Question

State the domain and range of the relation y = arcsin x

Answer 1

for arcsin to be considered a function the domain has to be restricted.

to envision arcsin x, first think of sin x

sin x looks like



if we mirrored this function about the line y = x (which is what the image of the inverse would look like), our domains and ranges would be reversed

since the domain of sin(x) is all real numbers and its range is [-1, 1] we might think that the domain of arcsin(x) is [-1, 1] and its range is all real numbers.

but if we look carefully, we can't just rotate the entire function sin(x) about y = x. we can only rotate a restricted amount of it, otherwise the oscillation of sin(x) will cause the inverse to map inputs to multiple outputs (which defies the definition of a function)

so the domain is [-1, 1] but the range would be [arcsin(1), arcsin(1)]
or [-pi/2, pi/2]