# What constitutes as a polynomial?

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.

• Depends what you mean by a polynomial here. Your example is a function which on its domain coincides with a polynomial function.
– Mark
Apr 15, 2023 at 17:09
• 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 $\mathbb R$)". Apr 15, 2023 at 17:13
• @Mark Isn't polynomial a well defined term in math? Or, is it more logical to consider that as a polynomial or not? Apr 15, 2023 at 17:16
• For $~x \in [-1,1],~$ let $f(x) = x.~$ Is $~f(x)~$ considered a polynomial? Apr 15, 2023 at 17:28
• Related
– Babu
Apr 19, 2023 at 0:52

$$\sin(\arcsin(x))$$ is a trigonometric function and is not an expression of any power of $$x$$, that is why it is not a polynomial.
• @Gonçalo $sin(arcsin(x))$ is a trigonometric function not a polynomial. Apr 24, 2023 at 23:27