What exactly does argument of a function mean?

Question

What does argument of a function mean? I read Wiki etc and I'm still not sure.

Is Argument and Domain one and same thing?

If Domain corresponds to Range? Do we have a term corresponding to Argument too?

Answer 1

The argument is an element of the domain, as for example $f(x):A\to \mathbb{R}$ any $x\in A$ is an argument for the function.

The argument $x$ is also denoted as input, the value $y=f(x)$ is denoted as the output or the image of $x$ by $f$.

Answer 2

For a function $$f:A \to B $$ we call the set $A$ the domain of the function. $A$ is a set from which $f$ takes its arguments, and $B$ is the set where $f$'s outputs live.

The argument $x$ is an arbitrary representative of an element of $A$. For example, in the case, where $f:\mathbb{R} \to \mathbb{R}$ and $f(x) = x^2$, $x$ is a representative for an arbitrary element $x \in \mathbb{R}$.

The terminology differs a little depending on where you look. I prefer to say "$x$ is a point in the domain" and "$f(x) $ is a value in the range". Points in the domain map to values in the range.

Given a fixed value $y$, the preimage of $y \in B$ is the set $$ \{x \in A \mid y = f(x) \}$$

For a non-injective function, there will be a $y$ such that its preimage contains several points.