I don’t like this sentence, but don’t have the math theory background to be sure I’m not making things up as I go along.
Shouldn’t it be “all functions are either linear or nonlinear”?
I know there are various types of functions (e.g., exponential), but aren’t they all either linear or nonlinear? Isn’t describing something as exponential saying that it is a specific type of nonlinear function? Can a function be neither linear nor nonlinear? If a function describes, say, a single point, isn’t that still nonlinear, or does nonlinear describe something specific?
Furthermore, by saying that a function can be linear and nonlinear, isn’t that a contradiction? Or can a function be both?
I think it’s just badly worded. As in “Functions can be linear and functions can be non-linear”.
A function could be linear over an interval [a,b] and nonlinear over the interval [b,c].
To my mind (and I’m pretty sure this is the standard meaning, but I don’t have a cite), “nonlinear” simply means not linear. So, functions which are not linear functions are, by definition, nonlinear functions.
Right – the original author knows what she’s talking about mathematically. The and is what initially drew my attention – which is why I wanted to change it to or. But like I said, when it comes to second-guessing real mathematicians, I thought to check.
The other thing that drew my attentions was can – if all were either, then it should be are. (Isn’t the determination of intervals’ behavior determined by looking at derivatives? Sorry that end of math is very far in my learning past.) Regardless, isn’t a such a function behaving linearly over an interval, but because it has a nonlinear component the function as a whole is nonlinear? Or do the words linear and nonlinear just not apply to the function as a whole. Is there a descriptive for such a function?
To me, “functions can be linear and nonlinear” basically means that it can be any function (or, depending on the context, any continuous function). It’s just a way to draw attention to the fact that the functions used may be linear, but they don’t necessarily have to be. Personally I would have written “functions can be linear or nonlinear”.
Was this a textbook or just someone’s notes on a website? They can get sloppy with their wording sometimes when it’s just informal notes.
I agree that your wording is more clear: “all functions are either linear or nonlinear". Or “A linear function has these properties: … A function that is not linear is called a nonlinear function”.
It’s for a teacher’s guide/lesson plan. Cool stuff–makes me wish I was in sixth grade again.
It’s horribly unclear to say this sort of thing, but a function can be linear and non-linear at the same time if you’re using two different but commonly used meanings of the word linear. The meanings are as follows:
[ol]
[li]A function f is linear if it is of the form f(x) = ax + b.[/li][li]A function f is linear if it satisfies f(ax + by) = af(x) + bf(y).[/li][/ol]
Every function which is linear in sense 1 is non-linear in sense 2, and every function which is linear in sense 2 is non-linear in sense 1. But if someone wrote that a function was linear and non-linear with a meaning like this, very few people would have an easy time figuring it out.