Well I feel ready to really show my ignorance here. Why does it matter if something is a function or not? Why do we teach relations & functions and function notation?
I’m teaching Algebra 1, which I’ve done for about 5 years, and for some reason this section always just annoys me because I really don’t know why (and don’t care either, which makes it worse). I hated function notation when I was in school, and never did quite get why it’s done that way. When I was younger, I just conteneted myself with “well, f(x) is just the same thing as y, so move on, it’s easy”, but I don’t think I teach this section well at all because I don’t get the underlying reasoning.
To be honest, not understanding this is bothering me a lot. Usually I totally get Algebra, and I can DO it with no problem. I just really don’t know why.
I think it’s because a function implies a one-to-one relationship. For every value of y, there is exactly one, and only one value of x (unless there’s a point at which y does not exist, i.e. y = 1/x at 0). y[sup]2[/sup] = x is not a function, because if I give you an x value, you can’t tell me for certain what the value of y is. Practically, you may be able to rule one out (if it’s not feasible), but in the real world, just about everything we plot has to be a function.
Presumably, at least some of your students will go on to take calculus in college, in which case functions are pretty much the basis of the whole thing.
It isn’t that saying y=f(x) is a big difference, especially when the only variables in the equation are y and x. But it has a pretty deep significance philosophically.
For one thing, what if I tell you that I=I[sub]0[/sub]e[sup]-t/τ[/sup]? It doesn’t mean much. But if I(t)=I[sub]0[/sub]e[sup]-t/τ[/sup], we know that I is a function of time, with a maximum at t=0 which decreases exponentially as t increases.
Furthermore, in science, the equations come from functions, not vice versa. You don’t say “y=5x+2, therefore y is a function of x”, you say “When I change x, y changes too, lets figure out exactly how much,” and proceed to write the equation from there.
Hope that helps, it was kind of off the top of my head. This is a subject that could probably take up several pages and there are lots of different perspectives, so if you want me to clarify or explain further, just ask.
How much I pay for cheese is a function of how much cheese I buy. How else do I and the seller know how much I need to pay?
The amount of insulation required in a building is a function of inside tempreature, outside temperature and other factors. How else am I going to know how much insulation I need to design?
The amount of current flowing through a diode is a direct function of the voltage and other factors. How else am I supposed to determine the amount of current flowing? Some guys got together, studied the dependencies and designed a mathematical function which approximates the behavior of the diode. Then others can use that function in their own calculations.
Maybe I am missing something or misunderstanding the OP but to me the thing is so obvious that I don’t understand how you can be teaching and asking this question. Maybe I am just not understanding the question.
Sorry, I got ahead of myself here. What I mean is that if you have more than one variable or quantity in the equation, you have to specify which is a function of what. It isn’t clear as in the case of y=x[sup]2[/sup] or y=3x+6.
Sure, it is obvious. Like a lot of math is once you already understand it. But while going from equations to functions is a small mathematical step, it is a big conceptual step whose full significance isn’t obvious to anyone when they first learn it. I know it wasn’t obvious to me in high school. But as it sunk in, it was the first step for me in thinking “Hey, this actually means stuff in the real world! Algebra is more than just a bunch of stupid symbols on a blackboard. It actually links two (or more) quantities in the real world together by a simple rule.”
This subtle and simple idea is actually the basis of how science is done.
The OP isn’t teaching construction design, electrical engineering, or advanced math. She’s teaching introductory algebra. Just because you understand it doesn’t mean you can demean people who ask; otherwise this forum would be completely empty, because I guarantee somebody finds any given question “too simple”.
ETA: Also, one does not need to understand the significance of the price of cheese being a function. As** DrCube** said, the full significance isn’t realized until one starts taking derivatives.
The “y” notation (such as “y = 5x+2”) is a simplification that’s adequate for teaching basic concepts in Algebra 1 such as slope-intercept. In this case, writing it as “f(x) = 5x+2” is adding additional wordiness for no benefit.
However, in more advanced math concepts, you may want to deal with multiple variables such as “f(x,y,z) = 5x + 3y + 4z + 2” or nested functions such as “f(g(x))”. Trying to rewrite these examples as basic “y=” notation would require multiple lines that referenced each other.
I’m not a mathematician so my examples come from perspective of practical utility. A real mathematician would probably offer additional philosophical subtleties.
Psst–I think you have that backwards. If y is a function of x (which is the way it’s usually set up), then for every value of x there is only one value of y, but not necessarily vice versa. (For example, y = x[sup]2[/sup] or y = sin(x) are functions, but you have more than one x giving you the same y.) A one-to-one function is a function for which there is exactly one y for each x and vice versa.
Getting back to the OP: As DrCube explained, a function is a particular kind of relation where one thing depends on (i.e. is a function of) something else. Other examples: If you get paid an hourly wage, the amount of money you make is a function of the number of hours you work. The cost to mail a package might be a function of the package’s weight. And for anything that changes over time (like the velocity of a falling rock, the level of pollutants in a lake, the temperature, or the population of a city), you can think of that quantity as a function of time.
Functions are a key concept in higher math. In fact, Calculus could almost be described as the study of functions. (Here’s a function. What can we learn about it? Where is it increasing or decreasing? Does it have a maximum or minimum? What are its zeroes? What does its graph look like, and what are its properties? Etc. etc. etc.)
