I saw a definition stating that given a function $f:X \to Y$ the natural domain of $f$ is the largest possible subset $D \subseteq X$ for which the rule defining $f$ is valid.
From what I understand, the $X$ in $f:X \to Y$ must be the domain of $X$. Let’s suppose $D$ is a subset of $X$ but is not $X$ itself, then that seems to imply that there is at least one element in $X$ not in $D$. But if $X$ contains an element not in the domain of $f$ then we can’t write $f:X \to Y$. If such a subset $D$ exists and is not $X$, then that implies $f$ is not a function in the first place.
My question is: is this definition flawed? Or, when written in arrow notation, is it acceptable for $X$ to be a set containing the domain; even if it is not the domain of the function itself?
Best Answer
After thinking way too long about this I'd say scrub the definition as given as you are correct: the notation "$f:X\to Y$ is function" does imply $X$ is the domain and the definition doesn't make sense.
But replace it with this:
That way we avoid saying the incorrect "$f:X\to Y$ is a function" (it isn't... that fails the definition of function) and avoid any weird and awkward "$f:X\to Y$ but not as function maps... we just mean some of the $X$ can be mapped"
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I guess what the book is getting at is if you take a naive idea of function being a "rule" and there is some universal set we are working in then saying things like
$f(x) = \frac {x^2 +7}{x-3}$ or $h(x) = \arcsin x$ or $j(x) = \sqrt x$
and we make the naive and blanket statement that all theses are real functions and map $\mathbb R$ to $\mathbb R$.
Then the "unnatural" super-general-not-much thought put into "domain" of all these functions are $\mathbb R$.
ANd so the natural domain is the one's where the function actually works. "Natural" domain of $f(x)$ is $\mathbb R\setminus \{3\}$. Naural domain of $h$ is $[-1,1]$. and natural domain $j(x)$ is $[0,\infty)$.
But the thing is a real "grown-up" mathematician would never say $\mathbb R$ is the domain. They'd simply say the if $f(x)$ isn't valid then $x$ is simply not in the domain. Period.
Unless.... well if this is practical math text.
Suppose I told you "I have function and it maps reals to reals and the function is $f(x) = \sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$ and I need to know what the domain is" well, the technical answer is the domain can be what I want it to be. If I want the domain to be $\{0,-1, -2,-3\}$ so that $f(x)= \begin{cases}\frac 1{\sqrt {431}}&x=0\\\sqrt{56}+\frac 1{\sqrt{430}}&x=-1\\\sqrt{106}+\frac 1{427}&x=-2\\12 +\frac 1{\sqrt{422}}&x=-3\end{cases}$
well, then I can have $f:\{0,1,2,3\}\to \mathbb R$ via $f(x) =\sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$. But that's not what I meant.
So then I say "C'mon.... you know what I mean.... I want the biggest set of real numbers that can be the domain.... I want the....um, let's call it 'natural domain'" and then .. well that is a legitimate question.
To define the function $f: X \to \mathbb R$ I must specify what the domain $X$ is.
But do I really have to? Do I really have to say... "well we need $x^3 - 57 \ge 0$ and we need $431 -x^2 > 0$ so the means $x$ must be between...." Do I even care?
Why shouldn't I just say: Let $f: NaturalDomain(\sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}})\subset R\to \mathbb R$. and not really worry about knowing what values of $x$ that $ \sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$ id defined on?