[MUSIC] Calculus study functions and this 
course is no exception. 
We're going to be studying functions in 
this course, alright? 
Functions like f of x or the sine 
function or the square root function. 
And that raises a question what are 
functions?If If we're going to be 
studying functions, we should know what 
they are. 
And that question is pretty metaphysical. 
I mean what are numbers? 
What are numbers for? 
I mean that question doesn't make a lot 
of sense does it? 
I, you know, it's not like a number four 
is a physical object that I can go look 
at it and see if it's got polka dots or 
it's striped or something, 
right? 
But I know four objects when I see them, 
right? 
I sort understand what numbers do more 
than what they are in some metaphysical 
sense. 
The same is true about functions, 
right? 
I'm not really going to tell you what a 
function is. 
What I'm going to tell you is what a 
function does. 
This is what a function does. 
A function assigns to each number in its 
domain another number. 
And this definition doesn't say anything 
about how the function does that 
assignment. 
Let's see an example. 
Just make up some example. 
Suppose I've got some function, I'll call 
it f, f for function. 
Maybe this function assigns to the number 
two, the number four. 
So, I'll write f(2) is 4 or f(3) is 9 Or 
f(4) is 16 of f(5) is 25. 
I'm just making this up. 
This is a function. 
And I'm telling you what number it 
assigns to each number, 
right? 
f assigns the number two, the number 
four. 
f(3) is 9, right? 
So, f assigns to the number three, the 
number nine. 
Well, look, 
I'm not going to just list off every 
single assignment that f makes. 
So instead, one way to talk about these 
assignments is to use a rule, like f(x) = 
x^2. 
And this single rule explains how all of 
these assignments are made, 
alright? 
This rule said that f assigns to the 
number x, the number x^2. 
So in particular, f assigns the number 
five, five squared or 25, or f assigns 
the number four, four squared which is 
sixteen or f assigns the number three, 
three squared which is nine, alright? 
So, a lot of times, when you actually 
want to talk about how these assignments 
are being made, use some sort of rule 
like this: and you write f of x is 
something, to compute the output value, 
right? 
So, a function assigns to each number in 
its domain another number. 
One way to do that is with a rule. 
Of course, this definition of function 
involves another word, domain. 
What is a domain? 
Unless, I say otherwise, the domain 
consists of all numbers for which the 
rule makes sense, 
right? 
A function assigns to each number in its 
domain another number. 
And the domain is just the numbers that I 
can plug in. 
Let's take a look to see what I mean by 
this. 
Suppose I've got a function f(x) = 1 / x. 
So, that's a function given by a rule. 
It takes an input x and produces the 
output 1 / x, 
right? 
It assigns the number x, the number 1 / 
x. 
But this rule doesn't always make sense, 
right? 
I'm dividing by x, 
alright? 
And to divide by x, what do I need? 
Well, 
I must have that x is not zero, 
not permitted to divide by zero. 
So, I can plug in any number for x except 
for zero. 
And that's when this rule makes sense. 
So, I'll summarize that by saying that 
the domain of f is all real numbers 
except zero. 
So, a function takes some input and 
produces some output. 
That's what it does. 
But how is that supposed to make us feel, 
you know? 
How are we supposed to imagine that? 
Well, here's one metaphor that you could 
use to try to think about what a function 
is actually doing, right? 
You could imagine a, a conveyor belt with 
the function, 
you know, 
And you could imagine the numbers coming 
in. 
Boom. 
Being hit by the function and then going 
out transformed somehow, by whatever the 
rule is of the function. 
Here's what I'm talking about. 
I've got a conveyor belt here, I got this 
big box, and imagine if that big box is 
the function. 
And in here, I've got the input, imagine 
the number five. 
Let's see what this machine is going to 
do. 
Maybe this is the function f(x) = x^2 + x 
and I've got this number five here. 
And I'll start the conveyer belt going so 
the number five starts moving through the 
machine and when the machine is done 
processing it, it comes out the other 
side and now, it's the number 30, 
alright. 
Because f(5) is 5^2 + 5, which is 25 + 5 
is 20. 
So, we have seen how you can write down a 
function by using a, a rule involving 
these, these mathematical symbols, 
right? 
x^2 + x. 
You can also write down a function just 
using English words. 
So, let's see an example of that, just 
make up a new function here. 
I'll call it R(x) and I'll define · R(x) 
to be thrice, that means three times, 
thrice the square root of x. 
And here, I've computed some, some values 
of the function, you know, like the 
function that four is six. 
Well, why is that? 
Well, R(4) would be three times the 
square root of four, which is three times 
two which is six. 
Okay. Now, of course, when I do the 
calculation this way, you know, it's not 
too surprising that instead of writing 
this out in words, thrice the square root 
of x, 
I mean I could have just written three 
times the square root of x. 
