For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### 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. ^{−1}operates on the output of f, it gives the original input that went into f. [f*Caution:*^{−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) = x^{3}+1

2. Swap f(x) for y; y = x^{3}+1

3. Interchange x and y; x = y^{3}+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

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 Basketball Factory
- 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
- Some Comments 25:01
- 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

### Math Analysis Online

### 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 factory*0038

*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 machine*0114

*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 way*0168

*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 means*0438

*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) = x ^{2}, then we plug in 3, and we get (3,9), (3,3^{2}).*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 draw*0490

*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 exponents*0612

*(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) = x ^{3} + 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 x ^{3} + 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) = x ^{3} + 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) = x ^{3} + 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) = x ^{3} + 1 in reverse.*1318

*f(x) = x ^{3} + 1; well, we move that over: f(x) - 1 = x^{3}; 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) = x ^{3} + 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 = y ^{3} + 1; so if x = y^{3} + 1, we solve for it.*1453

*We just move that 1 over: x - 1 = y ^{3}; 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, confusing*1603

*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 understand*1632

*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) = x ^{3} + 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) is*1752

*(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) = x ^{3} + 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) = x^{3} + 1.*1805

*So, f ^{-1} acting on x^{3} + 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

*x ^{3} + 1...that is our box...minus 1; so the cube root of x^{3} + 1 - 1...*1832

*+ 1 - 1 cancels; the cube root of x ^{3} 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) = x ^{3} - 2x^{2} - 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 (x ^{2} + 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 x ^{2} + 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 thing*2080

*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

*Now, we just go about this, and we solve it for y like we normally would.*2151

*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 x^{2} = 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 x ^{2} 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 x ^{2}? 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 about*2747

*"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

1 answer

Last reply by: Professor Selhorst-Jones

Fri Mar 25, 2016 5:24 PM

Post by Ru Chigoba on November 18, 2015

Hi I need help with this problem :

Find the inverse of each relation or function, and then determine if the inverse is a function.

1 f={1,3), (1,-1), (1,-3), (1,1)} f-1=

1 answer

Last reply by: Professor Selhorst-Jones

Tue Mar 17, 2015 11:24 PM

Post by thelma clarke on March 17, 2015

is there no easier way to solve this it seem very confusing

1 answer

Last reply by: Professor Selhorst-Jones

Mon Oct 20, 2014 11:27 AM

Post by Saadman Elman on October 18, 2014

It was a great clarification. Thanks,

However, Inverse function is not only has to do with horizontal line test passing but also has to do with vertical test passing. You only stressed on Horizontal test passing. My professor stressed both.

1 answer

Last reply by: Professor Selhorst-Jones

Mon Jun 16, 2014 9:18 PM

Post by Joshua Jacob on June 15, 2014

Sorry if this is slightly vague but I'm a little but confused on the last example. Could you explain it in other words please?