Common Core Algebra II.Unit 2.Lesson 6.Inverse Functions

Let's just substitute it into its rule. Just got to make sure we're doing our order of operations correctly. But that should be no problem for us. 22 divided by two when we get 11. So F of 5 is 11. Let's see what G of 11 is. Let's see what we're going to get two times 11 -7. Divided by three. 22 -7 divided by three. Let's see, 22 -7 is 15. Divided by three. Is 5. Huh. Look at that. F of 5 is 11. And G of 11 is 5. This X went to that Y and this Y went to that X or sorry, this X went to that Y all right, well maybe that's a coincidence maybe it's not. Let's take a look at what F of zero is. Let's see F of zero is three times zero plus 7 divided by two. Which will be zero plus 7 divided by two. And F of zero, 7 halves. Um. All right, let's take a look at what G of 7 halves is. We're algebra two students were not scared of all little fraction. So that's two times 7 halves -7. Divided by three. Let's see. Cancel and 7 -7 divided by three. Zero divided by three is zero. Well look at that. Take a look at those first two problems. Letter a and letter B F of zero equals 7 halves G of 7 halves equals zero. It's like these two functions are almost, I don't know, doing the opposite of each other. 

Let's take a look at letter C now, we have a composition problem we're doing F of G of negative one. Well, let's just do a little composition. How did that work? First, we figure out what G of negative one is. That's two times negative one -7. All divided by three. That would be negative two -7. Divided by three, which has got to be careful here. That's negative 9 divided by three and that's negative three. And then remember, we take that input and we put it into F F of negative three, let's see. That's going to be three times negative three plus 7 divided by two. A negative 9 plus 7 divided by two. Gives me negative two divided by two gives me negative one. Look at that. So F of G of negative one is negative one. Now let's do this one. G of 5. Let's see. What do we have? Two times 5 -7. Divided by three. Ten -7 divided by three. Three divided by three. So G of 5 is one. And remember that gets routed into F F of one. Let's see three times one plus 7. Divided by two, that'll be three plus 7. Divided by two, and that's ten. Divided by two, which is 5. So F of G of 5 is 5. Now, remember what functions do is they turn inputs into outputs. And what looks to be happening when we compose F and G is that they sort of cancel each other. I mean, think about that last one, right? It was like, it was like 5, went into G, it's output went into F, and that output went back to 5. Same thing happened in letter C down here. We'll get to letter E eventually. We'll just circle it. 

So we don't lose it. But in that case, negative one went into G, its output went into F and that went back to negative one. Right? So it's like F is undoing whatever has been done to the input. Therefore, F of G of pi, well, pi will go into G that'll go into F and that I'll go back to pi. I'm going to do that without any calculation. Now functions that do this functions that undo whatever the original function has done. Well, those are known as inverse functions. Now, I know inverse is way overused. But essentially, inverse kind of means opposite to to do the opposite of what has been done. Anyhow, we'll get into inverse functions a bit more in just a moment since they are the topic of the lesson. Pause the video now and copy down anything you need to. All right, clearing out the screen. Let's talk about inverse formally. Inverse functions. If F of X and G of X are inverse functions, then this is what it looks like. Here we get that domain of F and some element is in the domain of F some X value. It goes in as the input to F and out comes B, right? So here's my B and that's the range of F, right? The domain goes to the range. But then, for the inverse, the range of F becomes the domain of G and if we put that output into G as an input, it'll take us right back to where we started. It's like a circle of life kind of thing. Whatever F does to the input, G will undo and get us right back to where we started. That's the beautiful thing about inverses. Don't forget this diagram. 

It's very, very helpful to help think about what inverses are doing. I'm going to clear this out there. And let's actually get into some problems. All right. If the point negative three 5 lies on the graph of Y equals F of X, which of the following points must lie on the graph of its inverse. This is actually an amazingly important multiple choice exercise. So what I'd like you to do is really pause the video right now. Think about what we've just been talking about with inverses. Look at that diagram back in the previous page. And see if that can help you figure out what point lies on the graph of the inverse. All right. Well, without telling you what the answer is, let's take a look. So remember, negative three goes into F and what comes out is 5. Let's call its inverse G so now 5 goes into G and its output is negative three. In other words, G of 5 must be negative three. Which means the point 5 comma negative three must lie on the graph. In fact, if the point XY lies on the graph F of X actually, let me do this. If the point AB lies on the graph of F of X, then the point B comma a must lie on the graph of its inverse. All right. All inverse is really due is flip flop the roles of inputs and outputs. 

They really do. Inputs become outputs, outputs become inputs, but there's nothing about negating or flipping or reciprocals or anything like that. It's purely input becomes output output becomes input. All right, pause the video now and write down anything you need to. Okay, here we go. Inverse function notation. So obviously, if there's inverses, we're not going to keep calling them G because G doesn't have to be the inverse of F we've just been kind of using that so far. So inverse is have a special notation. And it's not my favorite notation. I've gotten used to it over the years. But if F of X is a function. With an inverse, actually, let me say with an inverse function. All right. Then F with a little negative one, all right. Is it's inverse. Now let me just be very clear. F of negative one X has nothing to do with one over F of X how 5 to the negative one equals one 5th. Well, that has nothing to do with it. Okay? It's just kind of an unfortunate notation. But if we see that little negative one is sort of an exponent on a function, it means the function's inverse. Something that undoes what the function does. All right, I'm going to clear this out. And let's do some problems. Exercise three. The linear function F of X equals two X two thirds X minus two is shown graph below. Use its graph to answer the following questions. All right. I want to figure out F inverse of two and F inverse of negative four. 

