Calculus: The Basics of Inverse Functions

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Elementary Functions and the Inverses
Inverse Functions
The Basics of Inverse Functions Page [1 of 3]
Okay, now suppose that you have a function, like f(x). Now how do you think of functions? You could visualize their graphs, if you graph it on the x-y plane, or you could think about it as a little machine, where you sort of input x, turn a crank and then output y. So if you think of it that way for a second, let's just do that. So here's a function and we call it f, and then what you do is you take an x and you put it inside. And then it spins around, and smoke comes out, all this stuff goes on, and then when you're all done what pops out is y, or sometimes f(x), depending on what you call it. So there's this input x, and you put it in, and it comes out. For example, if the function f(x) was like x^2, then if you put in 4, the out would come 4^2, which is 16. So you'd put in 4, this little machine would spit out 16, and so forth. And you can plot them and graph them and so forth.
Now, suppose that you had a function like this and what you'd like to do now is build a whole new machine. Here it is. Let me call it g. It's the g-machine, and the way this works is it takes as input the output of this other machine. And then what I want it to spit out is actually the original x. So what I want is another function that will actually untangle and undo all the complicated stuff that the f did. We started off with an x. It was happy, it was carefree, and it was singing and dancing on the prairie. And then we shoved it into the f-machine, cutting and so forth, and then it comes out and it's f(x). And it's all mangled and deformed now. But suppose you wanted to bring x back. Let's bring x back to the way it was. Can we find a machine that will actually take that thing and pull back x? Well, when you can do that, this g-machine, or this g function is what we call an inverse function. And so inverse functions are two functions that have the property that when you take x and plug it into one, and then take the answer you get as input into the second function, you get back x. And, conversely, if you took x and ran it into this machine, what you would get would be g(x). In fact, let me show you that with this dotted line. If you take x and put it into here, then out would come g(x). And if you stick g(x) back in here, what should come out should be x. When that happens, these two functions are called inverse functions. And the notation for this, by the way, there's even notation, but who cares about notation? But in case you actually want to talk to a mathematician - why would you ever in your life want to talk to a - I've talked to so many, well, okay.
Now, so this function g actually is called the inverse of f, and here's a way of writing it: f^-1, to indicate that it's the inverse of f. It undoes f, it's the opposite of f. So that's how we write inverse functions. Now I want to have a word of caution right now. This is really important, because the one thing that you might get confused with is you'll say, "Oh, so that means it's one over the function." That's not right. Don't be confused with reciprocals. So reciprocal, that's a one over. So if I have a function, like f(x), I can make a new function by writing . But that's not the inverse, necessarily. That's just taking the value you get and putting it one over it. So don't think of this as just one over it, this is genuinely a different and perhaps more exotic function. It's the function that does precisely this; it untangles the f-machine and allows you to get back to where you started. So, just as a little cautionary note, remember that inverse is not necessarily a reciprocal. So don't think of that.
Now, how could you write this out? Well, you could actually write this out in mathy ways. Let's try to write this out in mathy ways. So here we go. We take x and put it in. If we put x into the f-machine, we get f(x). So there's f(x). Now, if I take f(x) and use it as input into the g function, then I'm going to take g of that thing. I'm going to put the f(x) into the g function. So that means I would take g(f(x)). And what's the answer? Well, the answer should be x. Now, I want you to see that this is nothing more than this picture. In fact, I'll do it for you live. Here's the x. Then you run through the machine and you get f(x). Then you push that into the g-machine and you get x. You see how that works? And if you go backwards, you start off with x and plug it into g(x). And then if you take that as the input into f, then f of that should produce x. And that's actually the definition of two functions being inverses of each other if, in fact, g(f(x)) = x, and f(g(x)) = x. Okay, so there's a lot of pictures and colors and words and who know what any of this means. I don't know, but let's take a look at some examples, where you can actually see these ideas happening in front of our very eyes.
