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Topic: Simple question...or not


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Subject:   RE: Simple question...or not
Author: Craig
Date: Jun 1 2009
On Jun  1 2009, Susan wrote:
> Is a constant function a linear function?  I've seen it argued both
> ways. Please weigh in!

According to the definition on Wolfram's MathWorld site,
http://mathworld.wolfram.com , a linear function is any function for which f(x+y)
= f(x) + f(y) and f(ax) = af(x).  The constant function f(x)=c fits neither of
these criteria, since f(x+y) = c, which is NOT f(x) + f(y).  This definition is
probably not a good one for high school students, though, because the function
f(x) = 2x + 1, commonly recognized as a typical "linear" function, fails on both
criteria, as well!  Helpfully (?), MathWorld identifies a "linear equation" as
one of the form y = mx + b, going on to generalize with a linear equation of n
variables, of which the equation y = c is cited as an example.  I am not
satisfied with these definitions and examples!

The question arises, how, then, do we define a linear function in a way
satisfying to our students?  One approach might be a bit of a cop-out, by
claiming that a linear function is one whose graph is a line.  By this
definition, the constant function is linear, since its graph is a horizontal
line.

Another definition might be a linear function is a polynomial function of degree
one, which would exclude the constant function.

Still another definition might be "any function of the form f(x) = mx + b,"
where the constant function would be considered a linear function unless you add
"where m is not zero" to the definition, which might seem arbitrary.

How do you define an exponential function?  You might say "f(x) = a^x, where
a>0," which includes the constant function when a = 1, or you can add "but a is
not 1" to the definition.

An argument for includeing the constant function in the "family" of linear
functions is so that the algebra of linear functions involving addition is
closed and has an identity, f(x) = 0 (which is the only constant function
actually fits the MathWorld definition of a linear function).

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