So I have f(x)= arccos(cos(x)). When I graphed it on a graphing calculator I see that the f(x) is never lower then 0. So is there an explanation for this?.
The "angle whose cosine is cosine x" is any angle at all.
If you let x represent any angle, it's cosine will be cos x and the angle that has that cosine can be x OR -x, because of the symmetry of the cos x function, which is even. The computer will select the positive value, no matter which sign x has.
That's a very interesting question.