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### Inverse Functions

• A function does a transformation on an input. But what if there was some way to reverse that transformation? This is the idea of an inverse function: a way to reverse a transformation and get back to our original input.
• To help us understand this idea, imagine a factory where if you give them a pile of parts, they'll make you a car. Now imagine another factory just down the road, where if you give them that car, they'll give you back the original pile of parts you started with. There is one process, but there is also an inverse process that gets you back to where you started. If you follow one process with the other, nothing happens.
• It's important to note that not all functions have inverses. Some types of transformation cannot be undone. If the information about what we started with is permanently destroyed by the transformation, it cannot be reversed.
• A function has an inverse if the function is one-to-one: for any a, b in the domain of f where a ≠ b, then f(a) ≠ f(b). Different inputs produce different outputs.
• We can see this property in the graph of a function with the Horizontal Line Test. If a function is one-to-one, it is impossible to draw a horizontal line somewhere such that it will intersect the graph twice (or more).
• Given some function f that is one-to-one, there exists an inverse function, f−1, such that for all x in the domain of f,
 f−1 ⎛⎝ f(x) ⎞⎠ = x.
In other words, when f−1 operates on the output of f, it gives the original input that went into f. [Caution: f−1 means the inverse of f, not [1/f]. In general, f−1 ≠ [1/f].]
• We can figure out the domain and range for f−1 by looking at f. Since the set of all outputs is the range of f, and f−1 can take any output of f, the domain of f−1 is the range of f. Likewise, f−1 can output all possible inputs for f, so the range of f−1 is the domain of f.
• The inverse of f−1 is simply f. This makes intuitive sense: if you do the opposite of an opposite, you end up doing the original thing.
• Visually, f−1 is the mirror of f over y=x. This is because f−1 swaps the outputs and inputs from f, which is the same thing as swapping x and y by mirroring over y=x.
• To find the inverse to a function, we effectively need a way to "reverse" the function. This can be a little bit confusing at first, so here is a step-by-step guide for finding inverse functions.

1. Check function is one-to-one;    f(x) = x3+1
2. Swap f(x) for y;     y = x3+1
3. Interchange x and y;     x = y3+1
4. Solve for y;     y = 3√{x−1}
5. Replace y with f−1(x);     f−1(x) = 3√{x−1}
• While this method will produce the inverse if followed correctly, it is not perfect. Notice that in steps #2 and #3 above, the equations are completely different, yet they still use the same x and y. Technically, it is not possible for x and y to fulfill both of these equations at the same time. What's really happening is that when we swap in #3, we're actually creating a new, different y. The first one stands in for f(x), but the second stands in for f−1(x). This implicit difference between y's can be confusing, so be careful. Make a note on your paper where you swap x and y so you can see the switch to "inverse world".
• Taking inverses can be difficult: it's easy to make a mistake. This means it's important to check your work. By definition, f−1 ( f(x) ) = x. This means if you know what f−1(x) and f(x) are, you can just compose them! If it really is the inverse, you'll get x. Furthermore, since we know f( f−1 (x) ) = x as well, you can compose them in either order when checking.

### Inverse Functions

The table below shows how f(x) works. Is f(x) one-to-one?
 x
 f(x)
 apple
 a
 apricot
 a
 a
 banana
 b
 kiwi
 k
 kumquat
 k
 tomato
 t
• A function is one-to-one when different inputs always produce different outputs. In other words, there are no two inputs that produce the same output.
• For f(x), we see that there are cases when different inputs produce the same output. For example, f(apple) = f(apricot) = f(avocado) = a. Thus, because it is possible to have two distinctly different inputs result in the same output, the function is NOT one-to-one.
No, f(x) is not one-to-one.
g(x) = 3x. Is g(x) one-to-one?
• A function is one-to-one when different inputs always produce different outputs. In other words, there are no two inputs that produce the same output.
• For g(x) to be one-to-one, it must be that if we put in two different values for x, we will never get the same result.
• If we think about this carefully, we can realize that the only way to get the same output for g(x) is to start with the same x. For example, if x=3, we get g(3) = 9. But there is no other number we could possibly plug in other than x=3 to produce 9 as the output. [If we want to formally prove this, we can do it as follows: Let a and b be two numbers such that g(a) = g(b). Then 3a = 3b. Thus, by algebra, we have a=b. Therefore, if the output is the same, it must be that the inputs were the same, which proves that g(x) is one-to-one. You don't need to formally prove this, though: you can just think about how the function works.]
Yes, g(x) is one-to-one.
The graph of h(x) = |x−1| −3 is below. Use the Horizontal Line Test to determine if it is one-to-one.
• The Horizontal Line Test says that if a function is not one-to-one, you can draw a horizontal line somewhere on it that will intersect the graph twice (or more).
• There are many places a horizontal line can be drawn that will cut the graph twice. Thus, the graph is not one-to-one.
No, h(x) is not one-to-one.
The graph of f(x) = (x+2)3 −1 is below. Use the Horizontal Line Test to determine if it is one-to-one.
• The Horizontal Line Test says that if a function is not one-to-one, you can draw a horizontal line somewhere on it that will intersect the graph twice (or more).
• There is nowhere on the graph that a horizontal line can be drawn that will cut the graph twice. Thus, f(x) is one-to-one. [Near the center of the curve, it might look like f(x) is flat enough that you could cut it multiple times with a single horizontal line. This is not true, though. Notice that even when it looks fairly flat, it is still slightly sloped. It is not changing much, but it is still changing enough to pass the horizontal line test.]
Yes, f(x) is one-to-one.
What is f−1 ( f( 4) )? What is f ( f−1 (−27) )? What is f−1 ( f ( f−1 ( f(x)) ) )?
• f−1 is the inverse function of f: it cancels out whatever f does. In general, f−1 ( f( x) ) = x. By having the inverse operate on a function, it gets us back to where we started. Thus f−1 ( f( 4) ) = 4.
• The inverse of f−1 is f. Thus, just like f−1 cancels out f, f will cancel out f−1:  f ( f−1 (x) )=x. By having a function operate on its inverse, it gets us back to where we started. Thus f ( f−1 (−27) ) = −27.
• The above canceling can occur multiple times:
 f−1 ⎛⎝ f ⎛⎝ f−1 ⎛⎝ f(x) ⎞⎠ ⎞⎠ ⎞⎠ = f−1 ⎛⎝ f ⎛⎝ x ⎞⎠ ⎞⎠ = x
f−1 ( f( 4) ) = 4        f ( f−1 (−27) )=−27        f−1 ( f ( f−1 ( f(x)) ) ) = x
The function f(x) = 4x is one-to-one. Find the inverse f−1(x).
• We already know the function is one-to-one from the problem, so the next step is to replace f(x) with y:
 y = 4x
• Next, interchange the locations of x and y:
 x = 4y
• Solve for y from the new equation:
 1 4 x = y
• Finally, replace y with f−1(x):
 f−1(x) = 1 4 x
f−1(x) = [1/4] x
The function f(x) = [(x−3)/(x+3)] is one-to-one. Find the inverse f−1(x).
• We already know the function is one-to-one from the problem, so the next step is to replace f(x) with y:
 y = x−3 x+3
• Next, interchange the locations of x and y:
 x = y−3 y+3
• Solve for y from the new equation. This is kind of tricky: use the distributive property in reverse to pull out y once you have everything involving y on one side. xy + 3x = y−3   ⇒  xy−y = −3x −3   ⇒  y(x−1) = −3x−3
 y = 3x+3 1−x
• Finally, replace y with f−1(x):
 f−1(x) = 3x+3 1−x
f−1(x) = [(3x+3)/(1−x)]
Let f(x) = 2x+3 and g(x) = [(x−3)/2]. Show that f and g are inverse functions.
• Two functions f and f−1 are inverses when f−1( f(x) ) = x or, equivalently, when f( f−1(x) ) = x. This also means we can check to see if two functions are inverses by composing one with the other. Thus, if we can show f(g(x) ) = x or g ( f(x) ) = x, we have shown that they are inverses.
• This means we have two options. Let us show f(g(x) ) = x is true first:
 f ⎛⎝ g(x) ⎞⎠ = f ⎛⎝ x−3 2 ⎞⎠ = 2 ⎛⎝ x−3 2 ⎞⎠ +3
Simplifying the above, we find that f(g(x) ) = x, and therefore f and g are inverses.
• Alternatively, we can show g ( f(x) ) = x is true:
 g ⎛⎝ f(x) ⎞⎠ = g ⎛⎝ 2x+3 ⎞⎠ = (2x+3) −3 2
Simplifying the above, we find that g ( f(x) ) = x, and therefore f and g are inverses.
[The problem is answered by showing that f(g(x) ) = x or g ( f(x) ) = x. Look at the steps above to see how this is done if you are uncertain.]
Let f(x) = 2x3 −12 and g(x) = 3√{[1/2]x+6}. Show that f and g are inverse functions.
• Two functions f and f−1 are inverses when f−1( f(x) ) = x or, equivalently, when f( f−1(x) ) = x. This also means we can check to see if two functions are inverses by composing one with the other. Thus, if we can show f(g(x) ) = x or g ( f(x) ) = x, we have shown that they are inverses.
• This means we have two options. Let us show f(g(x) ) = x is true first:
f
g(x)
= f

