graph the inverse circular function\n\n$y = \\cos^{-1}(x)$

Explanation:

Step1: Recall the domain and range of (y = \cos^{-1}(x))

The domain of (y=\cos^{-1}(x)) is ([- 1,1]) and the range is ([0,\pi]). When (x = 1), (y=\cos^{-1}(1)=0); when (x=-1), (y = \cos^{-1}(-1)=\pi). The function (y = \cos^{-1}(x)) is a decreasing function.

Step2: Analyze each option

• Option A: The function in option A is increasing, which is not consistent with the property of (y=\cos^{-1}(x)) (since (y = \cos^{-1}(x)) is decreasing).
• Option B: The function in option B has an incorrect range (negative (y) - values are shown, but the range of (y=\cos^{-1}(x)) is ([0,\pi]\approx[0, 3.14])) and is also increasing.
• Option C: The function in option C is increasing and has a wrong range (negative (y) - values are present).
• Option D: The function in option D has the domain ([-1,1]), the range ([0,\pi]) (approximate positive (y) - values), and is a decreasing function.

Answer:

D.