Now, to do that what I want to do is consider this example. f(x) = x^3. And I want to show this to you visually now. So how can you think of x^3? Well, if you try to graph it, the cubed function sort of looks like this. That looks pretty good. There is the f(x) = x^3. Now, let's think about what we're trying to do here. I want to find a function that will untangle this. So basically I want to find a function so that if I compose those two functions, f(g(x)), I just get back to x. So what I really want to do is I want to go backwards. So I want to flip the roles of x and y. I want x to be y and y to be x. Now, how can you actually do that? Well, one way of doing that is just imagine taking this entire picture and flipping it over the y = x line. There's the line y = x. If I want to switch x and y, I should just flip this picture along that axis. And, in fact, this now is a wonderful opportunity for us to be artists and try to visualize what would it mean to take this red curve and flip it over this line? What would that look like? Well, one way to do that is just take a mirror and see what the mirror looks like. So here's the mirror, put it on the line, and you can see, or maybe you can't, that it sort of butts out. So part of it, you see part of it goes that way. So it seems to be going that way, and then part of it disappears. And then it bows out here, and then what happens down here? I guess we can't see that part. So it sort of comes out and then disappears and goes in. And if you now visualize that without the mirror, I think you'd see this picture. Look at that, isn't that beautiful? Wow! Now notice that the blue is exactly the red if you flipped it over this y = x line. See, this little wing here, if you flip, goes to that wing. And this little wing here would actually get flipped to this side, to that wing. That's why we couldn't see it in the mirror, because it was sort of hidden from the mirror. And then this little wing here gets flipped to the other side to here, and then this part here gets flipped to here. So this genuinely is the red curve flipped along this axis. We get this blue curve. And this is actually the inverse function. So graphically the inverse function is nothing more than the function itself, the graph of the function, but reflected over the y = x line. And you can do that with a mirror or like this.
Now, what is the equation of that? Well, it turns out the equation of that, I happen to know, the equation of that is the cube root function. So f^-1 = . And you can actually see that for yourself, because what happens if we actually take the two functions and put them together? So what if I took f f^-1(x)? If these are really inverses, this should produce just x. And let's see what I get. Well, the way I do this is just very carefully plugging everything in. What is f^-1(x)? That's just . So this is f . And then I go back to f and plug this in for x. So wherever I see an x, I'm just going to plug that in. So I see, well, f of stuff equals stuff cubed. So f of this number will be that number cubed. So it should be . And the is just x. And so, happily, I got the x back. And if you try the other thing, which you might just want to try on your own, we get the same thing. In fact, I'll do it for you really fast. That would be f^-1 of - well, f(x) = x^3 and f^-1 takes , so it's = x. So these are really inverse functions of each other. So that's pretty neat.
Now, when would a function actually have an inverse. Suppose you say, "Gee, my favorite function is blah, blah, blah function." Does it have an inverse? That's a great question. And you might want to know. I mean, for example, birthday gifts and stuff, you might want to give an inverse of the present. Can you always take an inverse? Well, the inverse might not be a function. Let's think about it. In this example here - you see, in fact the original function, which was the cubic function - you see, it's constantly going up. So what would be required of a function so that its inverse would still be a function? Well, what I'd need to have happen is that it can't go up and then come down, because what would happen if the function would go up and then comes down? Well, when I reflect it, I'm going to actually, in some sense, cross myself. Now let's actually see that in a real, live example. So let me draw some axes here and see if we can compute some inverses visually right on the fly.
