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[SOUND] So, who knows what a function is, 
right? 

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But I know what it does. 
It takes an input value, and produces an 

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output value. 
And we've got a whole bunch of functions, 

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right? 
And we can take these functions and start 

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asking questions about them. 
What happens when you plug in a really 

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big number, or a really small number? 
Or, or what happens when you plug in two 

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numbers that are nearby each other? 
How are the outputs related, right? 

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Those are the kinds of questions that are 
going to occupy us for the rest of the 

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term. 
But even before we start thinking about 

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questions like that, right? There are 
some things that we can still ask about 

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functions. 
Like, how do you know when two functions 

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are the same function? 
For instance, here's two functions. 

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f(x) = (1+x)^2, g(x) = x^2 + 2x + 1. 
Are these the same function? 

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Now, let's try. 
Look at the value like f(2). 

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f(2) = (1+2)^2, start replacing the x by 
two, 

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1 + 2 = 3. 
3^2 = 9. 

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Well, what's what's g(2)? 
Well, g(2) would be 2^2 + 2 * 2 + 1. 

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2^2 = 4, 2 * 2 = 4 + 1, 
4 + 4 = 8 + 1 = 9. 

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Look, f and g, when I plug in x = 2 give 
me the same output value of nine. 

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And that should be a little bit 
surprising, right? 

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Because the way that f and g are telling 
me to compute their output is totally 

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different. 
f takes the input two, adds one to it and 

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squares it to get nine. 
g takes two, squares it, doubles it, adds 

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those two numbers to one, to get nine. 
So, the method by which f and g are doing 

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the calculations is totally different, 
right? 

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This sequence of operations is not the 
same as this sequence of operations. 

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The, the rules are different. 
And yet, look at this. 

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f(x), for any value of x, right? Is 1 + x 
* 1 + x, 

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right? That's 1 + x^2. 
Well, I could expand this out, right? 

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1 * 1 + x, and then x * 1 + x. 
I could combine some of these terms, 

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right? 1 + x + x = 2x. 
x * x = x^2. 

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Look, 1 + 2x + x^2, that's g(x). 
this is really quite surprising. 

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f and g don't compute their output in the 
same way, right? 

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This one is doing something different 
than this function, and yet, for any 

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input value, f's output value is this, 
which is the same by expanding out as 

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g(x). 
Now, how we're going to deal with this? 

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We're going to say that f and g are at 
the same function, 

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right? 
Not because they have the same rule, 

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right? 
But because for every input value, they 

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have the same output value. 
Here's a much more subtle example. 

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Again, I got two functions. 
f is defined like this. 

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f(x) = x^2 / x, 
and g is defined like this, 

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g(x) is just x, the identity function. 
Same question, is f the same as g? 

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Are these the same function? 
Now, they're not the same rule, right? 

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This is not the same as this. 
So, you know, it's a little more subtle, 

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you know? 
But that's okay, right? 

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Two functions are the same if they have 
the same output for each input. 

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So, let's see if that happens here. 
let's just pick some value to get a first 

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test. 
Let's take a look at f(5), right? 

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f(5) would be 5^2 / 5, 
that's 25. 

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5^2 / 5, that's 5. 
Well, that's the same as g(5), right? If 

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I plug anything into g, I just get the 
same thing out. 

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So, plug in five, you get five. 
So, at least at the value five, f and g 

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agree. 
You might think this always works, 

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right? 
Because of something like this. 

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You might want to say, well, f(x) that's 
x^2 / x, 

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no matter what x is. 
You might rewrite this x^22 as x * x / x. 

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And then, you'd be tempted to say, cancel 
one of these xes with the x in the 

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denominator. 
And then, you'd write equals x. 

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And x, well that's, that's g(x). 
So, this looks like a pretty convincing 

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argument, right? 
Over here, I've got f of x, 

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I've got a bunch of equal signs. 
And over here, I've got g(x). 

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So maybe that means F and G are the same 
function. 

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Ha, but not so fast. 
What happens if you plug in zero? 

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What's f(0)? 
Well, I know what g(0) is. 

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g(0) is zero, right? 
Zero is in the domain of g because zero 

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makes sense for this rule. 
But, what's f(0)? 

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Well, that would be zero squared over 
zero, 

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whoa. 
Okay. 

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You see this is terrible, 
right? 

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I cannot divide by zero. 
This rule, x^2 / x doesn't make sense 

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when x is equal to zero. 
So, zero is not in the domain of f, 

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but it is in the domain of g. 
So, I'm going to say that these are not 

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the same function. 
They don't have the same domain, 

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right? 
f isn't defined at zero, 

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and g is defined at zero. 
In that sense, these are really different 

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functions. 
This example suggests that there's a real 

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richness to this theory of functions, 
right? 

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And we're going to be studying it a lot 
more this term. 

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