Is $f(x) = x^3 - x$ a mapping or function? I came across this question in the exam and from my knowledge, I chose that it's a function or a function.
But apparently it's not a mapping or function, which I don't understand why.
From what I know:
1.) If there's a rule in which we can assign $x \in \mathbb{R}$, a unique element $y = f(x)$, then such a rule is a mapping.
In the given relation we can assign different values of $x$ and get a unique $y = f(x)$
2.) A function is a mapping whose codomain is a set of numbers.
The relation also satisfies this definition.
So why is $f(x) = x^3 - x$ not a mapping or a function or is the answer wrong.
 A: The terms "function" and "mapping" are generally understood to be synonymous. However, the expression $f(x)=x^3-x$ is not a function. It is an equation that is understood to define a function $f$.
When mathematicians say things like "consider the function $f(x)=x^3-x$", this is intended as a shorthand for "consider the function $f:\Bbb{R}\to\Bbb{R}$ such that, for all $x$, $f(x)=x^3-x$". Notice that $f$ is the function, whereas $f(x)$ is the value of $f$ at $x$. Moreover, when you define a function for the first time, strictly speaking you should specify its domain (and its codomain, according to some definitions of the term "function"). If the domain is not specified, then usually it is assumed to be the largest set of real numbers for which the algebraic formula defining the function makes sense. In your example, the expression "$x^3-x$" makes sense for all real $x$, and so the domain of $f$ is assumed to be $\Bbb{R}$ unless otherwise specified.
A: It’s all a bit fuzzy. Generally a function is a clearly defined mathematical object that includes a domain, a codomain and has to satisfy certain conditions. A mapping or map is not clearly defined. Sometimes they are taken as synonyms, but mapping can also apply to broader concepts, such as mappings between categories or even metatheoretic mappings.
So it does depend on the definition of "mapping" you used.
Now, $f(x) = x^3-x$ is a statement, not a function. But even if we were to rephrase the question to "Let $f(x)=x^3-x$. Is $f$ a function?" the answer would still be no, because a function has to specify it’s domain and it’s codomain. So $f:\mathbb{R}\to\mathbb{R},\,f(x)=x^3-x$ specifies a function.
