Question 4

What are the domain and range of the function y=2cos⁡–1(2x)+2sin⁡–1(2x)y = 2\cos^{–1}{(2x)} + 2\sin^{–1}{(2x)}?

What is the minimum size of the choir to know that there must be at least 20 students in the choir from one of the schools?

A. Domain: [−0.5,0.5]\left[-0.5, 0.5\right] and Range: {π}\lbrace{\pi\rbrace}

B. Domain: [−0.5,0.5]\left[-0.5, 0.5\right] and Range: [−π,3π][-\pi, 3\pi]

C. Domain: [−2,2]\left[-2, 2\right] and Range: {π}\lbrace{\pi\rbrace}

D. Domain: [−2,2]\left[-2, 2\right] and Range: [−π,3π][-\pi, 3\pi]

Solution

The cos⁡−1\cos^{-1} and sin⁡−1\sin^{-1} functions are valid for input values between [−1,1][-1, 1], which means that we have a domain of

−1≤2x≤1−12≤x≤12\begin{align} -1 &\leq 2x &\leq 1 \\ -\frac{1}{2} &\leq x &\leq \frac{1}{2} \end{align}

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