[SOUND] So, who knows what a function is, right?
But I know what it does. It takes an input value, and produces an
output value. And we've got a whole bunch of functions,
right? And we can take these functions and start
asking questions about them. What happens when you plug in a really
big number, or a really small number? Or, or what happens when you plug in two
numbers that are nearby each other? How are the outputs related, right?
Those are the kinds of questions that are going to occupy us for the rest of the
term. But even before we start thinking about
questions like that, right? There are some things that we can still ask about
functions. Like, how do you know when two functions
are the same function? For instance, here's two functions.
f(x) = (1+x)^2, g(x) = x^2 + 2x + 1. Are these the same function?
Now, let's try. Look at the value like f(2).
f(2) = (1+2)^2, start replacing the x by two,
1 + 2 = 3. 3^2 = 9.
Well, what's what's g(2)? Well, g(2) would be 2^2 + 2 * 2 + 1.
2^2 = 4, 2 * 2 = 4 + 1, 4 + 4 = 8 + 1 = 9.
Look, f and g, when I plug in x = 2 give me the same output value of nine.
And that should be a little bit surprising, right?
Because the way that f and g are telling me to compute their output is totally
different. f takes the input two, adds one to it and
squares it to get nine. g takes two, squares it, doubles it, adds
those two numbers to one, to get nine. So, the method by which f and g are doing
the calculations is totally different, right?
This sequence of operations is not the same as this sequence of operations.
The, the rules are different. And yet, look at this.
f(x), for any value of x, right? Is 1 + x * 1 + x,
right? That's 1 + x^2. Well, I could expand this out, right?
1 * 1 + x, and then x * 1 + x. I could combine some of these terms,
right? 1 + x + x = 2x. x * x = x^2.
Look, 1 + 2x + x^2, that's g(x). this is really quite surprising.
f and g don't compute their output in the same way, right?
This one is doing something different than this function, and yet, for any
input value, f's output value is this, which is the same by expanding out as
g(x). Now, how we're going to deal with this?
We're going to say that f and g are at the same function,
right? Not because they have the same rule,
right? But because for every input value, they
have the same output value. Here's a much more subtle example.
Again, I got two functions. f is defined like this.
f(x) = x^2 / x, and g is defined like this,
g(x) is just x, the identity function. Same question, is f the same as g?
Are these the same function? Now, they're not the same rule, right?
This is not the same as this. So, you know, it's a little more subtle,
you know? But that's okay, right?
Two functions are the same if they have the same output for each input.
So, let's see if that happens here. let's just pick some value to get a first
test. Let's take a look at f(5), right?
f(5) would be 5^2 / 5, that's 25.
5^2 / 5, that's 5. Well, that's the same as g(5), right? If
I plug anything into g, I just get the same thing out.
So, plug in five, you get five. So, at least at the value five, f and g
agree. You might think this always works,
right? Because of something like this.
You might want to say, well, f(x) that's x^2 / x,
no matter what x is. You might rewrite this x^22 as x * x / x.
And then, you'd be tempted to say, cancel one of these xes with the x in the
denominator. And then, you'd write equals x.
And x, well that's, that's g(x). So, this looks like a pretty convincing
argument, right? Over here, I've got f of x,
I've got a bunch of equal signs. And over here, I've got g(x).
So maybe that means F and G are the same function.
Ha, but not so fast. What happens if you plug in zero?
What's f(0)? Well, I know what g(0) is.
g(0) is zero, right? Zero is in the domain of g because zero
makes sense for this rule. But, what's f(0)?
Well, that would be zero squared over zero,
whoa. Okay.
You see this is terrible, right?
I cannot divide by zero. This rule, x^2 / x doesn't make sense
when x is equal to zero. So, zero is not in the domain of f,
but it is in the domain of g. So, I'm going to say that these are not
the same function. They don't have the same domain,
right? f isn't defined at zero,
and g is defined at zero. In that sense, these are really different
functions. This example suggests that there's a real
richness to this theory of functions, right?
And we're going to be studying it a lot more this term.