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From the top of this page of Wikipedia-

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.

From the same page-

A mathematical function has one or more arguments in the form of independent variables designated in the function's definition.

Again from the same page-

The independent variables are mentioned in the list of arguments that the function takes.

So, I gets confused about the difference of these terms. It seems to me that they should be different but Wikipedia is not agreeing. Can you help me with this?

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  • $\begingroup$ If there is a difference, it's a very fine one. I honestly have no idea. $\endgroup$ – Arthur Sep 16 '16 at 5:36
  • $\begingroup$ I don't think these "definitions" are necessarily contradictory. In any case, the whole concept of "variable of a function" has no mathematical purpose and communication-wise, it's not useful enough to merit its existence, in my opinion. $\endgroup$ – Git Gud Sep 16 '16 at 5:39
  • $\begingroup$ Because I've never heard of the term "independent variable" to be replaced by the "argument" before this. Although the vice versa replacement looks ok since when we talk about $e^{i\theta}$, here books refer $\theta$ as an argument but it is like variable here. Also, in C programming we prefer the term argument for the things present after "," of printf(). So, it looks like that argument is a bigger, more general term. $\endgroup$ – ankit Sep 16 '16 at 5:40
  • $\begingroup$ function in math and function in programming do not mean the same thing. For example rand() is a standard C function, but doesn't qualify as a function mathematically. $\endgroup$ – dxiv Sep 16 '16 at 6:01
  • $\begingroup$ I've never heard of the term "independent variable" to be replaced by the "argument" before this That's highly dependent on the context, and usually easy to understand from the context. Consider this: "for any $z \in \mathbb{C} \text{Arg } z + \text{Arg } \bar z = 0$". It's generally obvious that $z$ is an independent variable (because of the unconstrained any $z$), and also an argument to the function $\text{Arg}$. $\endgroup$ – dxiv Sep 16 '16 at 6:07
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The terminology is a little arbitrary. One case where the two might not be used interchangeably is when there are two types of arguments - independent variables and parameters. Sometimes you might see a function written in a form like $f(x; \theta)$, meaning that $x$ is the "main" independent variable, and $\theta$ is an adjustable parameter to change what kind of function $f(x)$ looks like.

For example, you could write a generic quadratic function as $f(x; a, b, c) = ax^2 + bx + c$ - all of $x, a, b, c$ are arguments to the function, but for whatever purpose we happen to care about we want to treat it as a quadratic function in $x$, and $a, b, c$ are the tuning parameters to pick what shape the quadratic has.

It's possible there are other kinds of arguments to functions, but that's the one case that comes to mind for me.

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To add to the confusion, suppose $f(x,y)=x^2-y$, and consider the set of all $(x,y)$ where $f(x,y)=0$. Then, we can say that such $y$ are a function of $x$ (since, given $x$, the only $y$ which makes the function zero is $y=x^2$). However, $x$ is not a function of $y$.

Though, this can be cleaned up, I think:

  • A function takes arguments to produce a value associated with a particular set of arguments.

  • A function represents a relationship between the arguments and the value: given the arguments, one may calculate the value (hence we may say the value is a function of the arguments).

  • A function may be used to model independent and dependent variables, since a dependent variable is a function of the independent variables. In this case, the independent variables are the arguments of the function.

  • In the above example with $f(x,y)=x^2-y$, there are two functions I was speaking about. The first was $f$ itself, whose value is a function of both $x$ and $y$, and the second is the fact that we may solve $f(x,y)=0$ to get those $y$ which satisfy that constraint as a function of $x$.

  • Sometimes it is not clear which variables are independent and which are dependent: for $g(x)=2x$, sure $2x$ is a function of $x$, but $x$ is also a function of $2x$, so to speak. That is, given $2x$, one may obtain $x$ again by dividing by $2$. This is described by saying $g$ is an invertible function, with $g^{-1}(x)=\frac{1}{2}x$.

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A function has arguments, one or more : "square root" is a unary function (it has one argument), $+$ is a binary function (two arguments), and so on.

A variable is a syntactical object (a symbol) used in the expression of a function to designate an argument place: it is a "place holder".

We can write e.g "$f$( __)" to designate the fact that the function $f$ has one argument, like an "empty slot" to be filled with an input value in order to "calculate" the corresponding value of the function: the output.

With more arguments, it can be misleading to use "slots" :

$+$(_ , _)

and thus the use of variables for denoting argument-places has been adopted :

$+(x,y)$

i.e. by convention : $x+y$.

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