Question

consider the inverse cosine function, defined by ( y=cos ^{-1} x ) or ( y=arccos x ). complete parts (a) through (d)
0,\\( \pi \\)
(type your answer in interval notation. simplify your answer. type an exact answer, using \\( \pi \\) as needed. use integers
(c) is this function increasing or decreasing?
\\( \bigcirc \\) increasing
\\( \checkmark \\) decreasing
(d) ( \arccos \left(-\frac{1}{2}\
ight)=\frac{2 \pi}{3} \\). why is ( \arccos \left(-\frac{1}{2}\
ight) ) not equal to ( -\frac{4 \pi}{3} )?
\\( \bigcirc \\) a. because ( -\frac{4 \pi}{3} ) is not in the range of ( y=\arccos x ).
\\( \bigcirc \\) b. because ( \cos \left(-\frac{1}{2}\
ight) ) is not equal to ( -\frac{4 \pi}{3} )
\\( \bigcirc \\) c. because ( \cos \left(-\frac{4 \pi}{3}\
ight) ) is not equal to ( -\frac{1}{2} )
\\( \bigcirc \\) d. because ( -\frac{4 \pi}{3} ) is not in the domain of ( y=\arccos x )

Explanation:

The range of the inverse cosine function \(y = \arccos(x)\) is \([0,\pi]\). For the value \(-\frac{4\pi}{3}\), we check its membership in the range. Since \(-\frac{4\pi}{3}
otin[0,\pi]\), it cannot be a valid output of \(y = \arccos(x)\).

• Option B is incorrect because \(\cos(-\frac{1}{2})\) is a value related to the cosine of \(-\frac{1}{2}\) (a radian measure), not relevant to the range issue.
• Option C is incorrect as \(\cos(-\frac{4\pi}{3})=\cos(\frac{4\pi}{3})=-\frac{1}{2}\), but the key is the range of \(\arccos(x)\) not the value of the cosine function.
• Option D is incorrect because the domain of \(y = \arccos(x)\) is \([- 1,1]\), and \(-\frac{4\pi}{3}\) is not a domain - related value (domain refers to the input \(x\) values for \(y=\arccos(x)\)).

Answer:

A. Because \(-\frac{4\pi}{3}\) is not in the range of \(y = \arccos x\)