 This video is going to talk about functions and function notation. A relation is a set of ordered pairs. It could be any kind of ordered pairs. It could be X and Z Ys. It could be first names and last names. Anything that you can make an ordered pair out of. Function is every input has only one output. It's a relation because they're ordered pairs, but every input can only have one output. So if we look at this first example, these are students, and over here we have their degree that they're going for in college. So Elizabeth is going for education, Andrew is going for engineering, Dave is going for biology, Angie is going for business, and Mark is going for music. Now each one of those is only going for one degree, so that makes it a function. When we come over here to this other example, this is the birthday month, and this is the birthday day. In my family, one of us has a birthday on April 28, one of us has a birthday on May 10, one of us has a birthday on December 3, another one of us has a birthday on April 8, and one of us has a birthday on May 4. Now you can see that when you look at April, went to two different outputs, and when you look at May, the same thing happened. We have two birthdays in May. So this one is not a function, because those inputs of April and May had more than one output that was related to it. Now if you're looking at a graph, one way that you can tell whether you have a function or not is to do the vertical line test. Vertical line test says that if I move this vertical line anywhere across my graph, and it crosses my graph more than once, then it's not a function, because if you see this vertical line right here, this point right here, and this point right here, are on that vertical line. The input here looks like it's three, but it has two different outputs. It has an output of three, and it has an output of negative three. So it's not a function. If we take this vertical line and move it over here to this graph, and I move it all the way across my graph so far, everywhere I've moved it, it only has one place where my blue line is crossing my vertical line here. So this one is a function. So again, this one is not a function, and the other example was a function, because it passed the vertical line test. So function notation is often used when we're talking about functions. So f of x is the same thing as y. They are interchangeable. We've been looking at equations that are y equal mx plus b, and those are linear functions in reality. So we could say that f of x is equal to mx plus b. Just a different way of writing the same thing. f is the name of the function, and then inside the parentheses indicates the input. If it's a variable, then we know what our variable on the other side of the equal sign is going to be. Over here it's x. Or it might have a value in there, and then it tells you what to input for x. So let's suppose that we have y equal 2x minus three. Well, we can also write that f of x is equal to 2x minus three. So if we want to evaluate that f function when x is equal to negative one, it would look like f of negative one is equal to, and then wherever I see an x, I'm going to replace it with negative one. So f of negative one is equal to 2 times x, which is now negative one, minus three. So it says, what do you notice about the number in the parentheses and the x value? They are interchangeable. Every x becomes the number inside the parentheses. And what is f of negative one equivalent to? Well, again, that's going to be two times negative one, minus three. And two times negative one is negative two, minus three. So it would be negative five. So let's try it. Here we have a function x squared plus two. It's the g function. And we want to know when g is zero, what is equal to? So we would come in here, and we have an x to begin with. So we would put zero, and that's squared plus two. And zero squared is just zero, plus two will give us two. Now we're going to put in our negative two, because that's our x now. And we have to square it, because that's what the function told us to do to x, and then add two. Well, that's going to give us negative two times negative two would be four, plus two, and that's then going to be equal to six. Try again. G of negative one. Well, it's negative one, and we're going to square that, and then we're going to add our two. Well, if we do that, negative one times negative one is positive one, plus that two, and we can see that g of negative one is equal to three. One more. G of ten. So we have our ten as our x. We're replacing this x up here with ten, and then it tells us that we have to square that, so we will square it, and then add two. Well, ten times ten is one hundred, plus our two, so g of ten is equal to a hundred and two. This line is the f of x line. It's the equation that graphed it was the f of x function. And we want to find f of one. Well, we're going to use a graph to do that. Remember that in here is our x value. It used to be an x, but these are interchangeable. So the one is the x value. So I come over here to the one on my graph, and I go until I can hit my graph, and I find out that this is the point right here. I already know that x is one, so what's the y value that is across from my point, and it's equal to two. Now I want to do f of negative two. Again, this is my x, so I come over to negative two in the x, and then I go up to hit my graph and find out where I hit it. If I look across over here, I'm hitting it at y equal five, so f of negative two is equal to five. We can also use function notation and talk about a table. This is the h function, and we could say here y is equal to h of x if we really wanted to. And it's asking me to find h of negative two. Well, this is my x, so I go to negative two as my x, and I find out that it's equal to 9.8. Now you can have function notation that says, let's say, when is h of x equal to negative one? Well, I have an x in here, and I really need to remind myself is that h of x, function notation, is the same thing as saying y is equal to negative one. So I come over here, and I can see that when y is negative one, my answer is h of x is equal to negative one when x is equal to positive one. So let's use this and see if we can write a function and then use our function with our problem. So Lois has a new job with a math instructor at a local community college. Her starting salary is $41,500 with a regular teaching load of 30 credit hours. If she teaches beyond the regular load, she earns $950 per credit hour. So it wants us to write the linear function f of x for her yearly income where x is the number of extra credit hours she teaches. So she earns $41,500. That's the given. But if she works over that, she's going to add to that $950 for every credit hour she works extra. So it would be 950 times the number of hours that she works over. B then says if Lois teaches two additional four credit hour courses, what is her income? Well, two four credit hours is going to be how many hours? It's an x, and that means that she's going to have eight credit hours. So it's an x, so it's really asking us for f of eight. And how do we do that? Remind yourself it's $41,500 plus $950, but now instead of writing x, we write the eight that was in parentheses. Well this then gives us $41,500 and $950 times eight would be $7,600 and if we want to solve that then, we would say that we had $49,100 income. Well f of six then is the next problem that they're going to ask us. So f of six just means again this is x equal six credit hours. So plugging and chugging f of six is equal to that $41,500 plus the $950 times our six. Well working along this is $41,500 and $950 times six is $5,700 and if we add those all together we get $47,200 in income for working six extra credit hours.