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For example, are forms like $\sin(\arcsin(x))$ considered polynomials?

Yes it simplifies to $x$, but $x$ and $\sin(\arcsin(x))$ have very different domain and ranges.

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    $\begingroup$ Depends what you mean by a polynomial here. Your example is a function which on its domain coincides with a polynomial function. $\endgroup$
    – Mark
    Apr 15, 2023 at 17:09
  • $\begingroup$ I'd say "no". I'd describe that as "a function that reduces/simplifies to a polynomial (on its range, which isn't all of $\mathbb R$)". $\endgroup$
    – JonathanZ
    Apr 15, 2023 at 17:13
  • $\begingroup$ @Mark Isn't polynomial a well defined term in math? Or, is it more logical to consider that as a polynomial or not? $\endgroup$
    – N Kabir
    Apr 15, 2023 at 17:16
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    $\begingroup$ For $~x \in [-1,1],~$ let $f(x) = x.~$ Is $~f(x)~$ considered a polynomial? $\endgroup$ Apr 15, 2023 at 17:28
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    $\begingroup$ Related $\endgroup$
    – Babu
    Apr 19, 2023 at 0:52

1 Answer 1

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A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division. It does not include trigonometric functions.

$\sin(\arcsin(x))$ is a trigonometric function and is not an expression of any power of $ x $, that is why it is not a polynomial.

Trigonometric functions are periodic (a function that repeats its values at regular periods) and cannot be described by a polynomial.

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