus 3 divided by y-4.

And now we want to manipulate this equation in order to solve for y.

When we do that and solve for y, y will be equal to f inverse of f.

And we know we're going to be able to do that, because we're given that F is 1 to

1, and therefore the inverse will exist. So let's begin by multiplying both sides

of the equation by this denominator here, the Y minus 4,

which gives us X times Y minus 4 is = y + 3.

And now, let's distribute this x to both of these 2 terms,

which gives us, x * y - 4 x = y + 3. Now remember, we want to solve for y.

So let's bring all the terms with y in them to one side and everything else to

the other. And this will give us that x*y-y=3+4x.

And now on the left hand side we can factor out a y.

And if we divide both sides of the equation by this x-1 we will have solved

for y, so y = 3 + 4x / x-1.

And this is f inverse, so f inverse of x = 3 + 4x /x - 1.

So here is our inverse, however, we also are asked to find it's domain and range.

So let's start with the domain. Looking here at f inverse, the only issue

would be if x were equal to 1, because then we'd be dividing by 0.

So we need to exclude this value. That is the domain of f inverse, written

in interval notation is equal to negative infinity up to 1 open parenthesis,

because we want to exclude 1, union again open parenthesis at 1, cause

we want to exclude it, up to infinity.

All right, what about the range, though? Now this is where we can use the fact,

that the range of the inverse, is the domain of the original function.

That is, this is equal to the domain of f.

But what is this domain? Looking back up here at our original function, the only

issue would be, if X were equal to 4, because then we'd be dividing by 0.

That is we need to exclude this value in the domain of F.

That is the domain of F, written in interval notation.

It's the interval from negative infinity up to 4, open parentheses, because we

want to exclude 4. Union, open parentheses again at 4,

because we want to exclude 4, up to infinity.

Which is also the range of the inverse. That is this is equal to negative

infinity up to 4, union 4 to infinity. Now this is very useful to be able to

look at the domain of the original function to determine the range of the

inverse, and this is how we work with inverse functions.

Thank you, and we'll see you next time.

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