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Given a function $f$, is it technically accurate to ask "what is the domain of $f(x)$" and "what is the range of $f(x)$"? Or should we ask "what is the domain of $f$" and similarly "what is the range of $f$"?

Part of me thinks it's technically accurate to refer to $f(x)$ when talking about range as we are thinking about the outputs but that domain should refer to $f$ rather than $f(x)$ but seeking clarification as there is not much out there.

PhysicsMathsLove
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  • Possibly relevant discussion which you may find useful: https://math.stackexchange.com/questions/276650/what-is-the-difference-between-writing-f-and-fx#:~:text=Due%20to%20heavy%20abuse%20of%20notation%20%28that%20is,We%20often%20denote%20this%20unique%20value%20as%20f%28a%29. – J.D Jun 03 '24 at 19:17
  • Strictly speaking, the term $f(x)$ does not denote the function but rather its value at $x$, while the term $f$ denotes the function. That is, $f$ refers to the action or mapping (or a set modelling that, etc.), while $f(x)$ refers to a single instance of that function evaluated at a variable $x$. In the case of $x\mapsto x^2$, $f$ is the relation of squaring (furnished with the relevant domain and codomain), while $f(x)$ is the variable $x^2$. – Jam Jun 03 '24 at 19:27

1 Answers1

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  • The domain refers to the set of all possible inputs for the function $f$.
  • The range refers to the set of all possible outputs of the function $f$.

Using $f(x)$ focuses on the value of the function at a specific input $x$, whereas the domain and range pertain to the function $f$ as a whole.

So it is "better" to say $f$.

Antony Theo.
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  • Thank you. Is it then "incorrect" to write $f(x)$? – PhysicsMathsLove Jun 03 '24 at 19:05
  • Yes, it is technically incorrect. Think of it this way : $y=f(x)$ The range of the function are the values $y$ can take , you will never say "find the range of $y$". I suggest you read the pages linked bellow ! https://www.andrews.edu/~rwright/Precalculus-RLW/Text/01-04.html , https://mathresearch.utsa.edu/wiki/index.php?title=Domain_of_a_Function – Antony Theo. Jun 03 '24 at 19:37