 So an important skill in learning higher mathematics is the ability to read and express things in function notation. Function notation arises in the following way. Suppose f is a method that allows us to use a quantity x to determine a quantity y. We can express this idea in a couple of different ways. We write f takes x to y, or we write f of x is equal to y. And the second is what we usually mean when we refer to function notation. And something important, f of x should never, never, never, never, never, never be read as f times x. For example, let p of t be the population of a town in thousands of persons t years after 1950. Interpret p of 50 equals 350. So paper is cheap. It helps to copy down our definition. The problem tells us that p of t is the population of a town in thousands of persons t years after 1950. Now, since we have p of 50, we can replace the t with 50. So I have p of 50 equals the population of a town in thousands of persons 50 years after 1950. And since we also know p of 50 equals 350, we can replace p of 50 with 350. And this gives us one possible interpretation. And if we were a machine, we might read it like this. 350 is the population of a town in thousands of persons 50 years after 1950. But since we're a human being, we should rewrite this more colloquially. So instead of saying 50 years after 1950. We might say instead in 2000. And instead of saying 350 is the population of a town in thousands of persons. We might just say the population of the town was 350,000. We can also go the other way. So if I want to express the idea that in 2017 the population was 450,000. It helps to write down our definition. And if we want to write this in function notation, our starting point should be figuring out what t is equal to. And since t is the years after 1950, we observe that 2017 will be 67 years after 1950. So that means t equals 67. Now the thing to remember is that when we write something equals something. That means that every time we see the one, we can replace it with the other. So I'll replace t with 67. Well we can't just do a find and replace. The t we're replacing with 67 is the t that represents the number of years after 1950. And at this point we can make the observation that we have is the population of the town in thousands of persons 67 years after 1950. But we know the population 67 years after 1950 was 450,000. So since they're the same thing, we can replace the one with the other. And the only really tricky part about this is remembering that when we write down this number, this is a number in thousands. We're going to write 450 because that's the number of thousands. Now we can also talk about function expressions. A function can be given as a formula. For example, and here's a very important idea for later on. The type of function is determined by the last operation performed. So in this first expression, we have x squared. We have 7 times x and then we have a whole bunch of addition. And the order of operation says that the addition is done last. And so this first function is a sum. Meanwhile in this expression, we do the stuff inside the parentheses first and then we multiply the two factors together. And so this is a product. And finally for this horrible expression here, after all the dust settles, the last thing that we do is divide numerator by denominator. And so this is a quotient. We can also consider evaluating functions. If the function is given as a formula, we can use it to evaluate a function value. For example, if f of x equals 3x plus 7, then our formula tells us to take x, multiply it by 3, and then add 7. And what's useful to recognize here is that the x is a placeholder and whatever it is that's going to be done to x, that's what our function should do to the input value. For example, suppose our function is 8 minus x squared. We want to find f of 3 and f of 2x. Now one way we can evaluate f of x is to do what I call drop x. And what we're going to do is we're going to drop every appearance of x and replace it with an empty set of parentheses. And then whatever goes in one set of parentheses must go in all sets of parentheses. So here if I want to find f of 3, first of all I'll copy down the function definition. f of x equals 8 minus x squared. Next I'll drop every appearance of x and replace it with an empty set of parentheses. And here's the important idea. Whatever I put in any of these parentheses is going to be put in every one of these parentheses. Since I want to put a 3 inside this set of parentheses, I'll put a 3 inside the other set of parentheses as well, and this gives me an arithmetic expression I can evaluate, which gives me a function value of negative 1. This particular approach is important if we're going to evaluate f of something other than a number. So for example, if I want to find f of 2x, I'll do the same thing. I'll drop out every appearance of x and replace it with an empty set of parentheses. I want a 2x in here, so I'll put it in there, and everywhere else. I see an empty set of parentheses, and a little bit of algebra gives me my final expression. Now in the real universe, sometimes we may need to use two or more different formulas to find our function values, and this gives us what we might call a piecewise function. So here, f of x is going to be, well, it depends. If x is less than 5, we'll use the formula 3x. If x equals 5, we'll use the value 7. And if x is greater than 5, we'll use the formula x squared plus 3. And now if I want to evaluate the function, I want to use the relevant formula. So if I want to find f of 3, well, x is equal to 3, and 3 is less than 5, so we want to use the first formula. f of 3 is 3 times x, which is going to be 9. If I want to find f of 5, I have x equal to 5, so we use the second formula, which tells us that f of x is going to be 7. If I want to find f of 7, I note that x is equal to 7, and 7 is greater than 5, so we use the third formula, and that gives us our value 52.