Okay, let's see if we can understand functions in about seven minutes. And that's always my target goal with my videos here is to try to teach you something in about seven minutes. Sometimes, very often, I go a little bit over.
But in a short period of time, if you stick with me, you're going to learn a lot about functions. If you're struggling in this topic, don't feel bad. A lot of students struggle with functions. It's a huge topic in mathematics, and it's extremely important that you understand it. So.
Really the main goal for this video is just to get you to strengthen your fundamental understanding of what a function is. So that's kind of the objective here. But before we get going, let me go ahead and introduce myself. My name is John. I'm the founder of Tablet Class Math.
I'm also a middle and high school math teacher and I have many online math courses. I teach functions very in-depth in several of my courses, but I'm going to suggest that you check out my algebra course. Really get into functions heavily there.
I'm going to leave the link to that in the description of this video if you want to learn more, really master functions. But let's get into functions now. So here's an example of what a function looks like. So there's all types of functions. Again, it's a huge topic in mathematics.
But when you see a little notation like this, that indicates some sort of function. Now this... the way we state this or read this is f of x okay this is the name of this function and any function like so f of x is equal to the variable y all right so you can think of this is the function 3x squared plus 1 but it's equivalent to the equation 3x squared plus 1 okay so it's a little technical difference there but equations okay like this is a quadratic equation And we would refer to this as a quadratic function. Again, we say this as f of x is equal to 3x squared plus 1. But we could have all kinds of functions. For example, we could have something like this.
That would be g of x. That's the way we would state that. If I had this h of t, I would write it that way. So again, in mathematics, all sorts of functions.
But this is a real basic example to kind of stress what I want to teach you here. Okay, now what is a function? Well, a function is basically some sort of rule, okay?
A rule in the sense that we're going to input values into the function, all right? And then we have some sort of rule, okay? Now the rule basically says, okay, take your input values. The rule says do something with that value, like manipulate it in some way. OK, and when we're done with that, we're going to get some sort of output value.
OK, now the input value into a function, we're going to call this the domain. All right. So this word here. So all the numbers that we can put into a function is called the domain.
And then the respective output values is called the range. OK, so really important words again, depending on where you're at. With functions, you're going to study the domain and range at a much more advanced level. Again, functions are a huge topic, but let's just make sure you get the basics down in this video.
Okay, so let me go ahead and erase this. Alright, so we have the input, which is the domain, all the set of values that you're allowed to put into a function. It's called the domain. Okay, it's going to be associated with the x variable.
Let's go ahead and establish that. And then the output... is going to be associated with the y variable. You'll see that here in a second what I'm talking about.
Let's look at an example of how to evaluate a function. So let's go ahead and figure out what this function's value is at 2. So that means I want to evaluate, that's how we would state that, this function, we want to find f of 2. So here again is the input. So you can see I'm replacing this x with a 2. So over here in the rule part, I'm going to replace this x with whatever value I plugged in for this x here.
Of course, this is going to be 2 squared plus 1. So now at this point, I can go ahead and evaluate or finish evaluating this function by just being very careful. All of you out there are experts in order of operations. So you've got to be careful here because this is a common mistake where students make errors.
They'll go, oh, 3 times 2, that's 6, and they square that. Remember, you've got to be very careful with the order of operations. This is going to be 2 squared, so that's going to be 4. 3 times 4 plus 1, which of course is going to be 12 plus 1, or 13. Okay, so we have f of 2 was equal to 13. Now, again, let's go ahead and take a look at this function in a different way.
I want to really stress something here. So when we had the x... We had x equal to 2. Our y value is 13. Now, why is it the y value?
Remember, I said y is the same thing as f of x. So our f of x, our output, was 13. But you want to associate that with y. So whatever your y value here, let's just say y was 13 when x was 2. So you can construct a table like so, 2, 13. But But this corresponds to the ordered pair or point on the xy-axis to 13. That's a point. Okay.
Let's do it over here. Okay. So I have an X, I have a Y, let me erase this. So the point 2, 13, 1, 2, let's say it's up here. 2, 13 is on this function here, graphically speaking.
So we can look at functions in a variety of different ways. We can look at them in terms of their table, how they're represented, or sets of points, or algebrically by their actual function rule. So... This particular function happens to be a parabola. So it's going to start at one.
It's going to look something like so. OK, and all the points that are on this graph. All right. Are part of the domain and range. OK, now I don't want to go ahead and start taking this topic, you know, beyond seven minutes.
I'm probably already so beyond seven minutes right now, because again. Functions are a huge, huge part of mathematics. But before I end the video, let's talk about some other follow-on things that you're going to be learning with respect to functions. And each one of these are really important.
So first of all, you just have to understand what a function is. And oftentimes functions, let's just do this real quick. We have something in math called relations.
And a subset of relations are functions. So all functions are relations, but not all relations are functions. So one of the things that you need to understand about functions is knowing how to determine when you're dealing with a function or if something is a function or not. Let me state it that way.
That's a big part of what you should be learning in your math course. Another thing is function operations. So if I have two functions, let's say I have f of x as something and I have g of x as something else. Well, you can manipulate functions. You can add them.
You can multiply them. You can do all kinds of stuff. You can do what we call composite function.
All right. So F of or G of F of X can do all sorts of stuff. You can also find function inverse. All right. Which is going to be another huge topic.
So, again, many, many facets of functions. That's why, you know, it doesn't make sense for me to try to squeeze this in. In seven minutes, I would just. me uh i don't even know if i could do it but what i wanted to do was to stress the fundamentals of what a function is and even you know i could even taken more of my time to do that but let's go and wrap up this video hopefully this kind of um was a good little refresher for you um again uh at this time i've been on youtube for a good 12 years i have literally hundreds and hundreds of math videos posted all the time so hopefully you consider subscribing if you like this video definitely appreciate a thumbs up And leave me some feedback. Obviously you're interested in functions.
What math course are you taking? Algebra, Algebra 2, College Algebra. It's taught at many levels, even in calculus, pre-calculus. Excuse me.
So any feedback is good feedback. And I'm going to leave a link to my algebra course in the description of this video. Now I teach functions thoroughly in all my algebra courses. If you're in another course like Algebra 2 or something different, you can check out my course catalog and you can kind of go to that particular course. But if you want to know everything I was talking about with functions, you can definitely find it in my Algebra course.
But with that being said, I definitely wish you all the best in your mathematics journey and have a great day.