So, maybe you're not too impressed with 
this. 
The point is that you can define a 
function just by writing down what the 
function is supposed to do using English 
words. 
So, we've seen an example of a function 
that we defined just in terms of algebra, 
f(x) = x^2 + x. 
I've also seen an example of a function 
that I define entirely with just English 
words. 
We can kind of combine those two things, 
alright? 
We can use the English to sort of pick 
out different kinds of of algebra. 
Let's see an example of that now. 
So, here's another function I just made 
up. 
g(x) is x^22 if x is bigger than or equal 
to five. 
And twice x if x is less than five. 
The point here is that I'm using just a 
little bit of English. 
So, this word, if, in order to select 
based on how big x is, 
a different algebraic rule for 
calculating the function and this is a 
little abstract. 
Let me just do with some calculations and 
that might convince you, you know, how, 
how this notation is, 
this so-called piecewise notation works. 
So, what is g(1)? 
Well, let's take a look. 
One is less than five so that means I use 
this second rule for calculating g, 
so it's two times one which is two. 
What's g of, say, four? 
Well, four is still less than five so I 
use the second rule again. 
Twice the input, two times four and that 
means the output is eight. 
But what is, say, g(5)? 
Ooh. 
Five is not less than five. 
Five is greater than or equal to five. 
So, this if is telling me to use the 
first of the two rules for calculating g 
so I'm going to use five squared as the 
output and that will be 25. 
Or g(10). 
Well, ten is definitely bigger than or 
equal to five and it's bigger than five. 
So, I again use the first rule and the 
output to this function is computed as 
ten squared or 100. 
So, this is, you know, really more 
complicated than just using some algebra, 
right? 
I'm using these if statements to select 
which of these two rules g will use to 
compute its output. 
in principle, functions can be really 
complicated. 
I mean, all the examples we've seen are 
just doing various kinds of algebra. 
I mean, maybe different algebra, 
depending upon which value of x I'm 
plugging in. 
But it's all, you know, adding, 
subtracting, multiplying, dividing that 
sort of thing. 
But you can do really complicated things 
with, with functions. 
let's see an example of something that is 
you know really different than doing 
straight up algebra. 
I'll do a much crazier example. 
Alright. 
C(x) is the number of even digits in the 
number x, when x is a whole number. 
It is zero if x isn't a whole number, so 
otherwise. 
And it is a really different and crazy 
function, right? 
So, C of, say, 7236 is two, and why is 
that? 
This is a whole number so I use the first 
part of the rule. 
And the number of even digits in this 
number, 
well, I count them. 
That's odd, 
even, 
odd, 
even. 
So, two and six are the two even digits. 
The value of that function is two. 
Well, let's take a look at this number 
here, 60,202. 
That again is a whole number, so the 
value of the function at 60,202 is the 
number of even digits in x. 
And this number has how many even digits? 
One, two, three, four, five, they're all 
even digits so that value is five. 
Here's another example. 
Here's another 5-digit number. 
53,531. 
Again, it's a whole number so I use the 
first part of the rule, the number of 
even digits. 
I count how many of these digits are 
even. 
Five is odd, odd, odd, odd, odd. 
There are no even digits. 
So, the value of this function is zero at 
this point. 
Now, if I plug in a number that's not a 
whole number like this, this is not a 
whole number, then I use the otherwise 
part of the definition. 
And the function at that point is just a 
zero. 
So, we can use English to define a 
function but you got to be careful, 
alright? Sometimes, when you write 
another definition of a function, you 
might write down something that's more 
ambiguous than, than you intended. 
So, let's try to define a function. 
Just making this stuff up, I'll call the 
function B(x) for bad. 
And I'll say that the value of B(x) is 
some rearrangement of the digits of x. 
And it's okay that I'm using English to 
define my functions. 
There's nothing wrong with that. 
But this definition is too ambiguous to 
be the definition of a function. 
here's a definition of what goes wrong. 
What is the function's value at 352? 
Well, the function takes its input and 
rearranges the digits somehow, alright? 
So, you might think that the functions 
value at 352 is 325. 
Because 325 is some rearrangement of the 
digits of 352, right, I took the five and 
the two, swapped their positions. 
But then, the functions value at 352 
should also be 235 because 235 is also a 
rearrangement of 352. 
The function's value with 352 should also 
be 532 because 532 is some rearrangement 
of the digits of 352. 
This is terrible, 
alright? 
A function is suppose to take its input 
and produce unambiguously a single output 
value. 
But this so-called function takes this 
single input value and purportedly 
produces all these possible outputs. 
This thing here is not a function, 
alright? 
A function takes one input and produces 
one output. 
[MUSIC]