Introduction to inverses of functions
Let's think about what functions really do, and then we'll think about the idea of an inverse of a function. So let's start with a pretty straightforward function. Let's say f of x is equal to 2x plus 4. And so if I take f of 2, f of 2 is going to be equal to 2 times 2 plus 4, which is 4 plus 4, which is 8. I could take f of 3, which is 2 times 3 plus 4, which is equal to 10. 6 plus 4. So let's think about it in a little bit more of an abstract sense. So there's a set of things that I can input into this function. You might already be familiar with that notion. It's the domain. The set of all of the things that I can input into that function, that is the domain. And in that domain, 2 is sitting there, you have 3 over there, pretty much you could input any real number into this function. So this is going to be all real, but we're making it a nice contained set here just to help you visualize it. Now, when you apply the function, let's think about it means to take f of 2. We're inputting a number, 2, and then the function is outputting the number 8. It is mapping us from 2 to 8. So let's make another set here of all of the possible values that my function can take on. And we can call that the range. There are more formal ways to talk about this, and there's a much more rigorous discussion of this later on, especially in the linear algebra playlist, but this is all the different values I can take on. So if I take the number 2 from our domain, I input it into the function, we're getting mapped to the number 8. So let's let me draw that out. So we're going from 2 to the number 8 right there. And it's being done by the function. The function is doing that mapping. That function is mapping us from 2 to 8. This right here, that is equal to f of 2. Same idea. You start with 3, 3 is being mapped by the function to 10. It's creating an association. The function is mapping us from 3 to 10. Now, this raises an interesting question. Is there a way to get back from 8 to the 2, or is there a way to go back from the 10 to the 3? Or is there some other function? Is there some other function, we can call that the inverse of f, that'll take us back? Is there some other function that'll take us from 10 back to 3? We'll call that the inverse of f, and we'll use that as notation, and it'll take us back from 10 to 3. Is there a way to do that? Will that same inverse of f, will it take us back from-- if we apply 8 to it-- will that take us back to 2? Now, all this seems very abstract and difficult. What you'll find is it's actually very easy to solve for this inverse of f, and I think once we solve for it, it'll make it clear what I'm talking about. That the function takes you from 2 to 8, the inverse will take us back from 8 to 2. So to think about that, let's just define-- let's just say y is equal to f of x. So y is equal to f of x, is equal to 2x plus 4. So I can write just y is equal to 2x plus 4, and this once again, this is our function. You give me an x, it'll give me a y. But we want to go the other way around. We want to give you a y and get an x. So all we have to do is solve for x in terms of y. So let's do that. If we subtract 4 from both sides of this equation-- let me switch colors-- if we subtract 4 from both sides of this equation, we get y minus 4 is equal to 2x, and then if we divide both sides of this equation by 2, we get y over 2 minus 2-- 4 divided by 2 is 2-- is equal to x. Or if we just want to write it that way, we can just swap the sides, we get x is equal to 1/2y-- same thing as y over 2-- minus 2. So what we have here is a function of y that gives us an x, which is exactly what we wanted. We want a function of these values that map back to an x. So we can call this-- we could say that this is equal to-- I'll do it in the same color-- this is equal to f inverse as a function of y. Or let me just write it a little bit cleaner. We could say f inverse as a function of y-- so we can have 10 or 8-- so now the range is now the domain for f inverse. f inverse as a function of y is equal to 1/2y minus 2. So all we did is we started with our original function, y is equal to 2x plus 4, we solved for-- over here, we've solved for y in terms of x-- then we just do a little bit of algebra, solve for x in terms of y, and we say that that is our inverse as a function of y. Which is right over here. And then, if we, you know, you can say this is-- you could replace the y with an a, a b, an x, whatever you want to do, so then we can just rename the y as x. So if you put an x into this function, you would get f inverse of x is equal to 1/2x minus 2. So all you do, you solve for x, and then you swap the y and the x, if you want to do it that way. That's the easiest way to think about it. And one thing I want to point out is what happens when you graph the function and the inverse. So let me just do a little quick and dirty graph right here. And then I'll do a bunch of examples of actually solving for inverses, but I really just wanted to give you the general idea. Function takes you from the domain to the range, the inverse will take you from that point back to the original value, if it exists. So if I were to graph these-- just let me draw a little coordinate axis right here, draw a little bit of a coordinate axis right there. This first function, 2x plus 4, its y intercept is going to be 1, 2, 3, 4, just like that, and then its slope will look like this. It has a slope of 2, so it will look something like-- its graph will look-- let me make it a little bit neater than that-- it'll look something like that. That's what that function looks like. What does this function look like? What does the inverse function look like, as a function of x? Remember we solved for x, and then we swapped the x and the y, essentially. We could say now that y is equal to f inverse of x. So we have a y-intercept of negative 2, 1, 2, and now the slope is 1/2. The slope looks like this. Let me see if I can draw it. The slope looks-- or the line looks something like that. And what's the relationship here? I mean, you know, these look kind of related, it looks like they're reflected about something. It'll be a little bit more clear what they're reflected about if we draw the line y is equal to x. So the line y equals x looks like that. I'll do it as a dotted line. And you could see, you have the function and its inverse, they're reflected about the line y is equal to x. And hopefully, that makes sense here. Because over here, on this line, let's take an easy example. Our function, when you take 0-- so f of 0 is equal to 4. Our function is mapping 0 to 4. The inverse function, if you take f inverse of 4, f inverse of 4 is equal to 0. Or the inverse function is mapping us from 4 to 0. Which is exactly what we expected. The function takes us from the x to the y world, and then we swap it, we were swapping the x and the y. We would take the inverse. And that's why it's reflected around y equals x. So this example that I just showed you right here, function takes you from 0 to 4-- maybe I should do that in the function color-- so the function takes you from 0 to 4, that's the function f of 0 is 4, you see that right there, so it goes from 0 to 4, and then the inverse takes us back from 4 to 0. So f inverse takes us back from 4 to 0. You saw that right there. When you evaluate 4 here, 1/2 times 4 minus 2 is 0. The next couple of videos we'll do a bunch of examples so you really understand how to solve these and are able to do the exercises on our application for this.