3

 1 2 x+6

= 2

3

 1 2 x+6

3

−12
Simplifying the above, we find that f(g(x) ) = x, and therefore f and g are inverses.
• Alternatively, we can show g ( f(x) ) = x is true:
g
f(x)
= g
2x3 −12
=
3

 1 2 (2x3 −12)+6

Simplifying the above, we find that g ( f(x) ) = x, and therefore f and g are inverses.
[The problem is answered by showing that f(g(x) ) = x or g ( f(x) ) = x. Look at the steps above to see how this is done if you are uncertain.]
Let f(x) = √{x−5}. What are the domain and range of f? What is f−1(x)? What are the domain and range of f−1?
• The domain is the set of all values that the function can accept, while the range is the set of all values the function can possibly output.
• For f(x), it "breaks" when there is a negative in the square root, so the domain of f is x ≥ 5. The range of f is [0, ∞).
• To find f−1, we follow the same steps we did in previous problems. Working it through, we get f−1 (x) = x2 + 5. [Check it by plugging one function into the other if you're not sure.]
• At first glance, we might think the domain of f−1(x) is all numbers because x2 +5 never "breaks". However, we have to remember that f−1 is the inverse of f: it can only reverse values that could possibly come out of f. Thus, the domain of f−1 is the range of f: [0, ∞). Similarly, the range of f−1 is the domain of f: x ≥ 5 (Alternately, we can use the domain of f−1 to figure out what its range must be).
Domain of f: [5, ∞)    Range of f: [0, ∞)
f−1(x) = x2 +5
Domain of f−1: [0, ∞)    Range of f−1: [5, ∞)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Inverse Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:04
• Analogy by picture 1:10
• How to Denote the inverse
• What Comes out of the Inverse
• Requirement for Reversing 2:02
• The Importance of Information
• One-to-One 4:04
• Requirement for Reversibility
• When a Function has an Inverse
• One-to-One
• Not One-to-One
• Not a Function
• Horizontal Line Test 7:01
• How to the test Works
• One-to-One
• Not One-to-One
• Definition: Inverse Function 9:12
• Formal Definition
• Caution to Students
• Domain and Range 11:12
• Finding the Range of the Function Inverse
• Finding the Domain of the Function Inverse
• Inverse of an Inverse 13:09
• Its just x!
• Proof
• Graphical Interpretation 17:07
• Horizontal Line Test
• Graph of the Inverse
• Swapping Inputs and Outputs to Draw Inverses
• How to Find the Inverse 21:03
• What We Are Looking For
• Reversing the Function
• A Method to Find Inverses 22:33
• Check Function is One-to-One
• Swap f(x) for y
• Interchange x and y
• Solve for y
• Replace y with the inverse
• Keeping Step 2 and 3 Straight
• Switching to Inverse
• Checking Inverses 28:52
• How to Check an Inverse
• Quick Example of How to Check
• Example 1 31:48
• Example 2 34:56
• Example 3 39:29
• Example 4 46:19

### Transcription: Inverse Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about inverse functions.0002

A function does a transformation on an input; we have talked about functions for a while now.0005

But what if there was some way to reverse that transformation?0009

This is the idea behind an inverse function: it is a way to reverse a transformation, reverse the process that another function is doing.0012

It is a way to get back to our original input.0021

By way of analogy, let's imagine a factory where, if you give them a pile of parts, they will make you a car.0024

Now, if you take that car down the road to this other factory, you can give them that car,0029

and they will give you back the original pile of parts you started with.0034

There is one factory where they make cars out of parts, but then there is a second factory0038

where they take cars and break them down into the original parts that were used to make them.0042

There is one process, but there is also an inverse process that gets you back to where you started.0046

