Defining the rule of an arbitray function

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For any ##f:\Re \rightarrow \Re ##, is the only reason that we typically define the "rule" of the function with the the free variable, ##x##, as the argument of the function, e.g., ##f(x) = x^2 + 1##, because it's simply easier than having to do something like ##f(~) = (~)^2 + 1##? That is, does the use of ##x## not really serve a purpose when defining the function besides that of convenience and readability?

Also, as an extension to this question, if it is the case that the free variable ##x## is just there for convenience, then how would we right the differentiation operator ##\frac{\mathrm{d} }{\mathrm{d} x}##? Would you write it as ##\frac{\mathrm{d} }{\mathrm{d} (~)}## or something? Also, would one even be able to define the derivative ##\displaystyle\lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x ) - f(x)}{\Delta x}## without the use of the free variable ##x##?
 

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  • #2
jbunniii
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For any ##f:\Re \rightarrow \Re ##, is the only reason that we typically define the "rule" of the function with the the free variable, ##x##, as the argument of the function, e.g., ##f(x) = x^2 + 1##, because it's simply easier than having to do something like ##f(~) = (~)^2 + 1##? That is, does the use of ##x## not really serve a purpose when defining the function besides that of convenience and readability?
Pretty much, yes. The convenience and readability become even more important if you consider functions of multiple variables. E.g. ##f(x,y,z) = x^2 y z^3## vs. something like ##f((~), [~], \{~\}) = (~)^2[~]\{~\}^3##.
Also, as an extension to this question, if it is the case that the free variable ##x## is just there for convenience, then how would we right the differentiation operator ##\frac{\mathrm{d} }{\mathrm{d} x}##? Would you write it as ##\frac{\mathrm{d} }{\mathrm{d} (~)}## or something?
You could use the notation ##D##, e..g ##Df## is the derivative of ##f##. If ##f## is a function of ##n## variables, then ##D_1, D_2, \ldots D_n## are standard symbols for the partial derivative operators. See http://en.wikipedia.org/wiki/Differential_operator
Also, would one even be able to define the derivative ##\displaystyle\lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x ) - f(x)}{\Delta x}## without the use of the free variable ##x##?
$$f'((~)) = \lim_{[~]\rightarrow 0}\frac{f((~) + [~] ) - f((~))}{[~]}$$
 
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It is not what you have asked, but a symbol for a variable in the "operator" ##\frac{d}{dx}## is just useful for doing long calculations in a draft at most. It is not rigorous, it is not used in the definition and in fact is long time dead.

As a paradox, the same type of variable symbol is largely used in the definition of the limit of a function and is an ancient mistake. A much better notation would be ##\displaystyle \lim_{a} f## for the limit of a function ##f## at an accumulation point ##a## of the domain of ##f##. Again, the old notation is great for doing long calculations in a draft.

Now, the derivative of a function ##f : A \to \mathbb{R}## at a point ##a \in A \subset \mathbb{R}## is defined like this: consider the Newton's quocient function of ##f## at the point ##a## given by ##\displaystyle Qf_a (t) = \frac{f(a+t) - f(a)}{t}##, with an appropriate domain. Then the derivative of ##f## at the point ##a## is just ##\displaystyle \lim_{0} Qf_a##, if it exists. :-)
 

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