Scaling of functions

[mmwave]

In another thread Hallsofivy wrote

I think you are confusing things by always talking about "the function f(cx)". f(cx) does not represent a function, it represents a value of a function. f(x), f(y), f(cx) all refer to the same function, f.
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Not wanting to hijack someone else's thread I've started this to discuss the following:

I understand that in all cases f( ) is the same function. But if c is greater than one a graph of f(cx) is 'skinnier' than f(x) and when c is less than one f(cx) is 'wider' than f(x). In engineering these are useful tools especially in transforms. x here is the independent variable not a single value. Maybe mathematicians use a different notation to discuss this concept?

Also, if f() is linear function then
f(cx) = c * f(x) for any x and
f(a+b) = f(a) + f(b)

[Hurkyl]

You're implicitly defining a new function, g, given by:

g(x) := f(cx)

and it is the graph of this new function, g, that can be "skinnier" or "wider" than the graph of f.


Because of the common abuses of notation, there is some ambiguity in what precisely is meant.