Functions are going to be the main star 
of the course. 
So we should be building up in sort of a 
repertoire or library of functions that 
we might be interested in studying. 
Here's the first function in our library 
f(x) = x, 
the identity function. 
Whatever you plug in, this function 
outputs that same thing. 
Here's another function, 
a constant function. 
You pick some number c, 
c stands for constant. 
Alright? 
And you can define this function, 
f(x) = c. 
Whatever you plugin for x, f just ignores 
that and then outputs the original value, 
c. 
Here's a function, f(x) = 3x + 2. 
And if you're thinking about stuff like 
that, why not stuff like this? Pick two 
numbers, a and b, and then you can define 
a function like this, f(x) = ax + b. 
You can think about fifth f(x) = x^5 or 
nth power, f(x) = x^n for some fixed 
value of n. 
And about polynomials, like this 
complicated looking polynomial, f(x) = 
2x^3 + 5x^2 - 2x + 1. 
If you're thinking about polynomials, you 
might want to think about roots, 
f(x) equals, say the square root of x. 
You might remember the absolute value 
function f(x) equals the absolute value 
of x. 
You might have some experience with trig 
functions, 
like sines, cosines, and tangents, 
or with other transcendental functions 
like logarithms and exponentials. 
So now we've got our small library of 
functions, 
the identity function, constant 
functions, polynomials, some trig 
functions, 
and I want more functions. 
I, I want some way to be able to take two 
functions and produce a new function out 
of them. 
Okay. 
So in this setup I've got a conveyor belt 
and I've got two functions, 
a function here and a function here. 
Let's pick out what these functions 
should be. 
Maybe the first function I'll call f and 
f(x) would be 2X + 1. 
So I'll call this function f and maybe 
the second function I'll call g and g 
will take its input and square it, so 
g(x) will be x^2. 
So I'll label this function, g. 
And now here, I've got a number 3 and I 
am going to run that number through the 
first function and whatever comes out of 
the first function, I'm going to plug in 
to the second function to see what comes 
out. 
So let's take that number 3, let's start 
moving the conveyor belt. 
It's going to go through the function f, 
f(3) is 2 * 3 + 1 to 6 + 1, which is 7. 
So now we've got a 7 right there. 
So the 3 went into the function and came 
out as a 7. 
Now I'm going to take the output to f and 
put it in to the input of g. 
So g(7), well it's going to be 7^2 and 
that'll be 49. 
So here now, coming out of the function 
g, is the number 49. 
And I could have written this in a little 
bit a little bit of a shorthand way. 
I could have just written g(f(4)), 
right, f(3) is 7 and g(7) is 49. 
So once I've got this sort of conveyor 
belt metaphor going on in my head, I 
could do the following trick. 
I can take two functions. 
I can take the output to the first 
function and plug it in to the input of 
the second function.