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Functions are going to be the main star 
of the course. 

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So we should be building up in sort of a 
repertoire or library of functions that 

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we might be interested in studying. 
Here's the first function in our library 

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f(x) = x, 
the identity function. 

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Whatever you plug in, this function 
outputs that same thing. 

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Here's another function, 
a constant function. 

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You pick some number c, 
c stands for constant. 

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Alright? 
And you can define this function, 

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f(x) = c. 
Whatever you plugin for x, f just ignores 

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that and then outputs the original value, 
c. 

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Here's a function, f(x) = 3x + 2. 
And if you're thinking about stuff like 

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that, why not stuff like this? Pick two 
numbers, a and b, and then you can define 

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a function like this, f(x) = ax + b. 
You can think about fifth f(x) = x^5 or 

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nth power, f(x) = x^n for some fixed 
value of n. 

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And about polynomials, like this 
complicated looking polynomial, f(x) = 

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2x^3 + 5x^2 - 2x + 1. 
If you're thinking about polynomials, you 

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might want to think about roots, 
f(x) equals, say the square root of x. 

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You might remember the absolute value 
function f(x) equals the absolute value 

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of x. 
You might have some experience with trig 

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functions, 
like sines, cosines, and tangents, 

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or with other transcendental functions 
like logarithms and exponentials. 

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So now we've got our small library of 
functions, 

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the identity function, constant 
functions, polynomials, some trig 

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functions, 
and I want more functions. 

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I, I want some way to be able to take two 
functions and produce a new function out 

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of them. 
Okay. 

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So in this setup I've got a conveyor belt 
and I've got two functions, 

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a function here and a function here. 
Let's pick out what these functions 

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should be. 
Maybe the first function I'll call f and 

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f(x) would be 2X + 1. 
So I'll call this function f and maybe 

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the second function I'll call g and g 
will take its input and square it, so 

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g(x) will be x^2. 
So I'll label this function, g. 

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And now here, I've got a number 3 and I 
am going to run that number through the 

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first function and whatever comes out of 
the first function, I'm going to plug in 

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to the second function to see what comes 
out. 

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So let's take that number 3, let's start 
moving the conveyor belt. 

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It's going to go through the function f, 
f(3) is 2 * 3 + 1 to 6 + 1, which is 7. 

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So now we've got a 7 right there. 
So the 3 went into the function and came 

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out as a 7. 
Now I'm going to take the output to f and 

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put it in to the input of g. 
So g(7), well it's going to be 7^2 and 

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that'll be 49. 
So here now, coming out of the function 

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g, is the number 49. 
And I could have written this in a little 

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bit a little bit of a shorthand way. 
I could have just written g(f(4)), 

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right, f(3) is 7 and g(7) is 49. 
So once I've got this sort of conveyor 

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belt metaphor going on in my head, I 
could do the following trick. 

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I can take two functions. 
I can take the output to the first 

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function and plug it in to the input of 
the second function. 

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