 When we write an expression like f of x equals blah blah blah, the blah blah blah tells us what we're supposed to do with the independent variable. So for example, if we write f of x equals 2x, we mean take x and multiply it by 2. Or if we write g of x equals square root of x plus 7, we mean to take x, add 7, and then take the square root. Or if h of x equals this horrifying looking thing, well that means we're going to do some very complicated things to x. And there are three important things to remember about function notation. First, f of x does not indicate the product of f and x. Even more importantly, f of x does not indicate the product of f and x. But the most important thing to remember about function notation is that f of x does not indicate the product of f and x. And this leads to the following problem. To evaluate a function for a given input, rewrite the function replacing every occurrence of the input variable with an empty set of parentheses. And what goes in one set of parentheses should go in all sets of parentheses. So for example, let's say we want to evaluate f of x equals 3x minus 7. We want to find f of 7. So to begin with, let's rewrite our function. But we'll drop every occurrence of x and replace it with an empty set of parentheses. So paper is cheap, so let's copy our function down first. And now we'll rewrite dropping every occurrence of x and leaving behind an empty set of parentheses. So f of blank is equal to 3 blank minus 7. Now we want f of 7. And so that means we want a 7 inside this set of parentheses. So we'll put it in. But what goes in one set of parentheses should go in all sets of parentheses. So that means in this set of parentheses we should also have a 7. So we'll write one in. And we can compute this value to get our actual function value. This is going to be true no matter what our function looks like. So let's have this function. We want to find g of 0.3. So paper is cheap, let's copy down our function first. So again let's rewrite our function and we'll leave behind an empty set of parentheses. So that's g of blank is square root 12 minus 5 blank. Again we've dropped every occurrence of x and left behind an empty set of parentheses. What goes in one must go in all. We want a 0.3 of this first set of parentheses, so we'll write it. And we've got to put a 0.3 in all the others. And we'll finish off with some arithmetic. The important thing to remember is that this is true regardless of what we want to put in that first set of parentheses. So here we have h of x is 1 over 2x plus 7. We want to find h of x plus 2. So we'll rewrite our function leaving an empty set of parentheses any place we see an x. And again if it goes in one it has to go in all of them. So I want an x plus 2 in this first set of parentheses, so I'll write it there. And it's got to go in all the others. f of t is 2t plus 7, let's find f of t squared. Now since we're using a variable t this is totally different from everything else we've done. Well, actually it isn't. We're going to start out the same way. We'll drop our variable t and leave behind an empty set of parentheses everywhere. So f of blank is 2 blank plus 7. I want a t squared in the first set of parentheses, so I'll put it there. And if it goes there it goes everywhere.