Functional notation is useful because, if you’re going to be working with functions, it’s very convenient. Instead of asking “What is the value of y when x is 3?” you can just say “What is f(3)?”
All sorts of things are functions. There’s the square root function (What’s the square root of 9?), trigonometric functions (What’s the cosine of 60 degrees?), probability functions (What’s the probability of getting a royal flush?), even non-mathematical “functions” (Who’s the current Governor of California?) In a sense, whenever you talk about “The ____ of ____,” you’re probably dealing with a function.
By the way, I think a lot of your confusion is the textbook’s fault. I think beginning/intermediate algebra books introduce the concept of function because the students will need it later when they get to more advanced algebra or trig or calculus, but they don’t often do a very good job of motivating the idea or explaining why it’s going to be important. They talk about the Vertical Line Test or “a set of ordered pairs, no two of which has the same first element” (which is a concise and accurate definition from a mathematical point of view, but not very enlightening from a practical standpoint), without explaining who cares about such things, or why.
If there were no other reason, that last one–composition of functions–would be sufficient in and of itself to spend an entire year teaching function notation. This is especially important in advanced math, where you stop thinking of a function as a collection of values and just think of it as a point in some space.
I came to this thread because I had the same confusion as the OP. What’s wrong with “y=x^2”? (Sorry, I don’t know how to draw superscripts)
Then Ruminator’s post gave me a flash of inspiration. I think the key is that putting “y” on the left side of the equal sign introduces a needless variable. Why should something as simple as “y=x^2” need two variables? Absurd! There’s really only one variable here. There’s no “equation”, in the sense of two different things which are equal to each other. There’s only one thing, and we need a notation to reflect that.
So too in Ruminator’s example of “f(x,y,z) = 5x + 3y + 4z + 2”, The right side has only three variables, so there’s no sense in putting a fourth variable on the left side. Rather, the left side is “f(x,y,z)”, which demonstrates how the aggregate value changes along with any change to x or y or z.
While typing the previous paragraph, it occurred to me that the nomenclature might be partially at fault here. Instead of the word “function”, try the word “formula” and see if it makes it easier to understand.
I was thinking about this the other day, actually - the crappy way that functions were presented back in elementary school. Thudlow Boink explains it pretty well.
As I recall, they’d show us a line or curve drawn in the x-y plane and ask, “Is this a function? Yes or no?” What it boiled down to, of course, was that if for any given x there was at most one* associated value of y, then y was a function of x. They didn’t really explain it well, and it all sounded at the time like pointless tail-chasing.
During these exercises a circle was intimated to be of no use to man or beast because it was Not A Function: y = (1 - x[sup]2[/sup])[sup]1/2[/sup]. For x=0, is y 1 or -1?!? Or both? No, children! Avert your eyes and move on to the next problem, where we can look at a proper function like y=x. I exaggerate, but only a little.
Actually, if they had bothered to say that the multi-valued output were the problem, things would have made a lot more sense. A very low-level “You tell me an x, I tell you a y; that’s a function, boys and girls.” would have helped a lot.
There could be zero. If y(x) is only defined for, say, 0<x<1 then there is no y value associated with x=5. That’s not a problem as long as you’re fine with functions not having all of the reals as their domain.
Functions are used everywhere. Calculus is the most obvious area, but whole tracts of computer science study pretty much nothing but functions, how they behave and what you can do with them. Programming languages are one example (let’s ignore side-effects).
As to why we bother with function notation, it’s mostly for convenience. Most example are given with function names like f, but you can give more descriptive names to your functions, if you want. It’s also a lot easier to combine and work with functions if you use function notation, though.
As always, don’t get hung up on notation. You can use whatever notation you want, as long as you’re consistent. Mathematics isn’t about notation.
I disagree wholeheartedly, although I realize it is just my opinion. Most progress in math is simply newer and better notation. It is a big deal because it changes the way you think. Functions are a perfect illustration. y=3x is an equation. f(x)=3x is a function. A subtle change in notation completely changes the way we think about it, and suggests new questions to ask, opening up the entirely new area of math called calculus. In a lot of ways, math is notation.
I may even open another thread in IMHO or somewhere about this. I just need to think about how to phrase the question. The difference, if any, between “Math” and “Notation” is an interesting issue.
Thanks for your quick responses! I plan to sit down on my planning period and read through this better. Skimming through I saw some points that really came out clear to me, so I can’t wait to read and process this. Then, hopefully, this year when I teach it, I’ll be able to at least give them a little bit more than just “We are touching on this now, so you will have already seen it when it becomes important later.”
One of the problems with teaching only Algebra 1, which is what I’ve done the past several years, is I often forget about a lot of the stuff that’s done in higher level classes, other than broadly. I had totally forgotten composite functions, for example, and I loved them!
As I mentioned before, IANAM but I lean towards the idea that mathematics is more about “seeing and understanding patterns” as opposed to “notation”.
It’s somewhat analogous to music is more about the “underlying patterns of tones” instead of the “treble clef staff notation” of music.
However, it’s not black-&-white. For example, there is undeniable progression in expressing numbers as roman numerals vs base10 vs scientific notation. Maybe you could argue that some human innovations and inventions were caused specifically by convenient notation but I would think they’d be overshadowed by inventions of pure mathematics imagination (pattern recognition)–regardless of notation.