If you follow one process with the other immediately, it ends up as if you haven't done anything.0051

If you bring the pile of parts to the first factory, and then take that car to the second factory,0055

and they give you back the pile of parts, it is like you just started with a pile of parts and didn't do anything to it.0059

This is the idea behind an inverse function: it reverses a process--it reverses a transformation and gets you back to where you started.0063

We have used the analogy of a function as a machine before;0071

and it is a good image for being able to get across what is going on with inverse functions, as well.0073

So, a function machine takes inputs, and it transforms them to outputs by some rule.0079

So, what we are used to is: we plug in x into the function f, and it gives out f acting on x, f(x).0084

Now, we could plug it into another one; we could plug it into an inverse machine, an inverse to f;0092

and that would be called "f inverse," the f with the little -1 in the corner.0097

f-1 denotes the inverse of f; we call it "f inverse."0103

We plug f(x) into f-1, into that machine; we get right back to our original x.0108

It is as if we hadn't done anything; the first machine does something, but then the second machine0114

reverses that process and gets us back to where we started.0119

So, is there a requirement for reversing--can we make an inverse out of all functions out there?0123

No; let's see why by analogy first.0129

Imagine a factory where, if you give them a pile of wood or a pile of metal, they give you a basketball in return.0133

The basketball is the exact same, whether you started with wood or metal; it is always the exact same basketball.0139

It doesn't matter what you gave them; it is just a basketball.0145

Now, let's say you walk down the road to another factory; you give them that basketball;0148

you tell them to reverse the process; and then you walk away--you give them no other information.0152

Can that factory take a basketball and transform it back into a pile of wood or a pile of metal, if all they have is a basketball and no other information?0156

No, they have no idea what you started with.0164

Maybe they have wood; maybe they have metal; but the point is that they have no way0168

to be able to know which one they are supposed to give you at this point.0172

They don't have the information; the only person who has the information is you, when you brought the original wood pile,0175

or brought the original metal pile--because all they have is the basketball, and the basketball could indicate wood, or it could indicate metal.0180

They have no way to know what you started with; there is no way to figure it out.0188

The information about what you originally had has been destroyed (although you would know it, because you brought it to the factory).0191

But assuming you forgot, then the information has been destroyed--no one has the information anymore.0197

Another way to think about it would be if you took a piece of paper, and you burned that piece of paper.0202

You would be left with a pile of ashes.0206

Now, someone could come along and think, "Oh, a pile of ashes--it used to be a piece of paper."0208

But if you take two pieces of paper, and you write two completely different things on the two pieces of paper,0212

and then you burn the two of them, a person could come along and think, "Two piles of ashes..."0217

And they would know it was paper, but they wouldn't know what was written on them.0221

They wouldn't be able to get that information back; the information has been destroyed.0224

They know it was paper, but they don't know what was written on the paper.0227

The basketball one...you have given them wood; you have given them metal; you get the same thing.0231

The information about what you started with has been lost; the information has been destroyed,0235

unless you come along and also say, "Oh, by the way, that basketball came from ____."0239

The issue, in this scenario, is that we have two inputs providing the same output--whether it is metal or wood, you get a basketball.0245

So, if we try to have a reverse on that, we have no way to know which one to go back to.0252

We don't know if we are going to go back to metal; we don't know if we are going to go back to wood,0256

because we don't know what the basketball is representing.0259

So, to be a reversible process--for it to be possible to reverse something--the process has to have a different output for every input.0262

If you give them metal, they have to give you one color of basketball; and if you give them wood, they give you a totally different color of basketball.0271

Then, the second factory would say, "Oh, that is a wood basketball" or "Oh, that is a metal basketball."0277

And they would be able to know what to do at that point.0281

So, for a function f to have an inverse, it has to be that, for any a and b in the domain, any a or b that we could use in f normally,0284

where a is not the same thing as b (where a and b are distinct from each other--they are different),0291

then f(a) is different than f(b); f(a) does not equal f(b).0297

So, if a and b are different, then the function's outputs on a and b are different, as well.0302

So, different inputs going into a function have to produce different outputs; we call this property one-to-one.0308

If this function has a property where whatever you put in, as long as it is different from something else going in,0316

it means the two things will be different, that is called one-to-one: different inputs produce different outputs.0323

You give them metal; you get one color of basketball; you give them wood--you get a different color of basketball.0328

Here are some examples, so we can see this in a diagram.0334

Here is an example that is one-to-one: a goes to 2; b goes to 1; c goes to 3.0336

They each go to different things: different inputs each have different places that they end up going.0344

Something that is not one-to-one: a goes to 1 and b goes to 1.0350

It doesn't matter that c goes to 3, because a and b have both gone to the same thing, so they have different inputs that are producing the exact same output.0354

a and b are different things, but they both produce a 1; so it is not one-to-one.0365

We have that copy; we are putting in wood, and putting in metal, and we get basketball in both cases.0369

And then, finally, just to remind us: this one over here (hopefully you remember this) is not a function.0373

And it is not a function, because b is able to produce two outputs at once; and that is something that is not allowed for a function.0378

If a function takes in one input, it is only allowed to produce one output; it can't produce multiple outputs from a single input.0384

So, why do we call it one-to-one--why are we using this word, "one-to-one"?0391

Well, we can think of it as being because a has one partner, and b has one partner, and c has one partner.0395

Everybody gets a partner, and nobody has shared partners; everybody gets their partner, and that partner is theirs.0402

They don't have to share it with anybody else.0408

It is one-to-one: this thing is matched to this thing, and there is nobody else who is going to match up to that one: one-to-one.0410

All right, how can we test for this?0417

One way to test for this, to test if a function is one-to-one: we know, if we are going to be one-to-one,0420

that every input must have a unique output; that was what it meant to be one-to-one.0425

If we have different inputs, we have different outputs.0429

So, if we draw a horizontal line on the graph, it can intersect the graph only once, or not at all.0432

Remember, if we have some picture on a graph--like if we have this point--then what that means0438

is that this is the input, and this over here is the output.0443

We make it a point: (input,output); that is why it is (x,f(x)).0449

If it is f(x) = x2, then we plug in 3, and we get (3,9), (3,32).0458

The input is our horizontal location; the output is our vertical location.0465

The horizontal line test is a way to test if the function has the same output for multiple inputs.0469

We draw a horizontal line across, because wherever an output hits the graph, we know that there must be an input directly below it.0477

If a function is not one-to-one, you will be able to draw a horizontal line that will intersect it twice, or maybe even more.0485

