Statement "f(x) is just a fancy way of writing y" factually incorrect?

[Bruce Wayne1]

Hello!

One of the many aspects of math that I'm intrigued by is pedagogy. I find that many times folks grow up thinking certain mathematical statements that in the end may be inaccurate, or totally false.

I came across a little pic on the internet, and it says "f(x) is just a fancy way of writing y". To me, this is factually incorrect. If students are allowed to think this way, I think they are actually missing out on why function notation is so important, what it does, and why y is something else.

What are your thoughts? We can talk about it from a pedagogical standpoint, a pure mathematical standpoint, and a practical standpoint.
I'll start. I have tutored students in algebra many times before, and I found that function notation was difficult for many students to grasp. I would tell them that f(x) has to be read properly, namely "the function f in terms of the variable x" and explain that the ex can be replaced with stuff. This was crucial when doing function compositions.

 

[mathmaniac1]

f(x)=y is a special relation of two numbers x and y such that for every x there is one and only one y.

 

[Nono713]

f(x)=y is a special relation of two numbers x and y such that for every x there is one and only one y.

See, you need to be a bit more precise with this definition - every x of what? every y of where? why "numbers" and not something else? define relation?

I've seen functions defined as subsets of a cartesian product with particular restrictions (as you noted, for every "x" there must be one and only one "y" such that "f(x) = y"), which I think is the most common formal definition encountered in entry level mathematics. At least it tends to be for discrete mathematics, I'm sure calculus has its own ways of thinking about functions without involving sets.

 

[mathmaniac1]

See, you need to be a bit more precise with this definition - every x of what? every y of where?

why "numbers" and not something else?

This is a definition and "numbers" is a part of it.

define relation?

Given two numbers x and y,you can make y out of x by the application of operations on it.And what operations you did is relation of x with y.

 

[HallsofIvy]

This is a definition and "numbers" is a part of it.
Given two numbers x and y,you can make y out of x by the application of operations on it.And what operations you did is relation of x with y.

But Bacterius's point is that a "function" does NOT have to be from a set of numbers to a set of numbers. I can define a function, y= f(x), from the set of books in a given library to the set of words by "y is the first word on page 1 of book x".

 

[mathmaniac1]

But Bacterius's point is that a "function" does NOT have to be from a set of numbers to a set of numbers. I can define a function, y= f(x), from the set of books in a given library to the set of words by "y is the first word on page 1 of book x".

Really?

Oh,I didn't know it.

Thanks

 

[TheBigBadBen]

Hello!

I came across a little pic on the internet, and it says "f(x) is just a fancy way of writing y". To me, this is factually incorrect. If students are allowed to think this way, I think they are actually missing out on why function notation is so important, what it does, and why y is something else...

I'll start. I have tutored students in algebra many times before, and I found that function notation was difficult for many students to grasp. I would tell them that f(x) has to be read properly, namely "the function f in terms of the variable x" and explain that the ex can be replaced with stuff. This was crucial when doing function compositions.

Two issues that are bound to come up when teaching it:

1:
When it is taught, it is rare that a situation comes up in which functional notation is particularly useful. Honestly, I think the best way to make somebody understand and value a new notation is to put them in a situation where they need to express something like that. Since everything in early algebra is written in the form \(\displaystyle y = f(x)\), there's no misunderstanding that results from simply replacing \(\displaystyle f(x)\) with y. In a sense, it really is an unnecessarily fancy way of expressing something which is, when you have no greater context, unambiguous.

2:
From a purely typographical point of view, \(\displaystyle f(x)\) looks a lot like multiplication. I don't think this is a big obstacle to understanding the notation, but it's noteworthy.

 

[Fernando Revilla]

Related to this topic, I quote an old post of mine at another forum

There's an anecdote attributed to Picasso. After an exposition, a gentleman came up to the genius artist and said: My four-year-old son paints just like you.

Picasso replied: When I was four years old I painted like Michelangelo Buonarroti.

Even though the message was clear, it is obvious Picasso had an intelligent way to describe three states of knowledge (or sensitivity). Usually the third one appears similar to the first one, so those who only arrive at the second one cannot distinguish the third from the first one.

Something similar happened to me years ago whilst giving a lesson in Linear Algebra, a preparatory course for alumni of the School of Engineering (back then I didn't know the anecdote).

I usually named functions $f(x)$ instead of $f$. A student said to me: I know that the teacher who corrects the exams usually fails students who write $f(x)$ instead of $f$.

I advised them to write $f$, because if they wrote: $f$ differs from empty set and contained in $A\times B$ such that:

$(a)$ For all $x\in A$ there exists $y\in B$ such that $( x , y )\in f$.
$(b)$ If $( x, y )$ and $( x, y' )$ belong to $f$, then $y = y'$

and they abbreviated all that as $y = f(x)$ they were probably going to fail, too.

 

[Evgeny.Makarov]

"f(x) is just a fancy way of writing y"

I don't really understand this claim. To me it sounds like, "$x + y$ is just a fancy way of writing $z$". Suppose it is indeed often to write "$z = x + y$". Does it follow that $z$ and $x + y$ are interchangeable? What if we have to write $0 = x + y$, or $z' = x + y$, or $x = y + z$: can we replace $z$ by $x+y$ or vice versa? Similarly, sometimes we need to write, say, $x=f(y)$. The equality $y=f(x)$ is a statement that, given concrete $y$, $f$ and $x$ becomes true or false.

In fact, $f(x)$ is not some indivisible notation; it is the operation of applying the function $f$ to the argument $x$. Both $f$ and $x$ are entities by themselves and can be used in other contexts, e.g., $-x$, $f\circ g$, $f^{-1}$ or $\mathop{\text{map}}(f)$. The latter is an example of a function (called operator in this case) that itself accepts functions as arguments. I agree that the notation for the application operation—a simple juxtaposition of its arguments—may be confused with multiplication (which is deliberately used when working with linear functions), but usually the intended meaning is clear from the context.

 

[ModusPonens]

Pedagogicaly I think it's important to introduce the notation with a drawing. You draw a box with an entrance on top and an exit on the bottom and you say that the box is like a machine that transforms a number x entrying on top, into a number y exiting on bottom. You then designate that y by f(x) to remind that y is a transformation of x, transformation with the "name" f. You can then exemplify with $y=x^2=f(x)$

Of course this is not a general approach to defining a function, but I think it's the ideal form of introducing the concept to high school students. The general approach is only desirable when the student has maturity and experience to understand it.