\[\arcsin x \]
It is Inverse trigonometric function whose domain is [-1,1] .It means x (which is input here) can take any real values from -1 to 1.
Range is \[\frac{-\pi }{2} ,\frac{+\pi }{2} \] .That means output what we get by putting any real numbers from -1 to 1 will be in \[\frac{-\pi }{2} , \frac{+\pi }{2} \].
For ex- \[\arcsin (x-1) \] . In this case -1 \[\leqslant \] x-1 \[\leqslant \]1 since domain is [-1,1]
=> 0\[\leqslant \] x <\[\leqslant \]2.
so above function is defined only when input is given from 0 to 2.