Wow, that's kind of tricky. Now, you know, it'd be tempting to go over to two and pull something out, except this is the this is the output of F of X, right? I mean, remember, like we got something that goes in, I don't know what it is to F and what comes out is two. The two then goes into F inverse. And what comes out is a so I actually really need a Y value of two. Ah, there it is, right? And that goes to an output of one, two, three, four, 5, 6. Ah, that's 6. Right? So 6 goes into F of X and gives me two. When two goes into F inverse, it gives me back 6. See if you can do F inverse of negative four. Did you get it? So again, we're going to kind of come to the output side of things. And oh, okay. When I hit an output of negative four, it must have come from an input of negative three. Again, negative three goes into F outcomes negative four, negative four goes into its inverse and out comes negative three. The Y intercept of F inverse while that means F inverse of zero. But again, that's the output to F, right? It's got to be, you know, what's going in to give me an output of zero. Well, here's an output of zero. And what went in must have been three. So when three went into F, zero came out, when zero goes into F inverse three comes out. 

Finally, it says on the same set of axes draw the graph of Y equals F inverse. Well, this is really rather cool because to draw the graph of F inverse actually, it's not terribly difficult. What we're going to do is we're going to take every coordinate pair on F and we're going to switch them on eth inverse. So like watch, 123-456-1234, 5, 6. Well, that's not very helpful. Because if I take the pair negative 6 negative 6 and I flip flop them, I get negative 6 negative 6. That wasn't helpful. Let's do another one. One, two, three, one, two, three, four. So here I've got negative three negative four for the inverse. I'm going to have negative four negative three. Let me plot that in red. Well, actually, I'll wait until I've got all the coordinate pairs. Let's see, here I've got a coordinate pair of zero comma negative two for the inverse. I'll have negative two zero. Here I've got the pair three comma zero. So then I'll have the pair zero comma three. Here, one, two, three, four, 5, 6, two, 6, two, two, 6. The graph that in red really bring it out. Well, negative 6 negative 6 is right there. There's oops, that's not right. 

Let's try to go with negative four, negative three. Negative two. Negative two zero. Two, three. Zero three and there's my two 6. I'm going to connect them all with a nice red line. And maybe put some arrows on it. Extend it that way a bit. All right. And there's F inverse. Nice. All right. One thing that is true about all functions and their inverses. Is that they are symmetric across the line Y equals X, which is really kind of cool. It's a little bit of a trivia kind of issue. But if you draw the line Y equals exon, and then you take a look, this line reflected over would hit that line, and vice versa. So you remember reflections from geometry. We have reflective symmetry, a clock across the line Y equals X, which is kind of neat. It's kind of neat. Not that important, but kind of neat. All right, I'm going to clear this out. So pause the video now if you need to. Okay, here we go. All right. Last problem. Now, there's a bit of a mathematical mistake in this problem, but I kind of want to let that go for the time being, okay? And I'll explain what that mistake is in a bit. Normally, I don't try to design mistakes in, but I kind of like them here. 

So it says a table of values is given for the quadratic function F of X equals X squared. All right? And then let ray says graph the inverse by switching the ordered pairs. So for instance, I've got negative two four. That's going to be four negative two. Negative one, one. We're going to go with one negative one. Zero zero. Oh, that's boring. One one, again, kind of boring. And two, four becomes four two. So let me graph that. One, two, three, four, one, two, one, negative one, zero, zero, one, one, and four, two. All right. By the way, continues to be symmetric across the line Y equals X says, what do you notice about the graph of the function's inverse? It is not a function. Right? If I look at that, it's not a function. Whoops, violates the vertical line test. So let's talk about what my mathematical error was. And let's try to do it with slightly better writing. Let's try to go that again. I shouldn't use function notation. Shouldn't use. 

Function notation. For a non function. I wanted to kind of do it anyway. I didn't want to use Y at that point. I guess I could have, but I really wanted to emphasize sort of the function notation. But yeah, it's not a function. Now, the original function was all set, FX equals X squared. Why did its inverse not turn out to be a function? Think about that for a little bit. Why did the inverse not turn out to be a function? Pause the video and think about that. Did you come up with it? Here's the big reveal. A function will have an inverse that's also a function if and only if it is one to one. Hence, a quick way to know that a function has an inverse that's also a function is to apply the horizontal line test. So this original function is not one to one. And because of that, it fails the horizontal line test. The cool thing is that it's inverse then, will fail the vertical line test. So that's important. The only types of functions that have inverses that are also functions, all functions have inverses. But the only ones that have inverses that are also functions are one to one. And that's the main reason why we're concerned with one to one functions in this course. 

There's a lot of other reasons one to one functions are important in math. But in this course, it's mainly because they have inverses that are also functions. All right. Pause the video now and copy down anything you need to and then we'll wrap up. Okay, here we go. Cleared out, ready to go. All right. So today we saw what were known as inverse functions. Functions that undo the rule that the original function did. And we'll be seeing inverse functions throughout this course. Certain functions are simply defined in terms of the inverses of others. So it's really kind of an important topic. All right. I'd like to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler and until next time, keep thinking and  keep solving problems.