Math 100 lesson 81 is introduction to functions. So before we talk about a function, we need to know what exactly is a relation. A relation is a set of ordered pairs, so a relation usually comes in these little braces symbol, and so I might have the order pair negative 2315. 0 three.
And 611 so that represents a relation. It is a set of ordered pairs. Set notation is those little braces symbols. The domain is the X values of those ordered pairs, so the domain of this relation would be negative two zero one and six. Now I got those numbers from the.
values and I put them in order least to greatest. I'm not sure that my man lab cares if you put them in that order. It's just a good habit and the range would be the Y values, so those would be 3, 5 and 11. And so those Y values there they are 3, 5 and 11. Now notice there were two threes.
but I only put one of them in my range because you don't have to say it again. All you're saying is what numbers are the y values. Well they are 3, 5, and 11. Even though the 3 appears twice, we do not write it as in the range twice.
All right and then here we have our little idea of a function. What exactly is a function? It is a relation.
Okay so that would be a set of ordered pairs. Where no x values repeat. Okay, so if I look at the relation that I have listed here, all of my x values, negative 2, 1, 0, and 6, are different.
So they are not repeating. So this would in fact be a function. This relation represents a function, all right?
If I gave you the ordered pair, I'm sorry, the relation that looked like this. Okay, so I've got a relation again. And if I look at my x values, they are 1, negative 1, 1, and 5. Now, notice that there are 1s here and here. So my x repeated, which makes this not a function. It is still a relation, it just does not represent a function.
Sometimes my math lab will throw out some words for you, so I just want to make sure you know what they are. The independent variable and the dependent variable. So the variables that we're used to working with are x and y. The independent variable is x. That's the one that can stand all by itself.
The dependent variable is y. In other words, you can't have a y unless you have an x. Now function notation looks like this, and the way that we read that is f of x.
It doesn't mean multiply. It means The value of the function at the number x, whatever that number may be. Alright, so f of x means the value of the function at the number x. All right, so let me give you a potential problem that we could have. So it might say something like just find each value.
All right. Alright, so I've asked you to find the value of this function at 6. Okay, value of the function at 6. For the function that I have described for you. So what we are going to do is we are going to take 6 and we are going to plug it in where we see X in our function. OK, so my function at 6 is 4 times 6 plus 5. OK, 4X means 4 times the variable X.
So 4 times 6 I would get 24 plus 5. and i get 29 so the value of my function at 6 is 29. function notation doesn't have to just be f it could be g or h it changes that that does that is irrelevant it is still function notation All right, so this time we are going to take negative 5 and put it in place of x in our function. So the function at negative 5 is 3 times negative 5 squared. The negative 5 is what is squared. Please make sure the squared is on the outside of the parentheses there. It matters.
So now order of operations. I am going to do negative 5 squared and I get positive 25. Now multiply 25 times 3 and then we subtract our 10. So our function at negative 5 is 65. Right now, sometimes they do crazy things and they don't ask us to find the value at a number. They asked me to find the value at letters.
So our process is going to be the same. We're going to take a plus H and plug it in where we see X. In our function, so my function at a plus H. is equal to 6 times a plus H. Plus 9. I put a plus H in place of X.
So now I will distribute this 66A plus 6H plus 9. There are no like terms, so my function at a plus H. Is 6A plus 6H plus 9. Alright, I like to call those plug and chug types of problems. Alright, sometimes we will be given a table and then ask some questions about our table of values.
So you don't have to know where this table is coming from. It is something that will be given. It's not always this table, but it is something.
Alright. And then we'll answer some questions directly from here. So we're on question number four.
Our first question is, does the table represent a function? Or does it, it won't say represent. I think in my math lab it will say, does the table define a function?
Well. We said earlier that in order for it to be a function, no x values could repeat. So if we look at our table and we look at all of our x values, they are all different. Negative 2, negative 1, 0, 1, and 2, they are not repeating. So in this case, yes, because my x values do not repeat.
So it makes it a function. all right our next question would be you know what let's find the domain and the range of the values in this table well our domain remembers our x values so negative two negative one zero one and two And our range is our Y values. I like to go least to greatest, so 0, 1, 3, 4, 5. Those would be our range values.
Number six, if I said... Let's find the value of this function at negative 1. So that means that negative 1 is our x value. So x is negative 1. So we come over here to our table and we go, oh, x is negative 1. The value of the function at negative 1 is 0. In other words, the y value that goes with that is zero. Let's do another one like that.
So what is the value of the function at zero? So we're going to go look at our table. Remember, that means x is zero.
So we come over here to our table and we find x is zero. And the value that goes with it is 3. Alright, one other type of question. We're still going to use this same little table to answer our question.
Find x such that f of x is 4. Now this time it's not telling me that x is 4. All right, it says f of x is 4. So I come over here to my table. Let me choose a color. Come over here to my table and I find where f of x is 4. Let me slide our table down so we can see it a little bit better.
All right, so f of x is over here this column on the right. So f of x is 4 right here. Right, f of X is 4 right there.
So they said find the X value that goes with that. So in this case, the X value that goes with that would be X is 2. Just reading our little table and that is really all there is to Section 1. Kind of a short section. Section 2 is also a fairly short section so.
Make sure that you practice doing the homework and look for quizzes and we're getting really close to having the being at the end of the material for exam one. So I look forward to seeing you in a team's meeting.