Let's look at some examples: first, here is one that is one-to-one, because whenever we draw0490

any sort of horizontal line, it is only going to cut it once.0496

The only place that might seem a little confusing is if we draw it near here.0499

It might make you think, "Well, doesn't it look like those are stacked?"0503

Well, remember, we can't draw perfectly what is being represented by the mathematics.0506

We have to give our lines some thickness; in reality, the line is infinitely thin.0511

So, while it looks like they are kind of getting stacked, it is actually still moving through that zone; it is not constant.0517

It is increasing just a little bit, but it isn't constant.0522

Let's look at one that is not one-to-one: over here, this horizontal line (or many horizontal lines that we could make)--0525

it cuts it in two places; so we know that, here and here, there are two inputs.0532

We can produce the same output from two different locations.0537

We have two inputs making one output; so that means we are not one-to-one,0541

because this one is partner to that height, but this one is also partner to this height.0545

So, we are not one-to-one, because we have to share an output.0550

Now that we have all of these ideas in mind, we are finally ready to define an inverse function; we can really talk about them and sink our teeth into them.0554

Given some function f that is one-to-one (it has to be one-to-one for this to happen),0561

there exists and inverse function, which we denote as f-1 such that, for all x in the domain of f,0566

any x that could normally go into f, any value that could normally be input,0574

f-1 acting on f(x) becomes just x.0578

So, we have f acting on x like normal; and then, f-1 acts on that whole thing.0583

And it breaks the action that was done by f and returns us back to our original input.0591

In other words, when f-1 operates on the output of f, it gives the original input that went into f.0596

Caution: I want to warn you about something: f-1 means the inverse of f, not 1/f.0604

This can be confusing, because, if you have taken algebra and remember your exponents0612

(you might have forgotten them, but we will talk about them later in this course), -1 can mean a reciprocal for numbers.0617

So, 7-1 becomes 1/7; x-1 becomes 1/x.0622

But f is not "to the -1"; it is just a symbol that says inverse--"This is a function inverse."0630

So, in general, for the most part, f-1, "f inverse," is not the same thing as 1/f.0636

The inverse of f is normally not the reciprocal of f, 1/f.0643

This exponent thing, where 7-1 is 1/7, is not the case for functions.0649

On a function, the -1 does not represent an exponent; it is not an exponent.0656

But it instead tells us function inverse; it is a way of saying, "This is an inverse function"; that is what it is telling us, not "flip it to the reciprocal."0661

How can we get domain and range for f-1?0673

We can figure these out by looking at f; remember, the set of all outputs from f is its range.0675

The things that x can get mapped to by f, what f is able to map x to, is the range of f.0687

The domain of f is everything that x can be, everything that we can plug into f.0695

And then, the range of f is everything that can come out of f.0701

Now, f-1 has to be able to take any output of f.0704

It is not very good at reversing if there are some numbers that it is not allowed to reverse.0709

So, it has to be able to reverse anything from f.0713

If it is able to reverse anything from f, then that means the range of f has to be everything that we can put into f-1.0716

So, the domain of f-1, the things we can put into f-1, is the range of f.0723

The domain of f-1 is the range of f.0728

Likewise, because f-1 then breaks that f(x) and turns it back into original inputs,0732

we must be able to turn it back into all of the original inputs, because all of the original outputs have to be over here.0738

So, anything that we can make it to, we have to be able to make it back from.0745

So, since we are able to get back everywhere, that means that we can output all the possible inputs for f.0748

Since we can output all of the possible inputs, because we can reverse any of the processes,0755

then it must mean that we are able to get the range of f-1 from the domain of f.0760

So, the domain of f tells us the range of f-1, what we are allowed to output with f-1.0767

And the domain of f-1 tells us the range of f.0772

So, the domain of f-1 is the things that f is able to output; and the range of f-1,0776

the things that f-1 is able to output, is what we can put into f, the domain of f.0784

This idea is going to let us prove something later on.0790

Now we can get to that proof--the inverse of an inverse: what is the inverse of an inverse?0793

In symbols, what is (f-1)-1--what do we do if we are going to take the inverse of something that is already doing inverses?0797

Now, it might seem a little surprising, but it turns out that the inverse of f-1 is just f.0806

The inverse of f-1 is f; it seems a little surprising, maybe, but it makes a sort of intuitive sense.0812

If you do the opposite of an opposite, you get to the original thing.0820

If you do an action, but then you are going to do the opposite of that action,0824

but then you do the opposite of the opposite of the action, then you must be back at your original action.0828

So, we might be able to believe this on an intuitive level; it makes sense, intuitively, that two opposites gets us back to where we started.0833

But let's see a proof of this fact, formally; let's see it in formal mathematics.0840

So, how do we get this started? Well, by definition, f-1 is the function where,0845

for any x, f-1 acting on f(x) is going to just give us our original x.0849

If f acts on an input, and then f-1 comes and acts on that, we get back to our original input.0856

Now, consider (f-1)-1: by this definition of the way inverses work, it must be that f inverse, inverse,0862

when it acts on the thing that it is an inverse of...f inverse, inverse, is an inverse of f inverse...0869

I know it is complicated to say...but this one right here is going to be the opposite action of f-1.0874

So, if we take any y (don't get too worried about x and y; remember, they are just placeholders for inputs),0881

similarly, for any y, (f-1)-1, acting on f-1(y),0887

is going to just get us right back to our original y.0892

It is the same structure as what is going on here with f-1(f(x)) = x: we are just reversing a process.0895

So, it doesn't matter that one of the processes is already a reversed process, because we are reversing this other reversed process.0901

So, we get back to our original thing.0907

Now, we know that y is in the domain of f-1, because we are allowed to put it into f-1.0910

It is allowed to go into f-1; now, we know, from our thing that we were just talking about,0917

that the domain of f-1 is the range of f; so there has to be...0922

If f-1 is the range of f (the domain of f-1 is the range of f),0930

if you are in the range, then that means that there is something out there that can produce this.0940

That means that, if you are in the range of f, there must be some x in the domain of f;0944

there has to be some way to get to that place in the range, so that f(x) is equal to y.0949

There is some x out there in the domain of our original f, that f(x) is equal to y.0954

So now, we have what we need: we can use this f(x) = y, and we can just plug it in right here and here.0960

We will plug it in for the two y's up here, and we will see what happens.0967

Thus, f inverse, inverse, acting on f-1(f(x))...0970

because remember, we know that there has to be some way to get an x such that f(x) = y,0974

because of this business about domain and range; so we plug that in here, and we plug that in here.0979

