Answer:
C
Step-by-step explanation:
The arcsine ([tex]\sin^{-1}[/tex]) function of x is defined as the inverse sine function of x when -1≤x≤1.
So, when
[tex]-2\le x\le 2,[/tex]
we have that
[tex]-1\le -\dfrac{1}{2}x\le 1.[/tex]
This gives us the domain [tex]-2\le x\le 2[/tex] of the function [tex]y=\sin^{-1}\left(-\dfrac{1}{2}x\right).[/tex]
The range of the function [tex]y=\sin^{-1}x[/tex] is [tex]-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2},[/tex] so the range of the function [tex]y=\sin^{-1}\left(-\dfrac{1}{2}x\right)[/tex] is the same (options B and D are false).
When x=-2,
[tex]y=\sin^{-1}\left(-\dfrac{1}{2}\cdot (-2)\right)=\sin^{-1}(1)=\dfrac{\pi}{2}.[/tex]
So, option A is false and option C is true.
