Common Core Algebra II.Unit 2.Lesson 2.Function Notation

Hello and welcome to another common core algebra two lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit number two lesson number two on function notation. So function notation is something that you saw back in common core algebra one, but you probably didn't use that much in common core geometry. Maybe a little bit here and there with parabolas in the coordinate plane. But at the end of the day, what we'd like to do is we'd like to review this very important notation with you because you're going to be seeing it for the rest of this course. So let's jump right into it. Now, as we all know, functions are rules that convert inputs into outputs. So function notation is a lot of time written as Y equals F of X, we say it as F of X F of X and basically it's got everything there, right? You've got your input. You've got your rule. And you've got your output. One thing that should really be noted in all of this is that a lot of times we think of the function and the output as being interchangeable with one another because that's really what equality means is that the two things are the same. Why and the function really are the same thing. All right, even though in a certain sense, the function is itself the rule and I guess Y is the consequence of the rule. Okay? So I'm going to clear out that text. And then let's jump in and actually do some problems involving function notation to remember just how it works. All right, exercise number one. It says evaluate each of the following given the function definitions and input values. So in letter a, it says F of X equals 5 X minus two. What's F of three? Well, what they're really saying is that the rule is take X multiply it by 5 and subtract two to get the output. So F of three means take three, multiply it by 5, get 15, subtract two, and get an output of 13. So F of three is 13. Right? Why don't you do F of negative two? All right, let's do it quickly. So again, all it's saying, no multiplication here. Well, I mean, there's multiplication in the function itself. But all this thing is really doing is saying substitute in an input of negative two and tell me what the output is. And in this case, the output would be negative 12. Let her be in letter C are very similar. The only difference between them is that the functions are a little bit more complicated. Pause the video now and see what you can do, try to do it though without your calculator, okay? To help you review some important things. All right, let's take a look at G of three. Now G of X is X squared plus four. So G of three will be three squared plus four. Don't forget your order of operations. We'll do that exponent first. And we'll find that G of three is 13. Oh, there's a little bit of a coincidence. G of zero will be zero squared plus four. Zero squared is zero. And therefore we get four. All right, let's take a look at this one. This is two to the X, not two times X, so H of three will be two to the third, two times two times two. Gives me an output of 8. This may have been the hardest one, two to the negative two. Did you remember that negative exponents mean division? Right? So H of negative two is one fourth. We'll work more with negative exponents later on in the course, but hopefully you remembered a little bit about them. All right, so real simple. Function notation allows us to give the rule for the function, whether it's an equation or graphical form. We'll see that in a little bit. And then ask what the output is if we're given a particular input. All right, pause the video now, write down anything you need to. Okay. Clearing out the text. Let's deal with a table that's a function given in table form. Remember functions come in three primary forms, equations, graphs, and tables. I love tables. Here we've got boiling water at 212°F is left in a room at 65°F, and it begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature T is a function of the number of hours H and that makes sense, right? If I put in the time, the number of hours it's been cooling, there is only one temperature that the water can be. Look how easy it's going to be to evaluate the function. T of two, well, that's just saying, what is the temperature when the input is two? Well, the temperature is a 104. Likewise, T of 6 is the temperature when the input is 6. And that's 68. Now, what's interesting about letter B is it says, for what value of H is T of H 76, notice here what I mean given is the output, and I want to know what the input is. That's always a little bit trickier, but with a table. It's awesome because I scan along here. I find an output of 6 and therefore my input is a equals four. Make sure to put H equals four there. Let her see between what two consecutive hours will T of H be a hundred, right? Again, here they're giving me the output one of the things that's tricky about tables, though, is I don't see a 100 in there. But I do notice that since this is cooling down, I would have to hit a hundred some time in here. So it must be between the hours of two and three that I hit that 100° temperature. All right, pretty easy, pretty fast. I love tables. Pause the video now if you need to, and then we'll move on. Okay, here we go. Let's take a look at a function in graphical form. Remember, these function rules, equations, graphs, tables, any of the above. Now, this might be a little bit trickier, but keep in mind that the input are X values. So this, this, and this are all X