Int Algebra: Function Notation and Values

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Relations and Functions
An Introduction to Functions
Function Notation and Values Page [1 of 2]
Okay, so we have a sense of what a function of x means. It means you put in something for x, boom, and out pops an
exactly one value for y at most. But now how can we write this thing, this kind of object, in an algebraic way, so if you
want to actually plug in numbers and something you can actually do it very, very accurately rather than just trying to
estimate on a picture or something? Well, it turns out that one way of doing it is, as we’ve seen, we write x = 2x2 - 1.
But suppose that I say to you, “Okay, now I what I want you to do is find out what the y value is, like when x = 2.”
What would you do there? Well, you’d plug in 2 and you’d get 22, which is 4. You multiply that by another 2 out here
in front, and you get 8, you subtract 1, and you say, “Oh, it’s 7. y = 7.” So when x = 2, y = 7.
But look how difficult it was. It was almost like a mathematical tongue twister for me to actually say that question to
you, or ask you that question. In particular, I’d have to say the following. What is the value for y when x = 2? Well,
that was a long statement just to say, “Plug in 2 for x and see what you get.” So, in fact, mathematicians have come
together and realized that there’s a way of sort of saying that in a very short, short condensed manner. So what I’m
going to do now is give you a little inkling into the little window into the world of mathematics and the language of
mathematics. No new ideas, just new notations, new language.
Okay, so let’s think about what’s going on here. Remember, we’re thinking about this as a little machine here. You
think of it as sort of a machine. I’ll call it f for function, and what you do is you input something--in this case you input
x values--so you input some x value, and then you input the x value, it goes through this thing, and then something
comes out, just like what happens here. In this case it would be the following: I would take a number, square it,
multiply by 2, and subtract 1. That’s the little machine that’s going on doing the work, and then it spits out something.
And what it spits out is exactly some y value. So that’s what we have here. We have this input. It goes around, does
something, comes out here.
Now if I call that machine f for function, well then what could I say this target thing might look like? If I put in an x, let
me just say the output will be f(x). So that’s just the y value that I’m going to spit out. Now, it’s just notation. Don’t
think this is anything more than just a notational thing. But what it’s saying is the following: f(x)--that’s how you read
that, by the way, if you see this in a children’s story, you read this as “f of x.” You don’t say “f parenthesis x.” You say
“f of x.” And what does it mean? It means f must be some sort of function machine, and this entire thing here, that
number, is the number you get after you plug x through the machine. So you push x into the machine, and the answer
is f of x. So, in fact, I could now write this as f(x) = 2x2 - 1.
So these two things are saying the exact same thing, but this is my new notation. Instead of writing y, I’m going to
write it as a function of x where I’m very explicit, so y is a function f(x). So what that means is if you input a value for x
in here, to find out what this equals, I just plug it into here. So let me actually show this to you with very specific
examples. It’s one of these things that you first look at and you go, “I don't know what’s going on here,” but then once
you start seeing it again and again, you see it’s not a big deal.
Let me show you how to read this notation. If I write f(2) what does that mean? Well, that means the following. I
want to know the value of this function machine if I plug in 2 as the input for x. So all I have to do--it’s actually really
pretty neat--is go back to here and wherever I see an x, I’m going to insert 2. So I just do is sort of blindly. Wherever I
see an x, I plug in 2. And now I see what that equals. Well, now I can compute that. That’s going to be 22 is 4,
minus 2 is 8, minus 1 is 7.
So, in fact, look--remember before when I said the original question--find the value that x equals when x = 2. Well,
now all I can say is “find f(2).” You see how easy that is? It’s just notation. But what this says is find the value of this
whole thing if I plug in a 2 for x. That’s all the notation means. Let’s do a whole bunch more.
What does f(1) mean? Well, I go back to the function, and wherever I see an x I’m going to replace all x’s by just
whatever’s in this thing. In this case it’s 1. So I see 2 x 12 - 1. Well, if you compute that, it’s 1. I’m evaluating the
function in a really, really easy way. I’m just plugging in whatever’s in this parenthesis wherever I see x. So, for
example, what would f(0) be? Well, wherever I see an x, I plug in a 0. So I put a 0 in here; 02 is 0 x 2 is 0 minus 1 is -
1. So once you start getting the hang of it, not a bad thing.
Relations and Functions
An Introduction to Functions
Function Notation and Values Page [2 of 2]
Let’s try a really hard one. Now, this one actually really freaks people out, so I’m actually going to pose this to you
and let you try to answer this one yourself. But if you get freaked out, don’t worry by the way, you are with every
single other human being who has first seen this thing. But see if you can do it. See if you can defy the odds and
figure out what would this equal--f(a)? Try it right now.
Okay, well, if we keep our wits about us, all we know is the rule that wherever I see an x I’m going to replace it by
whatever’s in here. So since I see an a, all I’m going to do is wherever I see an x I’m going to put an a. So I see 2a2 -
1. Not a big deal. So I admit, I don’t know what number that is, but that’s the answer. That’s what the thing equals.
Okay, let’s try another example. How about if I give you this function--g(x) equals
3
1
x +
. So there’s a new function.
Now, let’s evaluate this in a few places. What’s g(2)? Well, g(2)--wherever I see an x I’m just going to insert a 2. So
that’s 1/5. See, once you get the hang of this, this really is sort of a piece of cake. I mean, you could put anything in
here. For example, suppose I want to know what is g of rubber ducky? Well, it doesn’t make a difference what I put
in here, whatever I put in here, I’m just going to replace all the appearances of x by that thing. So in this case I see
3
1
rubberducky +
. You see, it’s really just a substitution. This is just a symbol which means to find out where this
function sends x, you just do this. So if I put in a 2, you put in a 2 for all the occurrences of x. If you see a rubber
ducky, you put a rubber ducky in wherever you see an x. That’s all there is to it. Let’s try one last one. h(x), a
different function. By the way, notice I don’t always have to call them x, by the way. That’s another thing I wanted to
point out of you. You don’t always say f(x); you can call it anything. It’s just the name of the little machine. So maybe
one’s called f, one’s called g, one’s called h. It’s nothing at all complicated. Once I write g(x) = this, that means that
I’m just calling this whole thing g. It’s the g machine. The g machine in this case is
3
1
x +
. The f machine from earlier
on was 2x2 - 1, and so on. The h machine here--these are just literally random letters --is the square root of x2 - 1.
Notice, by the way, it’s not plus or minus the square root, it’s just positive square root, so it is a function.
Now, what would h(1) equal? Well, wherever I see an x I’m going to plug in 1. So I see a 12, 12 is 1 minus 1 is 0.
This is the square root of 0, which equals 0. What is h(2)? h(2) would equal the square root of--I’d put in a 2 here. So
I’d see 22 - 1, which equal 4 -1, which is 3, so the square root of 3. So this notation just allows us to quickly figure out
what values equal without writing out, “Find the value for y in this expression if we let x = 2.” Instead, I just say h(2),
and h(2) means exactly the same thing that I just said, “Find the value for y when x = 2.” So this function notation just
allows us to get rid of a neat, mathematical tongue twister.