 One of the most important skills you can have as a math student is the ability to write a function. Now if the universe were a kind and gentle place, functions would fall out of the sky. And indeed, sometimes you're given a function. But most times you have to construct the function. And here it's especially important to keep in mind, don't try to learn math by looking for examples of solved problems. This is especially true when trying to construct a function. Every problem is different. And what that means is that the skill of constructing a function can't be taught. The good news is it can be learned. And there are some strategies that will help you. One way to construct a function is to try and generalize the process of finding its values. So let's take a simple example. Express the area of a rectangle as a function of a type H and with W. So first, we need to come up with a clever name for a function that gives us the area. We'll use F because F is the first letter in the word area. What's that? Not by your spelling? Ah, okay. We'll use A because A is the first letter in the word area. It actually doesn't matter if we call it F or A, but it is helpful when our function names have some vague connection to what they're supposed to be. More importantly, since we want area to be a function of height and width, we have to express this as A of H, W, where our notation says that our area is going to be somehow related to H and W. And again, success in life is largely based on bookkeeping and similarly with mathematics. So what we'll want to do is we want to make sure to identify what our variables mean. So we have our function A of H, W. H is the height of the rectangle and W is the width of the rectangle. So there's many things we could do at this point. The worst way to learn mathematics is to look for examples of solved problems. And in this case, the problem of finding the area of a rectangle as a function has already been solved. You can look up that area formula. But don't do it. One way we can try and figure out what that relationship is is that we can set up a table and find the areas for different values of H and W. And here's a useful idea. It helps if you don't do the arithmetic. So for example, if I have a height of 5 and a width of 3, then my area is going to be 5 times 3. And again, we're not going to do the arithmetic. With height 2 and width 10, the area is. With height pi and width squared of 11, the area is. And fourth times the charb, what if we use our variables H and W? Well, looking at what we did in all of the other cases, we conclude that our area is going to be H times W. And so that gives us our area function. Let's see how this works in a different case. Here we want to write down a function giving the perimeter of the rectangle in terms of its height. So we'll come up with a clever name for our function. We'll call the perimeter P and indicate that it's a function of the height H. And so our function is going to be P of H, where H is the height. And again, let's see if we can set up a table to find our perimeter. So I'll pick a value for the height and find a value for the perimeter. So if the height is 5, the perimeter is... Well, that looks like it's going to be a little bit hard to calculate. If I want to find the perimeter of a rectangle, it's helpful if I also know the width. So I'll add in an extra column here. Now notice that our perimeter is supposed to be a function of the height only. It is not supposed to be a function of the width. Still, it's helpful to have this column even if it doesn't show up in our final step. It's important to remember paper is cheap. Write stuff down. So the thing to remember is that we are told that the width is 3 more than twice the height. So if we know what the height is, we should be able to calculate what the width is. So let's pick a value for the height. How about 5? We like 5. If the height is 5, then the width will be twice the height plus 3. And again, it helps not to do the arithmetic, so we'll leave this as 2 times 5 plus 3. How about the perimeter? The perimeter of a rectangle will be twice the height plus twice the width. And so that will be... And again, we're not going to do the arithmetic. Well, let's pick another value for the height. How about 3? If our height is 3, then the width will be 2 times 3 plus 3. And the perimeter will be... And again, we won't do the arithmetic. And if we pick another value, how about 10, then the width will be... And the perimeter will be... And finally, let's try to generalize if our height is h, then the width will be... And the perimeter will be... And so our perimeter function looks like it's going to be 2h plus 2 times 2h plus 3. And we could do a little bit of algebra to clean this up, but we don't have to.