values. And the outputs are a Y values. So if I want F of negative one, what I'm going to do is come over to an X value of negative one and the output is zero, the Y value is zero. On the other hand, if I come over to an X value of one, now my output is equal to four. Likewise, when I have an input of 5, one, two, three, four, 5, one, two, three, negative four. It is exceptionally important for you to be able to do this. For you to immediately know that F of 5 is a Y value. What's F of zero? Well, that means the input has to be zero. And the output, F of zero is three. And F of zero, always. Is the Y intercept. All this is the Y intercept. Letter C what values of X solve the equation F of X equals zero? What special features on a graph does the set of X values that solve F of X equals zero correspond to? But you have to watch out here a little bit. This one could be easily confused with letter B oh, I got zero. Got to put zero in. Oh, no, no, no. This is the output. So the output is Y equals zero. Let me circle those in red. That's here. Here, and here. So the values of X are X equals negative one, positive three, four, 5, 6, 7, and positive 7. These are known as the zeros. Or X. Intercepts. Finally, letter D says between what two consecutive integers does the largest solution to F of X equals three lie. Now again, that's the output. It's the Y value. The easiest thing to do is to graph Y equals three. So let me do that. I'm going to do it in red. One, two, three. I'm going to graph that. Here's Y equals three. And here is my largest solution. It's not exact. It's not like an exact integer. So one, two, three, four, 5, 6, 7, 8, must be between 7 and 8. All right? The other two locations where F of X equals three. X equals zero and X equals two. Well, those are easy. But that one's kind of tricky. All right, pause the video now, and then we'll clear out the text. Okay, here we go. Nice little multiple choice question very, very important. It says for a function Y equals G of X, it's known that G of negative two equals 7, which of the following points must lie on the graph of G pause the video for a second and think about this for a moment. I'm hoping it's obvious, but it may not be. All right. Well, I'm hoping you all kind of looked at that and said, that's just silly. That's easy. I mean, he's telling me that's the input. Which of course is X this is the output. And that's why. So the point negative two comma 7 choice two has to lie on that graph. And that's very, very important to be able to tie in function notation with what's going on with the function's graph. All right? I'm going to clear this out and pause the video if you need to. Okay. All right, last problem. Physics students drop a ball from the top of a 50 foot high building and model its height as a function of time with the equation H of T equals 50 -16 T squared. By using tables on your calculator determined to the nearest tenth of a second when the ball hits the ground. Provide tabular outputs to support your answer. All right. Well, let's bring out the TI 84 plus. Okay. Well, let's go into Y equals and put this equation in. Now, it's not going to be a little bit annoying because in Y equals, I'm going to have to put it in as 50 -16 X squared, but we're at the algebra two level. I bet we can handle it. So let's put that in. Clear on anything you might have and Y one, Y two, et cetera. And then in Y one, let's type this in. 50. Minus 16 X squared. Now, I should set up my table. Let's go into table setup right above a window. I'm going to start my table at zero, zero seconds. I'm going to make it go by ones, just to see what happens. You know, it might be this takes a long, long time to hit the ground. Let's go into our table. All right. Now how do we interpret this? Well, I mean, obviously at zero seconds, the ball is 50, what is it? 50 feet above the ground. At one second, it's at 34 feet above the ground, but look at this at two seconds, right? We're at negative 14, which means we're below the ground according to the equation. So at some point, between I guess X equals one and X equals two, we must hit the ground. But we want it to the nearest tenth. That's not good enough. So let's go back into my table. Actually, back into my table, set up my apologies. And let's start our table now, not at zero, but at one. And instead of making my table go by one, so I'm going to make you go buy one tenth. All right, let's go back into the table. Now, as we kind of scroll down what we now see is that there at 1.7 were above the ground because our H is positive. And there at 1.8 were below the ground because the height is negative. Now, which one is closer? Well, then it really, it's really a question of which one is closer to zero, right? Is the height at 1.7 closer to zero? Or is the height at 1.8? And it's 1.8. Not by a lot. But somewhat. So 1.8 is our final answer. All right. So what I'm going to do is I'm going to get rid of the calculator. We don't need it anymore. Bye bye. And now I'm going to clear out the text. All right. Let's finish up. So today what we did is we looked at what's known as function notation. Now sometimes function notation will be used when functions are out and about. And sometimes function notation won't be used. But when it is used, we have to feel confident and comfortable with what it means. All right, lots of practice on this as we move forward. For now, let me just 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, thank you solving problems.