 So how do we write about functions? So remember that a function is a relationship between variable. We have some set of input variables and an output variable. If our output variable is named f and our input variables are x, y, z, and so on, we use the notation f, parentheses, x, y, z, and so on, to indicate that we have a function of our variables, x, y, z, and so on. We say that x, y, and z are the inputs or arguments or independent variables, and the value of f is the output or function value or dependent variable. So for now, we'll focus on functions whose inputs are real numbers and whose output is also a real number. These are called, wait for it, real functions. Most real functions rely on several variables, but calculus with more than one variable is hard. So we begin by studying real functions of one variable. So with our notation, with one input variable, which we'll call x, and output variable f, we use the notation f, parentheses, x, which we read as f of x. Now there's three important things to remember about this notation. First, f of x does not mean f times x. Second, f of x does not mean f times x. And most importantly, f of x does not mean f times x. For example, let h of t be the height in meters of a balloon t minutes after it's released. Let's interpret h of 10 equals 40. So paper is cheap. We'll write down our definition of the function. And what we have is h of 10. So what we have has replaced t with 10. So we replace t with 10 everywhere. I mean the variable t should be replaced with 10 everywhere we see it. And so we get the next statement. h of 10 is the height in meters of a balloon 10 minutes after it's released. Now we're told that h of 10 equals 40. So equals means replaceable. So we'll replace h of 10 with 40. And we get our statement 40 is the height in meters of a balloon 10 minutes after it's released. Now this is a valid interpretation of our notation, but we'll rewrite this so it sounds more natural. And so we might say something like this. Well, that would have been natural for Newton. However, we might say something like this. Now it's possible that we might be given a formula for a function. So f of x equals x squared plus 3x plus 5. Let's evaluate f of 2 and f of 2x. So there's many ways we could evaluate a function. But success in mathematics and in life often relies on how good our bookkeeping is. So one way of keeping the bookkeeping organized is the following. To evaluate a function f of x, replace every occurrence of x with an empty set of parentheses. And then, whatever goes in one empty set of parentheses should go in all the empty sets of parentheses. So if I want to find f of 2, well, paper is cheap, so let's copy down our function definition. We'll replace every occurrence of x with an empty set of parentheses. And then, whatever we put in one set of parentheses should go in all of them. What we want to have in here is 2. So we'll put a 2 in here and in all the other places. And at this point, there's some arithmetic we have to do, so we'll do that. And equals means replaceable f of 2 is equal to 15. And if I want to find f of 2x, well, again, I'll copy down my function definition. I'll drop every occurrence of x and replace it with an empty set of parentheses. What I want in the first set of parentheses is 2x, and that should go in every empty set of parentheses, leaving me with a little bit of algebraic cleanup to do. As an applied example, suppose the population of a city is given by some function, and we want to find the population of the year 2025. So again, success in math and in life is largely based on how good you are at bookkeeping, how good you are at keeping track of things. So it helps to write down the meaning of our dependent and independent variables. So P of t is the population of a city t years after 2000. And just like we did when we evaluated a function, we can replace the independent variable with an empty set of parentheses. So we want to find the population of the year 2025. So we'll let t equal 2025. And again, whatever goes in one set of parentheses should go in all of them. And this is what we want, the population of the city, 2000 and 25 years after the year 2000. Maybe not. That would tell us the population of the city in the year 4025, which is not what we actually want. What we really need is P of 25, the population of the city, 25 years after 2000. So I want to evaluate P of 25. So let's copy down our function definition. We'll drop every occurrence of our independent variable and leave an empty set of parentheses. And what goes in one set of parentheses has to go in all of them. So what we'd like inside the set of parentheses here is 25. So we'll put 25 in all places. And at this point we have some arithmetic to do. Finally, it's a good idea to give your answer the same dialect that the question was asked in. So the question was, find the population of the year 2025. This is a complete sentence in English. And so we should write our answer as a complete sentence in English. The population will be 625,000 persons.