And we have that f inverse, inverse, on f inverse of f of x, must be equal to this over here on the right, as well.0985

We are just doing substitution.0993

But we know, by the definition of f-1(f(x)), that this just turns out to be x.0995

This whole thing right here just comes down to x--it simplifies right out to x.1002

So, it must be the case that f inverse, inverse, of x is the same thing as f(x).1007

If f(x) is the exact same thing as f inverse, inverse of x, it must be the case that (f-1)-1 is just the same thing as f.1012

And our proof is finished; great.1023

How can we interpret this graphically?--there is a great way to interpret inverses through graphs.1028

First, let's consider f(x) = x3 + 1.1032

Now, we know that this one has to be one-to-one, because it passes the horizontal line test.1035

We come along and try to cut this with any horizontal line; they are only going to be able to cut in one place.1040

Even here, where we have sort of seemed to flatten out, it is still moving, because we know it is x3 + 1.1045

And it never actually stops going up; it just slows down how fast it is going up.1051

And our lines have to have thickness, so while it kind of looks like they are stacked, they are not really.1055

So, we see that it passed the horizontal line test; so it must be one-to-one.1059

If it is one-to-one, we know it has to have an inverse; that is how we talked about this, right from the beginning.1064

Now, notice that the graph, any graph, is made up of the points (x,f(x)).1069

We talked about this before: 0 gets mapped to 1 when we plug it in as f(0) = 1; so that gives us the point (0,1).1073

That is how we make up our original graph for f.1082

Now, the graph of f-1 will swap these coordinates.1084

It takes in outputs and gives out inputs, in a way; so its input will swap these two things.1088

It takes in f(x), and it gives out x; so the points of f-1 will be the reverse of what we had for the points of the other one.1096

So, (f(x),x) is what we get for f-1.1105

Now, visually, what that means is that f-1 is going to be the mirror of y = x; and that is our line right here, y = x.1109

Why is that the case? Well, look: we swap x and y coordinates if we go across this,1117

because (-3,0) swaps to (0,-3); if we are going over y = x, if we are mirroring across this, we will swap the locations.1122

And so, if we mirror over y = x, we are going to swap x and y, y and x; we will swap the order of our points,1138

because y = x is sort of a way of saying, "Let's pretend for today that I am you and you are me."1145

y is going to pretend that it is x, and x is going to pretend that it is y.1153

They are sort of swapping places when we do a mirror over it.1156

So, that means our picture, mirroring f over y = x...we get the graph of f-1.1159

So, we look at this; we mirror over it; and we get places like this.1164

All right, we see how we are just sort of bouncing across it.1171

And this is going to happen with any of our inverses graphically.1179

So, any time f-1 is being looked at, we know it is going to be a bounce, a reflection through, a mirror over;1183

it is going to be symmetric to f with respect to the line y = x.1190

Since f-1 is swapping outputs and inputs, it is going to be sort of reversing the placements of these.1193

So, the graph of f-1 will always be symmetric to f, with respect to the line y = x.1201

It will bounce across, because when you bounce across y = x, you swap your coordinates.1206

Now, there are many ways to say this; I am saying "bounce across"; that is not really formal.1211

But we can formally say that it is symmetric to f, with respect to the line y = x.1215

You could also say that it reflects through y = x, or it reflects over y = x.1220

You could also say that it mirrors over, or it mirrors through, y = x.1224

There are many ways to say it; but in all of these things, the same idea is that we are going to bounce across,1227

and that that point will now show up at that same distance here.1232

So, let's see what it looks like: we bring them in, and indeed, they pop into those places.1235

They pop into being a nice, symmetric-to-the-line, y = x; and that makes sense.1240

We replaced the inputs with outputs and outputs with inputs; they have swapped locations.1247

We look at this one here, and the point (3,0) on the inverse is connected to (0,3) on the original function--the same sort of thing on both of them.1252

All right, so we have talked a lot about what is going on; we have a really great understanding of the mechanics behind an inverse.1264

But how do we actually find an inverse?1270

Now that we understand them, we are ready to actually go and find them.1273

How do we turn an algebraic function like f(x) = x3 + 1 into a formula?1276

Before we do this formula for f-1, consider that f-1 is taking the output of f(x); and it is transforming that into x.1282

To find a formula for f-1, we want a formula that gives x, if we know f(x).1290

Normally, f(x) = x3 + 1, for example--normally we have x.1298

We know x, and from that, we get our f(x); you plug in an x into a function, and it gives out f(x).1303

So, f-1 is the reverse of that; we know f(x), and we want to get x out of it.1312

So, to be able to do this, we are solving f(x) = x3 + 1 in reverse.1318

f(x) = x3 + 1; well, we move that over: f(x) - 1 = x3; so now we have 3√(f(x) - 1) = x.1323

If we know what f(x) is, we can figure out that that is what the original x that did it is.1338

We are solving it in reverse; we have reversed the function.1342

As opposed to solving f(x) in terms of x, we are solving x in terms of f(x).1345

Now, this is a little bit of a confusing idea; so instead, I am going to show you a method to do this.1351

The idea of reversing is really what is behind inverses; it can be a little hard to understand what to do on a step-by-step basis.1357

We are normally used to solving for f(x) in terms of x, having f(x) just on its own on one side, and having a bunch of stuff involving x on the other side.1365

So, at this point, it might be a little bit confusing for you to try to do it the other way.1372

And it would work; but let's learn a method that makes some of that confusion go away, and do things we are more used to doing.1376

Here is one step-by-step guide for finding inverse functions.1382

The very first thing we have to do: we have to check that the function is one-to-one.1384

It has to be one-to-one for us to be able to find an inverse at all.1388

Now, f(x) = x3 + 1...we just saw its graph; remember, it looked something like this.1391

So, we already know that it passes the horizontal line test; it does a great job; it is a great function.1397

It is one-to-one; great--we have already passed that part for this.1401

Next, we swap f(x) for y; this is going to be a little bit easier for us in solving.1406

We are used to solving for y's; we are not really used to solving for f(x)'s; so this will make it a little bit less confusing.1410

We switch out f(x) for y; great; in the next step, we interchange the x and the y.1415

In this one, we have x in its normal place and y in its normal place.1423

What we do on step 3 is swap their places: y takes the place of all of the x's, and x takes the place of y.1427

We swap x and y, interchange x and y; every time you had an x in step 2, you are now going to have a y;1436

every time you had a y (which is probably just the one time, since it was from a function), you are going to now have an x.1443

