Functions which relate to calculus: Questions about Notation

[richard9678]

I would not consider myself to be well educated, if I could only understand what a function is. I'd think it useful to grasp when a statement is bad form. My latest posting simply reflects this. For what it's worth, I did take f to be the name of a function (named: f), and took f(x) above all, to be a number and a shorthand for "Of x: square and add two", (or whatever).

If I am confused about anything, it would not be with the definition of a function, but some of the statements authors write. It is said x^2+2 is not to be referred to as a function, but is merely an expression. Here is what my book says - "Often instead of talking about function f defined by the rule f(x)=x^2+2, we shall simply say "the function X^2+2, or even "the function f(x)". Remember other notations are possible."

This shows that when I post what I think, it's based on something I've read, I've not just formulated my understanding from merely thinking about the issue.

So, the spanner in the works and the source of any misunderstanding or confusion is more to do with some of the statements authors use, which can be contested. Not confusion over what a function is. Probably true for all beginners.

Also, we should perhaps take into account what statement is seeking to do. If it's to define something or to establish equality in a mathematical sense, then that's one thing. It's another to simply give some other way to indicate which function is being talked about. In the light of this "..the function X^2+2" is probably OK. I'm thinking.

 

[richard9678]

This thread is finished I think. I'd say is was more about referencing a function, rather than a thread about what is a function. Thank you for all contributions. Rich.

 

[richard9678]

One more thing dawned on me. For some reason I've thought the introduction of the notation f(x) was in some way needed to get to grips with functions. But, as you know, we deal with a function when we write (say) y=2x. So, we don't need f(x) to help us understand functions, but the very fact that we use the notation f(x) indicates the equation or term involved is explicitly a function and not just any kind of equation or term.

 

[Stephen Tashi]

So, we don't need f(x) to help us understand functions,

Many people rely on f(x) type notation to understand important operations on functions. The chain rule in calculus is Df(g(x)) = f'(g(x)) g'(x). Students must understand the distinction between f'(g(x)) g'(x) versus f'(x)g'(x).

Of course, others rely on the Liebnitz notation df/dy dy/dx.

 

[fresh_42]

Many people rely on f(x) type notation to understand important operations on functions. The chain rule in calculus is Df(g(x)) = f'(g(x)) g'(x). Students must understand the distinction between f'(g(x)) g'(x) versus f'(x)g'(x).

Of course, others rely on the Liebnitz notation df/dy dy/dx.

I like the notation
$$
\left. \dfrac{d}{dx}\right|_{x_0}(f\circ g)(x)=\left. \dfrac{d}{dy}\right|_{y_0=g(x_0)}f(y)\cdot \left. \dfrac{d}{dx}\right|_{x_0}g(x)
$$
Too often are derivatives written when actually their value at a certain point is meant.

 

[richard9678]

Many people rely on f(x) type notation to understand important operations on functions. The chain rule in calculus is Df(g(x)) = f'(g(x)) g'(x). Students must understand the distinction between f'(g(x)) g'(x) versus f'(x)g'(x).

Of course, others rely on the Liebnitz notation df/dy dy/dx.

Absolutely. That dawned on me. I mean how you can really "soup up" f(x). You could write f="whatever" and it has certain degree of utility. But I can now see that someone (a long time ago) has recognised that if you employ the notation f(x)= "whatever", you can do a lot more with it. In other words, you can do, or express a lot more that just place an x in the brackets. So, the underlying format of the notation f(x) is really quite clever. I'm just beginning to realise that. Of course, this shows I have not studied very far with functions yet, but I'm beginning to appreciate how useful the notation f(x) is.

 

[Mark44]

So, the underlying format of the notation f(x) is really quite clever. I'm just beginning to realise that.

It's also pretty simple once you get past the notion that parenthesized expressions can represent a product, as in 2(x + y), or a function definition, as in $f(t) = 2(t + 1)$.