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When defining, or referring to, functions, I've seen both of the styles in the title. I was wondering which is considered to be more correct, or better style.

I've always found it strange to refer to $f(x)$ as a function, after all $f(x)$ is the value of the function at $x$. However, I suppose that we might agree that as long as $x$ is not declared anywhere $f(x)$ refers to the function itself, but as soon as $x$ is declared, then $f(x)$ is the value of the function at $x$.

Which do you prefer, and why? I am particularly interested in the context of undergraduate teaching, where the notation $f:X \rightarrow Y$ might be too foreign to be of use.

Bernard
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Peter
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    I like “let $f: X \mapsto Y$ be the function…” so that you explicitly mention the pertinent spaces $X$ and $Y$. – gfppoy Oct 07 '21 at 14:31
  • Prefer for what purpose? Depends. – Paul Oct 07 '21 at 14:32
  • I agree with you . Denoting $f(x)$ a function is an abuse of language – which may be acceptable only in some rare cases. – Bernard Oct 07 '21 at 14:35
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    Perhaps this post would be more appropriate for the math educators stackexchange. I would be happy to migrate it there if that is the consensus. – Peter Oct 07 '21 at 14:35
  • For a shortcut I actually like 'let $f()$ be' or sometimes 'let $f(\cdot)$ be'; they both make it clear that $f$ is a function without binding anything. – Steven Stadnicki Oct 07 '21 at 14:38
  • The argument of the function need not be on the right; $xf$ is used a lot in semigroup theory, for instance. – Shaun Oct 07 '21 at 14:43
  • See this post as well as this one – Mauro ALLEGRANZA Oct 07 '21 at 14:43
  • Sometimes it is correct to write "The function $f(x)$", although probably not for the reason you're asking. An example being: Suppose $X = \Bbb R$ and $Y = C([0, 1], \Bbb R)$, the set of real valued continuous functions on $[0, 1]$. In this case, given $x \in X$, $f(x)$ is again a function! – Aryaman Maithani Oct 07 '21 at 14:46
  • @gfppoy. It should be an ordinary arrow between the sets: $f : X \to Y.$ The arrow $\mapsto$ is used with a variable on the left hand side and an expression on the right hand side to construct functions: $x \mapsto x^2$ (in programming this is called lambda-expressions). – md2perpe Oct 07 '21 at 16:04
  • It's more correct to write "Let $f$ be ..." or rather "Let $f:X\to Y$ be ...". It's common though to write "Let $f(x)=\ldots$" which seldom is confusing. – md2perpe Oct 07 '21 at 16:08
  • I’m voting to close this question because it belongs on https://matheducators.stackexchange.com/ – postmortes Oct 08 '21 at 05:45

2 Answers2

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Declaring the operand of a function becomes important when it has parameters.

For example:

A map from $\Bbb N\times\Bbb N\times \Bbb N\to \Bbb N$ could be e.g. $f(a,b,x)=ax+b$ but if it's just a function $\Bbb N\to\Bbb N$ parametrized by two natural numbers, you will need to indicate which ones are the parameters and which are the operand like: $f_{a,b}(x)=ax+b$

it's a hire car baby
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Whether to use $f$ or $f(x)$ for writing a functon, really depends on the context.

As you pointed out, $f(x)$ looks like the value of $f$ at $x$. But, the term $f(x)$ actually means to say "$f$ as a function of $x$". So, sually when defining it, it's more elegant to write it as $f:A\to B$ since you are explicitly mentioning the domain and range of $f$.

On the other hand, writing something like $f=x^2$ or $\int_a^b f\;\text{d}x$ looks weird, whereas $f(x)=x^2$ or $\int_a^b f(x)\;\text{d}x$ makes much more sense.

Sayan Dutta
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    I disagree that 'the term $f(x)$ actually means to say '$f$ as a function of $x$'. That is a common understanding of it, but absent specifics of definition, etc. it doesn't necessarily have any clear, specific meaning. And the examples in your last paragraph aren't really at issue here; those are cases where we are explicitly binding the variable $x$, whether to define the function or to define the integral. OP is only wondering about cases where we don't have or need a bound variable. – Steven Stadnicki Oct 07 '21 at 16:01
  • @StevenStadnicki yes, I agree with you. What I meant to say is that writing a function as $f(x)$ is often quite necessary. – Sayan Dutta Oct 08 '21 at 05:44