That is how we are doing this step that is the reversing step.1448

Solve for y: at this point, we have x = y3 + 1; so if x = y3 + 1, we solve for it.1453

We just move that 1 over: x - 1 = y3; and we have 3√(x - 1) = y.1460

And you will notice that this actually looks pretty much the exact same as what we just did on the previous slide--1469

but perhaps a little less confusing, because it is what we are used to seeing.1473

So, we have y = 3√(x - 1); and finally, just like we replaced f(x) with y, we now do a reverse replacement.1476

But we are now going to f-1; so y now becomes f-1(x).1486

f-1(x) is equal to 3√(x - 1); f-1(input) = 3√(input - 1).1492

Great; now, while this method will produce the inverse if followed correctly, it is not perfect.1500

Now, remember steps #2 and #3; in that, we had to swap f(x) for y, and then we were told to interchange x and y.1507

Remember, they swapped places; now notice, these equations are completely different.1515

They are totally, totally different from one another; yet they are still using the same x and y.1520

Technically, it is not possible to have both of these equations be true with the same x and y.1530

x and y can't possibly fulfill both of these equations at the same time, because they are completely different equations.1534

So, what is going on here? When we swap in step #3, we are really creating a new, different y.1541

When we have "swap f(x) for y," it is really red y or something here.1548

But then, when we do the interchange, it becomes a totally different color of y; it becomes like blue y here.1553

So, we are creating a new, different y; the y when we first swap is different than the meaning of the second y.1560

The swapping y is a different y from our first time that we replaced f(x).1567

The first one is standing in for f(x); that was our red y.1572

And then, the second one stands in for f-1(x); that is really taking the place there.1578

This implicit difference between y's can be confusing; so be careful.1584

I would recommend making a note on your paper; make a note when you are working that says where you swap.1588

Use a note to see that swap of x and y, so that you can see the switch over to this inverse world,1593

where you are now in an inverse world, and you can solve for an answer.1599

This is a bit confusing; so why are we learning this method, if it has this hidden, confusing1603

implicit difference, when we really think about what is going on?1609

In short, the reason we are doing this is because everybody else does.1612

That is not because it is perfect; it is because everyone else out there pretty much learns this method for solving inverses.1616

Most textbooks, and almost all of the teachers out there, teach this method.1623

So, it is important to learn, not because it is absolutely, perfectly correct, but because it is standard--1627

so that you can talk to other people, and talk about inverses, and they will understand1632

what you are talking about, because they are doing the same method that you are doing.1635

If you do something different, they might get confused.1639

If they are really clever, or they really understand what is going on, they will think, "Oh, yes, that makes perfect sense."1641

But we want to go with the standard method, so that other people will understand what we are doing.1645

And if we are taking a course at the same time as we are watching this course,1649

the teacher will think that is correct, as opposed to being confused by what you are doing and marking your grade down.1653

But the important idea here, the really important idea inside of this thing, that is confusing, is the reversal.1657

That is what the moment is all about--that moment between #2 and #3--the #3 step where we reverse, and we create this new y.1663

We reverse the places; instead of solving for an output, we are solving for input.1671

We are reversing the places, so we can do this directly; I did that with f(x), where I did f(x) = y;1678

and I solved directly for if we know what f(x) is.1685

I'm sorry, f(x) equals stuff involving x; I solved it directly for f(x)...1687

We had f(x) = x3 + 1, and we figured out that it also is the same thing that the cube root of f(x) - 1 is equal to x.1694

We figured that out; so there is this direct way of being able to do this.1706

We can do this directly; but lots of students find this difficult or confusing, so we have this method of swapping x and y.1709

And also, it has just become the standard way to do things; so it is good to practice this way, even though it is not absolutely perfect.1716

It is not a perfect method, but it does the job.1723

As long as you are careful and you pay attention to what you are doing--you closely follow its steps--1725

you will be able to get the answer, and you will be able to find the inverse function.1729

Taking inverses can be difficult; it seemed a little bit confusing from what I have been saying so far.1733

And it is an easy one to make a mistake on; this means it is really important to check your work.1737

You really want to make sure that you check your work on this.1743

How do you do this? Well, remember: by definition, f-1(f(x)) is equal to x.1746

That means, if we know what f-1 is (we have figured out its formula), and we know what f(x) is1752

(we were probably told f(x), we can just compose them.1756

We know how to compose them from our lesson Composite Functions.1760

If you didn't check out Composite Functions, you will have to watch that before you are able to compose them and do this check.1763

But if it is really the inverse, you will get x; if you compose f-1 with f(x), it has to come out to be x,1768

because that is the definition of how we are creating this stuff, right from the beginning.1775

Furthermore, we also know that f-1, inverse, was just f;1779

so it also must be the case that f acting on f-1(x) will give us x, as well.1785

You can compose them in either order when you are doing a check; and you will end up being able to get it correct.1790

Let's see a quick example: for example, if f(x) = x3 + 1, and f-1(x) = 3√(x - 1)1796

(the ones we have been working with), how do we check this?1803

Well, let's start with f-1(f(x)); we compose this: we plug in f(x) = x3 + 1.1805

So, f-1 acting on x3 + 1...now, remember, we are going to plug that into f-1(x).1814

But it is f-1(input); whatever is in the box just goes to the box over here.1821

So, it is going to be that f-1 will become cube root...where does the box go?1825

x3 + 1...that is our box...minus 1; so the cube root of x3 + 1 - 1...1832

+ 1 - 1 cancels; the cube root of x3 equals x; great--that checks out.1841

What about if we did it the other way--if we did it as f(f-1(x))?1847

Hopefully, this will work out, as well (and it will).1852

So, what is f-1(x)? f-1(x) is the cube root of x - 1, so f(3√(x - 1))...1854

what is going to happen over here?--we know that you plug in the box; you plug in the box.1864

So, f(3√(x - 1))...we are going to take that, and we are going to plug it in right here.1869

It is going to be 3√(x - 1), the quantity cubed, because it has to go in as the box; plus 1--finish out that function.1875

The cube root, cubed...those are going to cancel each other; we will get x - 1 + 1, which is just equal to x; and it checks out.1885

So, we can check it as f-1(f(x)) or f(f-1(x)); sometimes it might be easier for us to do it one way or the other.1895

We could also do both ways, if you want to check and be absolutely, doubly sure that we really got our work correct.1902

All right, let's move on to some examples.1907

Using these graphs for assistance, which of the following functions are one-to-one?1909

The first one is f(x) = 1/x; we do the horizontal line test--it is going to pass any high horizontal lines.1914

