There’s always a billion and one ways to think about something. So here’s a fresh perspective on something basic: a function.
A function, according to any English dictionary, but really this definition is just off of the top of my mind right now, is basically what a thing does. Our very trusty friend, Merriam Webster, explains it as such:
func•tion n. \ˈfəŋ(k)-shən\ – 2: the action for which a person or thing is specially fitted or used or for which a thing exists : purpose
In math, it’s something, sorta, kinda like that, but not really.
A function is an equation that has an entire life on its own. Okay, so right now, I’m confusing you more. What I’m trying to say is basically, a function is an equation. It doesn’t matter what letter you use or how many variables you have, but when you put in or assign a value to one of your variables, then the others would have a value too. Usually.
Rene Déscartes (read it like, Day-Cart), a philosopher (“I think, therefore I am”) and mathematician devised what is known as a Cartesian Coordinate Plane. By assigning a value for an x (which usually shows horizontal movement or width, left to right) you get a corresponding value for a y (showing height, or vertical movement, down to up). And these values you have are coordinates, used to know where you put the points in a graph. Connecting these points, you get the graph of your equation. We know what an x is, but in functions, the y is explicitly represented as f(x) which means “function of x”. It means, that for every value of x, you have a certain value of y, which is its function.
The function of x is basically “what x does”. Told you it was sorta, kinda, a bit like the dictionary explanation but sorta not really like it.
If your teacher ever told you something about a vertical line test, then they’re partly right, and also partly wrong.
The vertical line test proves if something is a function. Of x. Just because it doesn’t pass for the vertical line test, it’s not a function anymore. It still is. But it’s a function of y. (If it passes the horizontal line test.) The vertical line test shows that for every value of x, you only have one value for f(x). Same goes for a horizontal line test. For every value of y, there is a one (and only one) corresponding value for f(y).
But for the sake of uniformity, everyone just says that the values of x is in your domain, and f(x) is your co-domain.
And as with all things mathematical, the best way to learn is by example.
Example: f(x) = 3x+5
||f(x) = 3x +5
As you can see, for every value of x, there’s only one value for y or f(x). Yup, yup. In functions, we just usually say that y is f(x). y=3x+5 is called “explicit”, because we really do show that y is the function. If we wrote that as 3x-y+5=0, then that’s called “implicit”, because we just imply and not exactly state that y is the function of x. In a sense, it can be written as f(x,y) = 3x-y+5. Now, it’s a function of both x and y. X is not the function of Y, Y is not the function of X. The entire thing is a function of both of them.
What is a domain? It’s not the name of a website. Well it is, but not in math. A domain is the set of all the values of x, or the values of your independent variable.
Oh, and by the way, x and y are called variables, because they vary. They change. It can take any value. The value of your x, however, is independent (in this case, and the case of most functions of x). But since y is implicitly the function of x, then that means your y only has a value if x has a value. What value y takes is dependent on the value of x. X basically dictates whatever y is.
And your co-domain or range is the set of the values of y.
If for every value of x in the domain, there is exactly one corresponding value of y in the co-domain, then the function is said to have a one-to-one relationship. The y values can have a lot of corresponding x’s. If for every x, there are a lot of corresponding y’s, then it is not a function.
In set theory, for the sets X and Y, and the function f: X–>Y, there are three correspondence relationships.
Surjective means that for every Y, there exists a corresponding X. Injective means that for every X, there is a corresponding value in Y. If there is a corresponding value for everything, which means it is both Surjective and Injective, we call the function to be Bijective.
Anyway, enough of this. I hope you have been confused.
There is an approach to teaching mathematics which uses less examples and applications. Sometimes, one of the best ways to learn math is to know the theorems by heart, and have them down packed upside down and inside out. Granted, it’s not the easier route to take, but learning that way is the way you practice your mind to think in loops and circles and find new ways and ideas and concepts, like theorems and corollaries, that may connect to each other.
This wouldn’t be a nice way to learn things if you’re vying to become an engineer or an economist. But it is, if you want to learn pure mathematics or be a philosopher.