Question

graph the inverse circular function

$y = \cos^{-1}(x)$

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.