$\displaystyle \mathfrak{hello}$
what is the domain of this function :
$\displaystyle f(x)=Arcsin(sin(x))$
Hi!
Domain is all real numbers.
This is because $\displaystyle arcsin(x) : [-1,1] \to [-\frac{\pi}{2},\frac{\pi}{2}] $ and we have $\displaystyle sin(x) : \mathbb{R} \to [-1,1] $
Therfore, the inner function, $\displaystyle sin(x)$ , will have codomain equal to the domain of the outer function.
Ok, this is because if you let $\displaystyle x=\frac{3\pi}{4} $ , then $\displaystyle sin(x)=\frac{1}{\sqrt{2}} $
But $\displaystyle arcsin(\frac{1}{\sqrt{2}})=\frac{\pi}{4} $ , not $\displaystyle \frac{3\pi}{4} $
This is why it is only true when $\displaystyle x\in [-\frac{\pi}{2},\frac{\pi}{2}] $