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What’s the difference between “the value $x$” and “the value of $x$”?

I’m from Poland. I read scientific articles and can’t figure out the difference. Some sample sentences: Thus, a function $f$ should be distinguished from its value $f(x_0)$ at the value $x_0$ in its domain. (…) since $f(x)$ and $x_2$ should both be understood as the value of $f$ at $x$. (…) valid for all real values of $x$ ". (Source: https://en.wikipedia.org/wiki/Function_(mathematics))

Thanks in advance,

Chris

Tanner Swett
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Krzysztof Potyrała
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    In english, "the value" and "the value of" have pretty much the same meaning. It has more to do with grammar than mathematics i think. – supinf Sep 07 '18 at 14:56
  • The relevant definition of value from the Oxford Dictionary: "Value: The numerical amount denoted by an algebraic term; a magnitude, quantity, or number." We sometimes refer to variables as values instead. I would point out that we would never say "the value $f$" here since $f$ is a function and not a number, it is a rule for mapping input numbers to outputs. On the other hand $f(x_0)$ is indeed a number and not a function, it is the output of $f$ after $x_0$ was applied to it. When referring to a number though, "the value $x$" and "the value of $x$" are interchangeable. – JMoravitz Sep 07 '18 at 15:05

1 Answers1

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Short answer: I don't think you should worry, as long as you understand this sentence, which says that a function is the whole rule, not a result of applying a rule.

Thus, a function $f$ should be distinguished from its value $f(x_0)$ at the value $x_0$ in its domain.

tl;dr

In the first quoted sentence each "value" is a named object, $x_0$ in the domain and $f(x_0)$ in the codomain.

The problem with "values" comes up because functions are often described using a formula with a "variable", usually $x$. The point of the discussion is to make clear that when the function is defined that way, as in "$f(x) = x^2$" , there is really "no $x$" in the definition,

since $f(x)$ and $x^2$ should both be understood as the value of $f$ at $x$.

Here you say "value of $f$" because "value $f$" makes no sense. The modifier "value" belongs before a number.

Then this last one is really tricky. Here you say "value of $x$" because you are thinking of $x$ not as a number but as the identity function on the domain.

valid for all real values of $x$

Ethan Bolker
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