Are they the same thing? No, they are not the same thing. Check out Wolfram Alpha for more of that stuff. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. I have to concur with that view. Whatever the local high school curriculum or textbooks has or lacks on relations or functions and so on, anyone who knows what they are talking about would point there as a good place to start.
Though I believe your state testing materials, textbooks, whatever probably could have mentioned things, like a vertical line test, maybe not realizing and conveying to students broader concepts. Honestly the calculus teacher ought to know and teach students reasonably relevant definitions for the material those students will learn, given things like the fundamental theorem, though I know those are older and more advanced students.
I cover that in an earlier post. Thus, I think there are quite a few very good explanations of function notation and equation notation and the reasons for the differences in some of the online guides to function creation within said CAS packages. I mention this mainly because the differences between the two was something I had to recently relearn which, given that I am a phd student in nuclear engineering, was more than a little embarrassing as I was attempting to teach myself to use Maple.
An example of one of the guides I used is below. Click to access 07functionsExpressions. I do all of this in the context of linear and quadratic equations, which takes precedence.
I need time to warm up to novelties like that. And you can also treat a variable just like a number. Is that true? Asking, not asserting. Partial Differential Equations. One day I tried to illustrate a problem for my classmates, using colored chalk to indicate the different stages of the solution. Oh, I see. You can add, subtract, multiply and divide constants by functions and the reverse. You can add, subtract, multiply and divide functions by each other. You can find the inverse for functions that have an inverse ….
No, I know that you can treat a function just like a number. The question is whether you can treat an equation like a number in the same way. An equation e. You can change an equation but you are bound by the properties of equality. This got a huge discussion online. I need to link it in; it was really helpful and interesting. As were the comments here. See, just to jump in on this, nothing was learned here other than whether DensityDuck likely had a lazy or irresponsible calculus teacher; remaining commenters can be given the benefit of the doubt for not having any specific formal studies.
Whether or not various operations and properties apply to a function depends on more than just the definition of a function. Differentiability is not an inherent property of functions, for instance, everyone in this comment chain should get that.
A word you are looking for, at least what Mark Roulo was looking for because several of his sentences make no sense otherwise, is expression. A better offhand description, and possibly vaster more useful to give your students, is that an expression can be treated like a number, and when a function, defined by an expression, satisfies various properties, relevant at the depth of what the students are learning say again a function of one variable on the real plane certain operations can then apply.
But here, consider the Caesar cipher, a function defined piecewise like so:. This function maps letters of the english alphabet to letters of the english alphabet it shifts them forward three points. You can still work with expressions like f x , and extend the function by defining using. If f is some other kind of function, all bets are off. In computer programming, there is a useful distinction between named functions and anonymous functions.
We give the function a name so we can refer to it later, even if the problem involves many different functions. Yes, I think computer programmers use and understand the concept of functions much better than most math teachers teach it.
Look, functions have a mathematically rigorous definition, often introduced in some kind of formal setting. They are a mapping from one domain into another. You take an object or, getting technical, an ordered n-tuple from one domain, and there is exactly one object corresponding to it in the other domain often called the range.
None of this has anything to do with equations per se. In fact, getting a bit technical, there are only countably many equations, but uncountably many functions from numbers to numbers. But often functional notation is the best, partly because that notation can be ornamented to carry with it its meaning.
I think though that the real problem that you have here in is just in your discomfort using functional notation. The thing is, functional notation is in many ways whatever its contriver specifies it to be. Certain common practices, however, are pretty much assumed. Not sure where you can go to develop a sense of those common practices. This is not the introduction to functions. I have another post on that which I link in. For the record this guy appears to making the same general point I made.
Good for him for not reading any of the comments. If you are required to teach students certain shortcuts to meet the standards of some state test, absolutely do that up to what is necessary. And the reason we can tell is that the material you are considering for excercises is not germane to the underlying issue. If anything, for the general reader, it illustrates structural problems for American schools, with state curricular orgs, with insane insular notions of IP and so forth.
You mention this is a first in presenting or focusing on some of this material though, perhaps without the paper, non-electronic books and resources being valuable to the students.
And for that matter it is completely on the shoulders of the computer science teachers and book authors to take responsibility for their material and be clear that constraints of programming langauge functions, return types, call structure and all are distinct from mathematics. Throughout much of the discussions here there is a persistent confusion of language and metalanguage.
I think that I remarked on the futility of centuries of effort to make things clear. He said something to the effect that someone who could not keep these two things separate should give up on mathematical logic and take up something else — he suggested bee-keeping. You are being pedantic. A typical math major, even at a good college typically takes calculus, differential equations, and then proceeds to major in linear algebra taking it many times.
More likely, they hit a conceptual brick wall three courses away from completing the major. Then the class becomes about a set of problems. Even if those problems were initially designed to test conceptual understanding. But perhaps someone who majored in a difficult liberal arts subject philosophy or English at the right university and is open to learning math might be better in some respects.
But he could draw a pizza pie and break it up into a certain number of equal pieces. A function should be thought of as an oracle that given certain input, gives certain output. Functions are often expressed in terms of elementary functions tied together by the usual operations of arithmetic and function composition this is your function algebra ; not all functions have an expression in this way. The function algebra expression of a function f x is code for: to evaluate the function at a certain value, substitute in that value for x.
More conventionally in mathematics, there is an algebra of polynomials. There is also a more rigorously defined algebra of functions if differential algebra. Indeterminates which are symbols should be distinguished from variables which are stand-ins for a value that you intend to substitute later.
BTW, I do teach less, and try to teach it conceptually. I do understand what you are saying about introducing a concept not to work problems, but simply to understand a concept. So I'm going to draw-- if this is the world of equations right over here, so this is equations. And then over here is the world of functions. That's the world of functions. I do think there is some overlap. We'll think it through where the overlap is, the world of functions.
So an equation that is not a function that's sitting out here, a simple one would be something like x plus 3 is equal to I'm not explicitly talking about inputs and outputs or relationship between variables. I'm just stating an equivalence. The expression x plus 3 is equal to So this, I think, traditionally would just be an equation, would not be a function.
Functions essentially talk about relationships between variables. You get one or more input variables, and we'll give you only one output variable. I'll put value. And you can define a function. And I'll do that in a second.
You could define a function as an equation, but you can define a function a whole bunch of ways. In Depth In. What about f -x? First of all, it is important for you to know that y and f x mean exactly the same thing. They are interchangeable. They BOTH mean the same thing.
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