This question is difficult to put into words, but here we go...
1) If the domain of y = sin(x) is all possible x values...
2) And if the range of y = sin(x) is all possible y values...
...When looking at the inverse of this function...
3) Wouldn't the domain of x = sin(y) be all possible
y values?...
4) And, similarly, the range of x = sin(y) be all possible
x values?...
...Regardless of the notation being y = arcsin(x) or y = sin^-1(x)
I ask because I'm confused as to why the y values of x = sin(y) are always referred to as the range in this situation (and x values as the domain). Maybe I'm missing something huge, but it seems to me that the common notation (y = arcsin(x)) should not take precedence over the actual
meaning (x = sin(y)) when deciding upon the domain and range of a function. Wow, that was difficult to express!