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Consider a function,

$$ f(x)=2x-4\sin x, $$

having a given domain of $[0,2\pi]$.

Is it incorrect to say that its range is,

$$ \{f\in [-1.37,13.93]\}. $$

Please also comment on the notation. I am trying to improve my precision in writing mathematical notations.

rayank97
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    I don't like the notation, $f$ is a function, not a numerical value. You could just right that the range is the interval $[-1.37,13.93]$. Also, just as a technical note, I see the max as $13.936\dots$ so, if you are rounding, I'd have a $13.94$ instead. of course it's easy enough to have "exact" values (in terms of $\pi$ and some square roots) for both min and max so maybe that's preferable. – lulu Oct 18 '22 at 10:30
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    There are a number of standard ways to specify the range, but as lulu said the notation you wrote isn't standard. You could write "the range is $X$" or "$\text{range}(f) = X$ or perhaps least ambiguously $f([0,2\pi]) = X$ (also not the term image is often used instead of range in higher maths). Your notation is off since $f$ itself doesn't lie in the range (it's values lie in the range, so that $f(x) \in \text{range}(f)$ for all $x$ in the domain). – Brevan Ellefsen Oct 18 '22 at 10:35
  • One can formalize functions as ordered pairs of values, one from the domain and one from the codomain (such that each domain value lies in exactly one pair) and you could then view $f$ as an element of this set, but this is a perspective rarely used in practice and is more useful just for defining what a function is set-theoretically – Brevan Ellefsen Oct 18 '22 at 10:39
  • @lulu If I defined the function as $y=f(x)$, then would it be correct to say that the range of the function is $y\in [a,b]$ ? – rayank97 Oct 18 '22 at 14:10
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    That's certainly an improvement, thou I still think it's best to simply say that the range of $f(x)$ is the given interval. – lulu Oct 18 '22 at 14:15

1 Answers1

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Better is $R=[\frac{2\pi}{3}-2\sqrt{3},\frac{10\pi}{3}+2\sqrt{3}].$ If I calculated correctly.

Bob Dobbs
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