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Dependent and Independent Variables

Date: 05/24/2005 at 22:22:27
From: Miguel
Subject: Is the Z variable dependent or independent?

In a function, X is considered independent and Y is dependent.  What 
is the Z variable? 

It's obvious that X is independent and Y is dependent...but Z makes 
no sense to me. I guessed "codependent", but maybe not.

Date: 05/24/2005 at 23:33:00
From: Doctor Peterson
Subject: Re: Is the Z variable dependent or independent?

Hi, Miguel.

It depends on the function you have in mind!  The variables x and y 
are not necessarily independent and dependent respectively; it is just
tradition that calls for that.  There is nothing wrong with defining a
function as

  f(y) = 2y - 1

so that if x = f(y), y is independent and x is dependent.

Note that in the definition of the function, x is really just a
placeholder; I could have written it as "f(t) = 2t-1" or "f(z) = 2z-1"
and the function would still be the same thing.  Presumably you are
thinking of an equation like

  y = 2x - 1

as representing a function you want to graph.  But again, you can just
as well graph the function

  x = (y + 1)/2

in which y is independent and x is dependent.

Similarly, z can come into an equation in various ways.  For example,
you might define a SURFACE

  z = f(x,y)

where, for each value of the pair (x,y), there is one value of z
"above that point" on the surface; then x and y are BOTH independent,
and z is dependent.  But you could instead define a CURVE this way:

  y = f(x), z = g(x)

so that for each value of x, there is a pair (y,z) on the curve.  Then
x is independent and y and z are BOTH dependent.  And you can also
define a curve this way (a parametric equation):

  x = f(t), y = g(t), z = h(t)

so that ALL THREE variables are dependent on a "time" variable t.

So there is nothing inherent in any variable that makes it dependent
or not; and even convention does not dictate what role each will play.
But z does tend to be dependent, and x to be independent, just by

Did you have some particular function in mind?

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Functions

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