 Calculus is the Latin word for a small pebble, and computations were once done by setting down piles of pebbles, which is why we use the term calculus for any sort of computational mathematics. So let's talk about function calculus. The idea here is that it's possible to combine several functions into a single function. So for example, if f of x equals 2x, and g of x is x plus 5, then I could write down f of x plus g of x equals means replaceable, so I can replace f of x with 2x, and g of x with x plus 5, and get my sum f of x plus g of x. Likewise, I could write f of x minus g of x, f of x times g of x, and so on. We'll introduce the following notation. If f and g are functions, and c is any real number, then c f of x is c times f of x, well, that's hardly earth-shattering, f plus g of x, well that's the notation we use to mean f of x plus g of x, f minus g of x, well that means f of x minus g of x, f g of x, that's f of x times g of x, f over g of x is f of x over g of x, and then this last one, and then there's this last one where we use this special circle symbol. This is called the composition of the functions f and g, and we sometimes read this as f of g of x, and this is what happens when we apply our function f to the function g of x. We sometimes use the notation f of g of x for clarity. For example, let f of x equals x squared, and g of x be x plus five, let's find f plus g of x, and f over g plus two f of x. So f plus g of x is f of x plus g of x, so equals means replaceable, so I'll replace f of x with x squared, and I'll replace g of x with x plus five, and so my function f plus g of x is x squared plus x plus five. So f over g plus two f of x, that's a quotient of two functions f of x and g plus two f of x. g plus two f of x is a sum g of x plus two f of x, and equals means replaceable, so every place I see f of x, I'll replace it with x squared, every place I see g of x, I'll replace it with x plus five. So there are no real tricks when we multiply a function by a constant, or we add, subtract, multiply, or divide functions, but what about this composition of functions? So given our functions f of x and g of x, let's find f of g of x and also g of f of x. And it helps to drop out the independent variable and leave an empty set of parentheses, so we'll rewrite both of our functions this way. So f of g of x is f applied to g of x. So remember our f function is f of squared, and what we want in the parentheses is g of x. Well, what goes in one set of parentheses should go in all of them. So we'll put a g of x in both our sets of parentheses, and equals means replaceable, g of x is x plus five, so we'll replace that. And that gives our composition f of g of x. As for the other g of f of x, well, that's g applied to f of x. Our definition says g of is plus five. What goes in one set of parentheses should go in all of them. We want an f of x there, so we should put an f of x in all of them. And equals means replaceable, f of x is the same as x squared.