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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)