Differentiating Variables from Functions
Date: 02/03/2004 at 19:11:55 From: Kathy Subject: Function Notation What letters are used to note/name a function? We've been learning about variables and finally thought that we had them figured out. Now we start learning about functions today and they too use letters to show a function. The kids are asking how in the world they are supposed to know when a letter stands for a function and when the letter is a variable? This is one of those things where I just always did it that way and can't explain why. I thought all functions were noted with an f as in f(x), or "f of x", or "for function f, what is the value of y every time you enter x". I haven't had any calculus and wasn't aware that a function could be written g(x) or b(x). The kids really want to know how they can differentiate variables and function notations.
Date: 02/03/2004 at 23:43:51 From: Doctor Peterson Subject: Re: Function Notation Hi, Kathy. Any letter can be used for a function, though we tend to use f and g when we are talking about generic functions or temporary ones, just out of habit, as we tend to use x as a variable when there's no reason to give it another name. Functions that have special names usually are NOT called f, to avoid confusion! For example, Greek letters are used for the phi function in number theory and the gamma function in analysis. It can, I suppose, be tricky to distinguish a function from a variable if you were to see an equation with no context, like r(t) = 2s(t) + t(t+1) (I just made that up to make it hard!) I know, because of what I meant when I wrote that, that r and s are functions and t is a variable; but they could all be variables and it would still make sense. (It wouldn't make sense if t were a function.) You should also be aware that we can use a variable as a function: if we've defined y in terms of x, we can write "y(x)" to mean the value of y for a particular x, while still just writing y when we don't need to emphasize that relationship. So what makes the difference between a function and a variable? Definitions! When we write anything in algebra, we have to first define our variables and functions. We tell the reader what things mean before we use them. Ultimately, that's all that stands between comprehension and chaos. And that's why I constantly emphasize the need to clearly state the definitions of variables at the start of a student's work on an algebra problem. You'll see the same thing in legal papers: when you want to use general terms (like "plaintiff" or "Company") with specific meanings, you have to say what you mean by them before you start writing. Pretty simple, isn't it? Communication 101. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 02/04/2004 at 08:51:19 From: Kathy Subject: Thank you (Function Notation) Oh my goodness! Now I understand :-) Thank you so much for taking the time to answer thoroughly. Being the teacher in the class, I felt really silly writing to you, but I've always told my students that the only truly stupid questions are the ones you want/need answers to and do not ask! I appreciate your efforts on my behalf as I help my young students answer their 'burning' questions.
Date: 02/04/2004 at 09:09:21 From: Doctor Peterson Subject: Re: Thank you (Function Notation) Hi, Kathy. I like this kind of question more than anything else: the ones nobody stops to think about, or dares to ask, because they seem too simple! It's a great way to stretch my mind. I'm glad my rambling, until I realized how simple the answer was, didn't confuse you. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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