Answer:
D
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]-4\le x\le 4,[/tex]
we have that
[tex]-1\le \dfrac{1}{4}x\le 1.[/tex]
This gives us the domain [tex]-4\le x\le 4[/tex] of the function [tex]y=\sin^{-1}\left(\dfrac{1}{4}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}{4}x\right)[/tex] is the same (options A and C are false).
When x=-4,
[tex]y=\sin^{-1}\left(\dfrac{1}{4}\cdot (-4)\right)=\sin^{-1}(-1)=-\dfrac{\pi}{2}.[/tex]
So, option B is false and option D is true.