So let me just a function here. I'll draw one sort of just at random. How about this function right here? It'll go up and it'll come down. Now, if that's the function, f(x), what does the inverse look like? Well, all I have to do is take the line y = x, which I could denote as green. So I just draw on the y = x line, and then I visualize. If I had the mirror, I could just put the mirror here and we'd take a look and see that, in fact, I would just see this reflection. Now, so what would that look like? What would the reflection look like? Well, I just have to take this whole image and then reflect it over. So it would look like this. That would be the inverse. Now just look into the mirror. Does that mirror image actually look like a function? No, because it seems to actually fail the vertical line test. I'll try to draw this image for you right now live and see if I can actually show you - and drawing the inverse function is sort of tricky. It's a very visual thing. You've got to now imagine this thing coming here, so it would look something like this. If you think it's easy, by the way, you are sadly mistaken. I'll try to do it live right now for you. Oh, it's so hard to do, by the way. Not bad, not bad, just taking this image and reflecting it over to here. This wing goes to here and so forth. This piece goes out to here. And you could see that is not a function, because it actually fails the vertical line test. To be a function, for every x value you could only have, at most, one y value. And in this region here, you see that for one particular x value I've got three y values. So this thing here is not a function. So the inverse of this is no longer a function. So we say this is non-invertible or it doesn't have an inverse function, because the inverse is just some sort of relation like this, it's not a function.
So when does a function have an inverse? Well, what's required? Well, in some sense, it not only has to be a function, which means pass the vertical line test, but when you flip it over the y = x line, what you've got to have happen is the new thing has to still be a function. That means it wouldn't be a vertical line test, but what would be this vertical line after I flip it back to the original function? Well, I'll do it for you right now live and you can see what happens. If you flip a vertical line over this line, it looks like this. It becomes a horizontal line. So, in fact, you have to see if, in fact, the original function satisfies the horizontal line test, which means that no matter where I put a horizontal line, it only crosses at, at most, one point. And here it fails. And so since it fails the horizontal line test, the original function, that means that when I flip, the inverse is going to fail to be a function for that same reason. So, in fact, this is the horizontal line test. And functions that actually pass the horizontal line test are called one-to-one. So a one-to-one function is a function that actually not only is a function, but passes the horizontal line test for every y value that came from, at most, one x value. That's sort of the point here. Here there's this y value that came from a lot of different x's. So when I flip it, I see this thing is not a function.
So what functions are one-to-one? Well, let's think about that. A function is one-to-one if it keeps climbing. If I keep climbing and climbing, that means that when I go over the line y = x, the thing will still be climbing. So, in fact, as long as there's no ups and downs - it can't be like life. It has to be something like fake, where you just have ups or you just have downs. So let me try to show you one. Here's an example. You could think about a function that's increasing. So a function that's just going up is a function that we'd say is increasing. And notice that here, well, it passes the horizontal line test, because at every horizontal point, I only cross, at most, one point. So, for example, if you're going up, you're always going up, which means that if a function is increasing, then, in fact, it has an inverse. The function is increasing. Similarly, if a function is decreasing, this is a little bit more exciting, then also it has an inverse, because it passes the horizontal line test. The point is this thing actually has an inverse. So if functions are increasing always or decreasing always, then those functions are invertible. They have an inverse.
So when is a function decreasing? Well, a function is decreasing whenever you have a situation where the derivative is always negative, because that means that the function is always falling. Now, in fact, you actually could have the derivative equal 0 at one point, but you can't have it equal 0 at a lot of points. And where it's not 0, we must, in fact, have it be negative. It could level off if you have, for example, a point of inflection.
So when is a function increasing? When the derivative is positive everywhere or, at most, at finitely many points, where, fact, you equal 0. Again, points inflection and we've seen that right here. If we go back to the original function we were thinking about, you can see that this function is increasing everywhere, but right at the origin we see that the derivative there is 0. So you can have a point where the derivative is 0, but you can't have a lot of points that are 0, just a finite number of so. And we have to positive everywhere else to be increasing, and then, in fact, you absolutely do have an inverse.
So, who's invertible? Functions that are monotomically increasing, that means you're going up, or monotomically decreasing, you're always going down. Those are invertible functions. If you have an up and then a down, then, in fact, you're out of luck. That will not be invertible, because it fails the horizontal line test. So if you have ups an downs in life, I'm sorry, you can't take an inverse of yourself. But if you're always going up, or even if you're always going down, absolutely fine, you can find the inverse, you can find the function that untangles you. But if you have ups and downs in life, then I'm sorry to say, no inverse. I'll see you at the next lecture.