Start  So, let's start talk about how we might represent functions.
So we often write or call our function by the letter f and we write that f of x equals x squared plus 1 (x^2 +1), for example. This is the name of funcion. This is our sample input. And this is how we compute the output. 
0:40  So the way that we would deal with this function like x^2 +1 is by we replace x with what our input is. So in this case f(3) would be 3^2 +1 which is 9+1 which is 10. But we don't have to 
1:04  be so restricted as to just put 3 and there we can put anything in there. Now we could say this is exacly the same funcion if we compute f of y. This is the same f just operating on a different letter instead of operating on x it's going to operate on y, and so it would give me y^2 plus 1. Now, sometimes students get a little bit annoyed because they 
1:33  think functions should always be called "f" and the inputs should always be called "x", but we can put anything in there. We can also compute what f(x+y) would be. And that would be whetever we see the x in our function we replace by (x+y). So that would be (x+y)^2 + 1 which we can leave this way or we can simplify: (x+y)^2 is (x+y) times (x+y) which would be 
2:13  I get x^2 by multiply this guy to this guy and xy from this times that and I get
another xy from this times that, and I get y^2 here, plus 1.
So, we can do all sorts of various thinks with plugging in 
2:34  different inputs it's still the same function. Sometimes we might represent our function with a longer name. So for example the sine function and let's use theta for our letter instead of x. So this represents, this is the number which is the sine so let's call 
3:14  the number y so that y is the sine of the angle theta for example the sine of 45° is 
3:43  the square root of 2 divided by 2. So again, the notation here is that the name of the function is on the outside, and the stuff in the parentheses is what goes inside. Let's do one more example 
4:01  which is slightly more complicated but it is still the same idea just to make sure you got it down. So one more example here. Suppose I have a function g(x) which is: x squared minus 2 x (x^2 2x) and I wanna figure out what g(x+2) should be. So here 
4:46  everywhere I seen an x  because this is just the place holder remember everywhere I see an x, I put an (x+2). So this would be (x+2)^2  2(x+2). In that case that x^2 +4x +4 
5:09  because that's what (x+2)^2 works out to be; and then minus 2 times x, minus 2 times plus 2 is 4. And I can clean that up if
you want: as x squared plus well there are 4x and minus 2x, which is 2x, and the fours cancel.
