I know that a function is typically defined as a mapping such as ##f: x\rightarrow f(x)##, but I am confused about how we really "define" functions. In many common texts, we define functions as ##f(x)##, such as ##f(x) = 2x##. And we say that we take the derivative of a function ##f(x) = 2x## by ##\frac{\mathrm{df(x)} }{\mathrm{d} x} = 2##. My question is, why do we define functions based on the output, ##f(x)##, rather than the actual function ##f##? And why do we call ##f(x)## the function when really ##f## is the function? Also, why do we say that a derivative is an operator that maps functions to functions, when the input to the operator is the output of the function ##f##, ##f(x)##, and not the actual function? These are all rather pedantic questions, but I really need to get to the bottom of this seeming abuse of notation. (I made a post on this not to long ago, but these are more specific questions).(adsbygoogle = window.adsbygoogle || []).push({});

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# Abuse of function notation

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