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I know that functions and dependent variables are two fundamentally different things, one "produces" results based on its input(s), and the other "represents" values.

But is there any practical differences between them, since both represent a value based on another.

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Also note the relation of the notion of dependent variable to algebraic (or linear) dependency between variables. For example, if we have $x^2+y^2=1$, then the variables $x$ and $y$ are not independent. For jointly distributed random variables $X$ and $Y$, we also can define the notion of independence. Here, algebraic dependency precludes stochastic independency, but absence of stochastic independency doesn't entail algebraic dependency. And being dependent is not necessarily a causal relationship. – Thomas Klimpel Feb 14 '12 at 7:51
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Attempt at an analogy:

Think of a function as being the entire telephone directory, while "independent variable" refers to the name on any particular line of the directory, and the "dependent variable" to the phone number in any particular line of the directory.

(Now the analogy is not perfect, but I think it points in the right direction; and note that a particular phone number is not the same thing as the telephone directory)

An informal definition of function is:

A function is a rule that assigns to every valid input one, and only one, output.

A slightly less informal defintion:

A function $f$ from $A$ to $B$ (where $A$ and $B$ are sets) is a rule such that:

  1. For every $a\in A$ there is a $b$ in $B$ such that $f$ "associates" $a$ to $b$.
  2. For every $a\in A$, if there are elements $b$ and $b'$ of $B$ such that $f$ "associates $a$ to both $b$ and $b'$, then $b=b'$.

(A more formal definition would require that we go into the "basement" and deal directly with sets, ordered pairs, etc., which will probably not give you any more insight).

In the special case in which $A$ and $B$ are subsets of the real numbers, and the function $f$ can be described via a "formula", we often refer to the input as the "independent variable" and the output as the "dependent variable". We think of being "free" (independent) to choose whatever we want for the input, but once the input is chosen, the output is forced (depends on the input, is not 'free').

If you think of a function in which $A$ and $B$ are sets of real numbers as its graph,, then the "independent variables" are the $x$-coordinates of points on the graph; the "dependent variables" are the $y$-coordinates.

The important thing is not to confuse the function (which is the entire association) with the formula for the function, or with the values of the function. (By the by: "Variable" and "function" were introduced by Leibnitz (who lived in the late 17th and early 18th century); the terms "independent" and "subordinate variable" occur in a translation of a textbook of Lacroix in 1816; "dependent variable" seems to appear for the first time in 1831.)

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Thank you so much for this detailed explanation, and for emphasizing the difference between a function and a formula. Your effort is greatly appreciated! – seininn Feb 14 '12 at 5:09

First, the boring answer. If $x$ and $y$ are (real-valued) variables, then $y(x)$ never makes sense. You cannot evaluate a real number at a real number. If $f$ is a variable of type 'function-whose-domain-is-the-reals', however, then $f(x)$ does make sense.

Of course, people abuse notation. e.g. they might name one of their real-number variables $y$, and then name one of their function-variables $y$. I don't like it, though. :(

Now for the more interesting answer. Dependent variables are more difficult to define since it's an issue of syntax and grammar -- things usually (and IMO unfortunately) glossed over at an introductory level.

Dependent variables tend to arise naturally when trying to express things. For example, in the course of solving a problem, I may introduce a variable $P$, whose domain is the unit circle. I will likely find it convenient to introduce other variables such as $x$ and $y$, and insist on an algebraic dependence: $P = (x,y)$. I may want other variables, such as $\theta$, with the obvious meaning intended.

Now, $P$ is just a point, and $x$, $y$, and $\theta$ are all just real numbers, and they have various interrelationships, such as $x^2 + y^2 = 1$. i.e. dependent variables. Dependent variables are often very useful for setting up problems and doing a variety of calculations.

However, they're not functions -- at least not in the usual sense. I can always consider a model -- a functions $\xi$ that transform the variables into specific numerical values that satisfy all of the dependencies; e.g. the function defined by $\xi("x") = 0$, $\xi("y") = 1$, $\xi("\theta") = \pi/2$, and $\xi("P") = (0,1)$. Of course, one often turns this on its head, and instead thinks of $\xi$ as living in some hypothetical state space, and turning the variables into functions: $x(\xi) = 0$, $y(\xi) = 1$, $\theta(\xi) = \pi/2$, and $P(\xi) = (0,1)$. Occasionally, you can conflate one of the variables with this "state", but it doesn't always work out nicely.

This idea is often needed for doing calculus. However, it is not needed for differential forms (i.e. $dx$, $dy$, $dP$, $d\theta$) -- $dx$ makes sense as a 'differential number' in the same way that $x$ makes sense as a "number". But that's another topic.

Anyways, relations among variables are all fine and dandy, but they're often awkward to work with -- in various situations functions are much easier to work with. So, I might be interested in writing down a bivariate function $f$ with the property:

If y > 0, then y = f(x)

And now that I have expressed the dependence of $x$ and $y$ on the domain $y>0$ in this fashion, I can use all of the nice properties I know about functions to solve problems. Similarly, I might do the same on the other half-circles: e.g. find another function $g$ such that

If x > 0, then x = g(y)

Since functions are much easier to work with, you usually see things first introduced in terms of functions. e.g. calculus is introduced as ways to work with functions. Of course, one usually work with dependent variables too, but it's learned through example -- and, unfortunately, often with the implicit suggestion that you're "supposed" to think of it as being short-hand for something involving functions. It's not until differential geometry do you get some rigorous foundations for doing things the way you would like with dependent variables. (of course, it often comes with the implicit understanding that they're really functions, such as the on 'state space' I described above, but meh)

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A dependent variable is the variable which results from applying a function to the independent variable.

"In mathematics, a function is a correspondence[1] that associates each input with exactly one output." -wikipedia

That is the dependent variable is the output generated by putting the independent variable into some function.

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A dependent variable is a result while a function is what gives that result. Thank you for this explanation. – seininn Feb 14 '12 at 5:13

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