Abuse of function notation

[Mr Davis 97]

I know that a function is typically defined as a mapping such as $f: x\rightarrow f(x)$, but I am confused about how we really "define" functions. In many common texts, we define functions as $f(x)$, such as $f(x) = 2x$. And we say that we take the derivative of a function $f(x) = 2x$ by $\frac{\mathrm{df(x)} }{\mathrm{d} x} = 2$. My question is, why do we define functions based on the output, $f(x)$, rather than the actual function $f$? And why do we call $f(x)$ the function when really $f$ is the function? Also, why do we say that a derivative is an operator that maps functions to functions, when the input to the operator is the output of the function $f$, $f(x)$, and not the actual function? These are all rather pedantic questions, but I really need to get to the bottom of this seeming abuse of notation. (I made a post on this not to long ago, but these are more specific questions).

 

[Stephen Tashi]

why do we say that a derivative is an operator that maps functions to functions, when the input to the operator is the output of the function $f$, $f(x)$, and not the actual function?

What do you mean by that statement? Are you saying that if I only know that f(2) = 7 that I can find the derivative of f evaluated at x = 2?

The derivative operator (on a particular domain) is a function from the domain (which is a set of functions) into another set of functions. "The derivative of a function f" is function resulting from evaluating the derivative operator at f. A notation like f'(2) refers to the value of the function which is derivatve of f evaluated at 2. The notation f(x) is ambiguous. It may mean the function f, or the function f evaluated at the specific value. You have to understand the context surrounding the notation f(x) to understand how to interpret the notation.

These are all rather pedantic questions

A pedantic question should be precise. You are using terminology like "The input of the function" and "the output of the function" without relating it to standard mathematical terminology. Even in computer programming, "The input" and "the output" is not precise terminology. Functions in computer programs have "input variables" and "return values".

, but I really need to get to the bottom of this seeming abuse of notation. .

It's more than a seeming abuse; people who write mathematics do abuse notation.

 

[Mr Davis 97]

What do you mean by that statement? Are you saying that if I only know that f(2) = 7 that I can find the derivative of f evaluated at x = 2?

By my previous statement, I mean why is it necessary to have a closed-form expression for any number in the range of f, i.e. f(x) = 2x, in order to find f'.

The notation f(x) is ambiguous. It may mean f, or the function f evaluated at a specific value.

Yes, it does seem ambiguous. I can never tell when an author is talking about one or the other. But my main question is still, to talk about the function f, why is it (usually) always necessary to have a closed-form expression of f "acting on" the variable x, i.e. f(x)? Why do authors tend to designate f(x) as the function, when f is actually the function?

 

[Stephen Tashi]

By my previous statement, I mean why is it necessary to have a closed-form expression for any number in the range of f, i.e. f(x) = 2x, in order to find f'?

The derivative of a function is defined in a "pointwise" fashion. - i.e. the definition of finding the derivative involves a finding a limit involving specific value in the domain of the function, which usually called "x" and doing this for each value x in the domain. (The limit may fail to exist at some values in the domain of f.) The result of $\lim_{h \rightarrow 0} {\frac{f(x+h) - f(x)}{h}}$ depends on the values of the function at numbers in the domain that are in the vicinity of $x$ . For example, the functions $f(x) = x + 2$ and $g(x) = x^2$ each have he value of 4 when evaulated at $x = 2$ , but $f'(2) = 1$ and $g'(2) = 4$.

But my main question is still, to talk about the function f, why is it (usually) always necessary to have a closed-form expression of f "acting on" the variable x, i.e. f(x)?

It isn't always necessary, but it's customary.

Why do authors tend to designate f(x) as the function, when f is actually the function?

Your question does not have an abstract mathematical answer. It's a question about cultural tradition and human behavior. It can be discussed in a subjective manner like other questions of human behavior. For example, the way things are written in mathematics today is affected by the history of mathematics, both the concepts and the habits of notation that were used in the past.

Take some passages that involve functions from a calculus textbook and rewrite them without using a symbol for the argument of the function. See if you find writing in that manner an effective style.

 

[Fredrik]

I know that a function is typically defined as a mapping such as $f: x\rightarrow f(x)$, but I am confused about how we really "define" functions. In many common texts, we define functions as $f(x)$, such as $f(x) = 2x$. And we say that we take the derivative of a function $f(x) = 2x$ by $\frac{\mathrm{df(x)} }{\mathrm{d} x} = 2$. My question is, why do we define functions based on the output, $f(x)$, rather than the actual function $f$?

A function $f$ is, loosely speaking*, a "rule" that associates exactly one element of a set Y (the codomain) to each element of a set X (the domain). So to define a function $f$ with domain $X$ and codomain $Y$, you have to specify, for each $x$ in $X$, the $y$ in $Y$ that $f$ associates with $x$. This $y$ is called the value of $f$ at $x$ and is denoted by $f(x)$. (This is typically a number, not a function).

For example, the function that takes every real number to its square is the $f$ defined by $f(x)=x^2$ for all real numbers $x$. Note the "for all" statement at the end. It's an essential part of the definition that's often omitted out of laziness. The "for all" part is what ensures that we have specified the output for every possible input.

And why do we call $f(x)$ the function when really $f$ is the function?

Good question. I find it annoying when people do. It doesn't make sense and causes a lot of confusion in many types of problems. But I must admit that I'm sometimes tempted to abuse the terminology in that way too. For example, if I'm asked for an example of a bounded function that doesn't have a limit at 0, I'm inclined to say $\sin\frac{1}{x}$. The proper ways to say it include "the $f$ defined by $f(x)=\sin\frac{1}{x}$ for all real numbers $x$" and "the function $x\mapsto\sin\frac{1}{x}$ with domain $\mathbb R$".

Also, why do we say that a derivative is an operator that maps functions to functions, when the input to the operator is the output of the function $f$, $f(x)$, and not the actual function?

You got this one wrong. The derivative of a function is a function. In particular, the derivative of the function $f$ defined by $f(x)=x^2$ for all real numbers $x$, is the function $g$ defined by $g(x)=2x$ for all real numbers $x$. A number like $f'(1)$ isn't determined by $f(1)$. This is just the number $1$. $f'(1)$ is determined by the behavior of $f$ on an open interval (any open interval actually) that includes $1$.

A notation that reflects the fact that the derivative of a function is a function is $f'=Df$. The operator $D$ takes a differentiable function as input, and produces a function as output. To define $D$, we must specify the output $Df$ for every differentiable $f$. So let $f$ be an arbitrary differentiable function (with domain $\mathbb R$). We define the function $Df$ by
$$(Df)(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ for all real numbers $x$.


*) So what exactly is a function? It's perfectly reasonable (even though it may not seem that way) to leave the term undefined, and just view what I said as a good way to think about functions. Such an explanation of a term that's deliberately left undefined is called an elucidation.

However, there's a branch of mathematics called ZFC set theory that's rich enough to include all the math you've ever heard of, and probably all the math you ever will hear about. In this branch of mathematics, all definitions of terms associate a class of sets with the term to be defined. So in this branch of mathematics, the definition of "function" must specify what sets to call "functions". This is a good way to do it: (Unfortunately you must know a bit of set theory to understand this).

A triple $(X,Y,f)$ such that $f\subseteq X\times Y$ is said to be a function if

(a) For all $x\in X$, there's a $y\in Y$ such that $(x,y)\in f$.
(b) For all $x\in X$ and all $y,z\in Y$, if $(x,y)\in f$ and $(x,z)\in f$, then $y=z$.