College Algebra: Identifying Functions

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03/23/2011

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Very easy to understand. In your face teaching and easy to understand. Thank you!!!

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Okay, so we have a sense of what a function is--it's basically a little machine where you take an input and you spit out one particular output. And we've also seen how you can recognize whether something's a function or not by just looking at its picture. So if you look down here, for example, is this a function? Well, we can give it the old vertical line test and make sure that for every value of x there's at most one value of y. So we use the vertical line test to see if the curve only hits the vertical line at most one point, and you can see it's always one point. And what exactly does that mean? It means that for this x value there's only one y value and it's that one, so it's a function of x. Notice, by the way, if I do this, just make it a little more interesting. That's even a more interesting curve; it's an even more interesting graph. Is it a function? Oh yeah, it's a function here. Sure, no problem here, no problem here, no problem here. Okay, now there's a problem, because you see, for this particular value of x, if I were to plug it into the machine, what would it spit out? Well, on the one hand it should spit out that number, but on the other hand it also has to spit out that number. So there's not one value--there's no one unique value of y that will be spat out by this particular value of x. That means this is not a function of x. And you see if I keep going it gets even worse. Look at that. For this one value of x, this curve has three values for y. One, this height, two, that height, and three, that height. So this is definitely not a function of x, because in fact, this fails the vertical line test. As I bring my vertical line through there are points where the vertical line crosses at more than one point. So that's what it means graphically to be a function of x or not.
But what does it mean algebraically? What if someone gives you an expression and asks you, "Hey, is this a function of x or what?" Well, how would you resolve that question? Well, it turns out that one way of resolving it is just to look at the formula or the equation they give you, and ask yourself can you turn that into a function machine, which means can you solve for y and have y equal stuff with just x in it?
Let's take a look at a particular example and you'll see exactly what I mean. Let's suppose we have 2x + 3y and that equals 7. The question is, is this a function of x? Well, what does that mean? It means, can I take this formula and convert it into a little machine where I input x and it spits out one value of y? If the answer is yes, we know it's a function of x. If the answer is no, we know we can't do it. So let's see, how would I do that? Well, the trick would be to solve for y. See if you can write y just in terms of x. That means that x is the only input that you need to spit out one y value. So take this and ask, can I solve for y. Well, sure. I'll take this 2x and bring it to the other side by subtracting it from both sides, so I see 3y = 7 - 2x, and if I divide through by 3, I would see y = 7 - 2x all divided by 3. So is y a function of x? The answer is, yes, it is. And why? Well, because if you give me any one x value at all, any one x value, you see, I can plug it in right here and compute, and that gives me one particular number, right? If you say, "I'm thinking of a number," like maybe you're thinking of -2. So if I put a -2 for x, this gives me a -2 x -2 is 4; 4 and 7 is 11; 11 divided by 3 is 11/3. So I have 11/3. That's one number. So when I plug in -2, this machine will spit out 11/3. So the point is, it's always one number per one value of x. So when you look at this you can see this is really a function of x, so the answer is yes, it's a function of x.
Okay, let's try another example. How about this one? y = x^2 - 4. Well, I want to ask can I solve this for y, but notice it's already solved for y. y = all this stuff. If I give you any x value will this always represent one unique value? The answer is yes. Because if you think of any number at all, if you square it you get a unique number, right? For example, here, let's say 5. When you square it you just get one answer--25. Now I could subtract off 4 and get 21, and that's one value. So when I plug in 5, this thing spits out 21. So, in fact, what I'm seeing here is that this is really a machine where I input one value for x and one value of y comes out. Another way of thinking about it algebraically is I was able to solve this for y, so this really is a function that just depends upon x, the function of x.
Okay, let's try another example. -x + y^2 = 2. Okay, is this a function of x? So what do I do? I want to know can I solve this for y. So let's try to solve it for y. If I bring this x over to the other side, I would add it to both sides, and so I'd see y^2 = 2 + x. And now to get rid of that square what I could do is take plus or minus the square root of both sides. So what I could see here, if I were to keep the action going, I would see y = + or - the square root of 2 + x. Now, is that a function? It looks that way because I have y equals stuff with x. But let's just think about the input/output thing. Suppose, for example, that I put in 7 for x. If I put in 7 for x, would there be one number that would be spit out? Let's see? 7 and I add 2, that's 9. The square root of 9 is 3. But I have + or - 3. So, in fact, the output would actually be two values, +3, -3. That's actually two values. It's not just one value associated with x = 7, so this is not a function. This is not a function. Right? Because for some values of x, we actually have two values for y. It turns out, though, this is a function of y, which means that I could actually solve this for x if I wanted to and see that actually x is a function, which depends only on y. But it's not a function of x. I can't solve this for y and get one answer. So, in fact, this one is not a function. Now, graphically, by the way, you can graph all these things and you'd be able to see that with a vertical line test. Let me show you that really, really fast here.
This first thing that we saw, which was a function, it turns out if we graph it, to be a straight line, and you can see that by noticing that we just have x's and y's to the first power and we saw already that if you just plot some points you'll be able to see what that looks like, and the line, roughly speaking, will look something like this. I'm not drawing this at all to scale, but the idea is it's sort of a downward-sloping line. We'll talk about this more in detail later. But anyway, you can see it does pass the vertical line test; only touches at one point at most.
This next one, I remind you that, in fact, when you see an x^2 that's going to be some sort of parabola-like thing, so in fact, this is going to be sort of like a parabola, and it'll be somewhere out in space, but it'll look roughly like this shape, and you'll notice again, it's a function. It passes the vertical line test, and that's what we found.
Well, what about this one? Well, actually, this has a square, too, so it's also a parabola, but since it's y^2, what we actually have here is it's a parabola that opens this way. And you can see now that the plus square root wing is actually this positive part above the x-axis, so y = positive square root of x is this top part; it's all this part up here. That's the positive square root of x. And the minus square root of all that stuff is way down here. So actually, when you put it all together though, you can see it's not a function, because it fails the vertical line test dramatically. Boom--there's not one y value; boom--there's not one y value; boom--they have two y values. So since this fails the vertical line test, this curve cannot be a function of x. So there's the idea of how to figure out if a particular algebraic expression is a function or not of x. If you want to find out if a function is a function of x, see if you can take it and solve for y just in terms of x and have one answer. One answer; not many answers. If you get one answer, then y is a function of x. If you get many answers, then y is not a function of x.
Okay, now on with that idea. What I want to do next is actually take a look at some new notation, just a little bit of new language that will make it easier for us to study and write out functions. I'll see you there.
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Identifying Functions Page [2 of 2]