Are $f(x,y,z)=x$ and $f(x)=x$ equivalent?

Question

Suppose I have a function $f(x,y,z)=x$, is it equivalent to $f(x)=x$? Is it a correct abbreviation?

Or can I maybe write $g(x)=f(x,y,z)$ so $g(x)=x$?

What is mathematically correct?

Thanks!

Answer 1

Technically, this is not correct. Although you're capturing all the values of $f$ by just looking at the $x$ component, a function comes equipped with three pieces of data: first, the domain $D$ (in this case I assume you mean the domain of $f$ to be $\Bbb R^3$), the codomain $C$ (most likely $\Bbb R$, at least if my domain assumption is correct), and the rule itself (here $(x,y,z)\mapsto x$, more generally a subset $G\subseteq D\times C$ such that if $(d,c), (d,c')\in G$, then $c = c'$ and such that for every $d\in D$, there exists $c\in C$ such that $(d,c)\in G$). These three pieces of data are important to take into consideration when determining properties of $f$: two functions given by the same rule but with different domains/codomains might fail to have the same properties. You could define $g(x) = f(x,y_0,z_0)$ (for any choice of $y_0,z_0\in\Bbb R$), but then you have a function $g : \Bbb R\to\Bbb R$ which is now surjective and injective, whereas $f$ was surjective but far from injective.

In fancier language, you might capture your idea by saying the value of $f$ only depends on the first coordinate of $\alpha\in\Bbb R^3$, and so $f : \Bbb R^3\to\Bbb R$ descends to a function $\tilde{f} : X\to\Bbb R$ on the quotient space $X = \Bbb R^3/\sim$, where $(x,y,z)\sim (x',y',z')$ if $x = x'$, and this $X$ is homeomorphic to $\Bbb R$.