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I've seen it written 'the function $f(x)$', is it not the function $f$ and $f(x)$ is it's value at $x$ e.g. for some number $x$ there exists a value $f(x)$ depending on $x$?

I also see used to describe that 'the function $f(x)$ allows us to know the value $f(a)$ for any $a$' Again, surely the value is $f(x)$ and we can determine $f(x)$ for any $x$ in the domain, and this is the same? A textbook I have uses this language and also describing $a$ as a constant? Surely if we say 'for any $a$' we are allowing $a$ to change in this context as the 'for any' is a universal quantifier over the domain?

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    Saying that "the function $f(x)$" is sloppy but is used often. The function is $f$ and $f(x)$ is its value at $x$. – Gary Mar 06 '22 at 13:51
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    People use that expression a bit casually. Sometimes it denotes the function, other times the value of the function at $x$. Context ought to make it clear which is intended. Note: writing "we can determine $f(x)$ for any $x$" seems awfully strong. It's easy to produce examples of functions for which some particular value is beyond our ability to compute. – lulu Mar 06 '22 at 13:52
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    @lulu are you saying defined functions on a point can be hard to compute? what do you mean by beyond our ability to compute? – AyamGorengPedes Mar 06 '22 at 13:55
  • let me rephrase that as 'there exists f(x) for any $x$ in the domain. –  Mar 06 '22 at 13:55
  • Where there are different variables about $f(x)$ can be used to emphasise the functional dependence on a particular variable of interest. Essentially, this usage is so common that you will need to get used to it. It is occasionally useful to know that a finer distinction has to be made. – Mark Bennet Mar 06 '22 at 13:57
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    Consider the function on the reals given by $f(x)=1$ if $x\in \mathbb Q$ and $f(x)=0$ if $x\not \in \mathbb Q$. We can't compute $f(e+\pi)$, for example. And so on. Any unsolvable problem could be coded as the value of a function. Being able to compute values of a function should not be linked to the notion of "function". – lulu Mar 06 '22 at 13:57
  • @lulu ah I see what you mean. I interpret functions as a look up table anyway, but I never though of this example. Nice! – AyamGorengPedes Mar 06 '22 at 14:01

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