What about as we get lower? Well, we know that 1/x continues to move--it never freezes and becomes constant.1921

Does it ever cross this x-axis, though? No, it doesn't.1928

We haven't formally talked about asymptotes yet; we will talk about asymptotes in a later lesson.1932

But 1/x...as we go positive (f of a positive), 1 over a positive is going to also have to be positive.1936

So, it never crosses the x-axis; the same thing goes with the negatives--f of a negative is going to be a negative.1945

So, when it goes to the left, it never manages to cross this x-axis; as it goes to the right, it never manages to cross this x-axis.1951

And it keeps changing; so the two things never cross over each other.1958

So, yes, this is one-to-one.1961

What about the blue one, g(x) = x3 - 2x2 - x + 1?1968

It is easy to say it fails: we cross lots of places in the middle here, and it is able to have multiple points at the same time.1973

So, any one of these hits here and here and here; there are three points that all give the same output of 0; so it fails the horizontal line test.1982

It is not one-to-one.1992

Finally, (2x - 1) and (x2 + 1); 2x - 1 is just a line that is going to keep going on this way forever and ever and ever.1998

2x - 1, when x is less than or equal to 1...this is from piecewise functions; if you haven't checked out piecewise functions, this might be a little confusing.2007

But hopefully, you have watched that lesson already.2013

2x - 1 is x ≤ -1; it is just going to keep going on down and down and down, to the left and left and left.2015

And x2 + 1 is the right side of the parabola; if we plug in higher and higher numbers, it just keeps curving up and up and up to the right.2020

So, that means that we are never going to cross; the parabola is never going to double back and manage to touch itself again.2027

The parabola might eventually do this, but that part isn't on it.2033

And the line is never going to be able to go down to have itself crossed horizontally.2037

So, if we do any horizontal line crossing on this, it is never going to hit twice; so it is one-to-one.2042

One thing I would like to make a special comment on: notice that right here there is an empty space.2055

There is this gap where it jumps; is that a problem for a horizontal line test?2060

No, it is not a problem at all, because the horizontal line test is allowed to hit no points, as well.2064

It is allowed to hit one point or zero points; in this case, if it goes through that gap, it hits no points; but that is OK.2070

We are only worried about having multiple inputs for the same output.2076

It is OK if there are no inputs to make an output; the important thing2080

is that there are no double sets of inputs that all make the same output.2082

Like, in the blue one, where we had multiple different places where we could plug in some number--2087

plug in different numbers, but they would all produce zeroes.2092

All right, let's actually find an f-1: f(x) = -3x/(x + 3).2096

They told us, right from the beginning, that it is one-to-one; so we can jump right to figuring it out: what is f-1(x)?2102

And then, after it, we need to check our answer.2108

OK, so what is f-1(x)? Remember all of our steps, one by one.2111

f(x) = -3x/(x + 3): they told us, right from the beginning, that it is one-to-one, so we are already checked out.2115

We have already checked out the first one.2122

The next step: we swap y for f(x): y = -3x/(x + 3).2124

Now, that is not the important part of when we reverse, though; we reverse into inverse world.2131

So, here is when we go into inverse world; we reverse the place of x and y.2136

So now, it is x where y was, and it is -3y/(y + 3).2144

Multiply both sides by y; we get x times (y + 3) equals -3y; let's distribute this out: xy + 3x...let's also move the 3y over, so + 3y = 0.2154

OK, at this point, we will pull out the y's from these two things; we will move them together, so we can see it a little bit easier at first.2170

xy + 3y + 3x = 0; let's subtract that 3x to move it over; -3x, -3x here.2176

So, then we will pull out the y's to the right; so we have x + 3, times quantity y, equals -3x.2186

Finally, we divide by that x + 3, and we get y = -3x/(x + 3).2195

And now, finally, we can plug in f-1 for this y; so we plug it in, and we get f-1(x) = -3x/(x + 3).2202

Great; now, let's check and make sure that we got this right.2216

We check this in red; here is our check--let's check it by plugging f into f-1.2219

So, we want this to come out to be x; it should be x, if we got everything right.2232

So, f-1(f(x)); what is f(x)? f(x) is this; and here is something funny to notice.2239

Notice -3x/(x + 3); amazingly, it just so happens that for -3x/(x + 3), f(x) and f-1(x) are the exact same thing--kind of impressive.2245

We plug this in; we have f-1(f(x)); f(x) is -3x/(x + 3); now, over here, we plug it in; what is in the box?2260

The box shows up here; the box shows up here; it shows up twice, so it is f-1 on -3x/(x + 3).2274

It is going to be -3...what is in the box?...-3x/(x + 3), over (-3x/(x + 3)) + 3.2281

Great; so the first thing that is going to be confusing is that we have this x + 3, and we have this x + 3 here.2301

So, let's take that out by multiplying the whole thing by (x + 3)/(x + 3).2307

We can get away with that, because it is just the same thing as 1: (x + 3)/(x + 3) is just 1.2312

So, (x + 3)/(x + 3)...multiply that here; the (x + 3) will cancel out here and cancel out here.2317

But remember, it also has to distribute to the other part, because they are not connected through multiplication on that; they are connected through addition.2324

So, we have -3, -3x, over -3x plus 3 times x plus 3.2330

These two negatives cancel out; so we have 3 times 3x on the top, -3x plus 3(x + 3)...so we have 9x on the top,2341

divided by -3x plus 3x plus 9; -3x plus 3x...they cancel each other out; we have 9x/9.2350

9 over 9...those cancel out, and we have just x.2362

So, that checks out--great, we have the answer.2367

All right, the next one: we have, this time, a piecewise function.2370

This is a little confusing: we didn't talk about this formula, but we will see how to do it.2375

f(x) = -x + 1 when x < 0, and -√x when x ≥ 0.2378

This is confusing; we don't know what to do about the different pieces of the piecewise function.2387

We don't know what to do about these two different categories: we have x < 0 and x ≥ 0.2393

We didn't learn that when we learned how to do inverses; but we could still figure out these two.2397

We could figure out what is the inverse of -x + 1 and what is the inverse of -√x.2402

We were told, explicitly, that this is one-to-one; so we can go ahead and do this, and then we will think about it.2407

First, we will do inverses on these two rules; and then we will figure out how they fit together--what are the categories for these two rules?2413

So, first, -x + 1; we will have y = -x + 1; we swap them, so we now get into our inverse world.2422

Swap their locations; we interchange them, and we are now at negative...sorry, not -x; the negative does not swap.2439

We are at x = -y + 1; we move the y over and move the x over; we get -x + 1; so y = -x + 1,2446

which is going to give us f-1 for at least the first rule here.2458

Now, what about the other one?--let's do that, as well.2464

So, y = -√x; we go into inverse mode; we reverse their locations; and we are now at x = -√y.2467

So, how do we solve for y? Well, we move this negative over: -x = √y.2482

Square both sides; we get (-x)2 = y; and then (-x)2...the negatives will cancel out, so we get just x2 = y.2487

And so, this is the inverse rule for this part.2498

Now, here is the part where we start thinking: we know that f-1 is going to break into a piecewise function using these two different things.2502

y = -x + 1...so it will be -x + 1 for the first rule, and then x2 for the second rule.2513

But the question is that we don't know what the categories are.2520

How do we figure out what the categories are?2524

Well, remember: if f goes from its domain to its range, let's call that a to b, then f-1 does the reverse of that.2526

f-1 goes from b to a; it does the reverse.2543

What that means is that the domain...the thing that determined which rule we used...we need to do the range to determine which way to get back.2548

The range on these two rules...now we are back to using f, so range on f...for -x + 1:2557

well, -x + 1 was x < 0; that was the category, so it has to be within those.2571

So, what can it go to? Well, if we plug in a really big negative number, like, say, -100, we will get -(-100) + 1; so we get 101.2577

So, as long as we keep plugging in more and more negative numbers, we get bigger and bigger numbers.2584

We are able to get all the way out to positive infinity, as we are really far in negative numbers.2587

What is the lowest that we can get to? Well, we could get really close to 1, as we plug in -0.00000001.2592

We are really close to being to 1, so we can get right up to 1; but we can't actually touch it.2601

We have to exclude it, so we use parentheses.2605

So, the range for the first rule is this: -x + 1 becomes this.2607

So, I will put a red dot on that, because that matches to this rule here.2614

Now, what about the range for the other rule?2619

The range on this rule is -√x; it has x ≥ 0 as its domain; what are the numbers we can get out of this?2622

What is the largest number we can get out of it?2630

The largest number we can get out of it is actually 0; why?--because, when we plug in any reasonably large positive number,2632

like, say, 100, then -√100 is -10; so as we get bigger and bigger positive numbers that we plug in,2639

we actually get more and more negative.2649

So, we can actually go to any negative number we want; we can go all the way down to negative infinity.2651

Can we actually reach 0? Yes, we actually can reach 0, because it is greater than or equal to.2655

So, if we plug in x = 0, we get -√0, which is just 0; and we put a bracket to indicate that we are actually allowed to do it.2660

This one is the range for -√x, that rule; it is going to get a green dot on it, because it matches to the green rule.2668

That means that -x + 1 is allowed to take in...what values? It is allowed to take in the range values.2677

It is allowed to do a reverse on anything that shows up in the range (1,∞).2684

Also notice: these two ranges, (1,∞) and (-∞,0], don't have any intersection.2689

They don't overlap at all, so we don't have any worries about pulling from one versus pulling from the other.2695

They will never get in each other's way.2700

So, for this one, -x + 1, if it is going to be allowed to go from 1 to infinity, then that means we can plug in anything into f-1,2703

where x is greater than 1, which is to say input; it is not the same x that was up here.2711

It is now just saying "placeholder--anything that we are plugging in."2719

What about x2? Well, that was the green dot--that was allowed to go from negative infinity up to 0.2722

So, it is allowed to have x ≤ 0; it is allowed to go all the way up to negative infinity, but it can only just get to touching 0.2727

It is allowed to actually have 0, though; x > 1 is not actually allowed to touch 1, but it is able to get as close as it wants.2735

And there is our piecewise inverse function.2741

It is a little bit difficult, but if you think about it, you do each of the inverses, and then you think about2747

"How do I get the domain for the inverse? I get it from the domain of f becoming the range of our inverse,2750

and the range of our f becoming the domain of our inverse."2760

So, what the original function was able to output to is what the inverse is allowed to take in.2766

And that is how we figured out these rules, these categories--what the categories were for these two different transformations.2773

All right, the final example: f and g are one-to-one functions; now, prove that f composed with g, inverse, is equal to g inverse composed with f inverse.2779

This might be a little daunting at first; these are weird symbols; we are not used to using these sorts of things.2791

So, if that is the case, let's remind ourselves: from composition, f composed with g, acting on x, is equal to f(g(x)).2795

Now, I said before: it makes things always, always, always easier to see it in that format.2806

What we want to show is that g-1 composed with f-1 (which would be g-1(f-1(x)))...2811

we want to show that this one here is an inverse to that one over there.2823

That is what we are trying to prove, that f composed with g inverse...2829

We know, by the definition of how this symbol works, by how inverses work...2833

f composed with g-1, acting on f composed with g, on x, is going to just leave us as if we had done nothing,2837

because we are putting an inverse on something.2845

So, we want to show that this means the exact same thing as this right here.2848

So, let's just try it out: we will set it up like this: f composed with g-1, acting on f composed with g, acting on x.2854

OK, so what does that become? Well, we know that f composed with g, acting on x, is the same thing as f(g(x)).2871

All right, what is f composed with g-1? Well, we know (from what we did over here)2881

that we can bring that into g-1 acting on f-1, acting on whatever is going into it.2887

What is going into it here is this whole thing; so, it is going to be g-1, acting on f-1, acting on f, acting on g, acting on x.2891

And then, we close up all of those parentheses.2908

That is a little bit confusing; but we are seeing inverses right next to functions: f-1 acting on f, acting on whatever is in there.2912

It just cancels out and gets us right back to what we originally had in there.2922

So, f-1 acting on f...that cancels out, and we get g-1, acting on whatever was in there, which was g(x).2926

So, g-1 acting on g(x)...the exact same thing: we get down to x; so we have proved it.2933

g-1 composed with f-1 is how we create f composed with g, inverse.2940

Great; we have proved it.2948

All right, I hope you have a much better idea of how inverses work at this point.2949

They can be a little bit confusing, but you have that method to be able to guide you through it.2952

Just follow it really carefully, step-by-step.2956

The danger is if you break from those steps and do something else; that is where you can make mistakes.2958

If you really understand what is going on, you don't even have to use that method.2963

But it really is the standard method, so it is a good idea to stick with it, just because it is what a lot of other people are used to using.2966

And you can find it in a lot of textbooks.2972

All right, we will see you at Educator.com later--goodbye!2974