What is the domain of $y={\cos }^{-1}x$ ?

• A. $-1\le x\le 1$
• B. $-\infty <x<\infty$
• C. $0\le x\le 1$
• D. $-1<x<1$

Answer: (A)

Verified step by step guidance

1. Recognize that the function given is the inverse cosine function, written as \(y = \cos^{-1}(x)\), also known as arccosine.
2. Recall that the domain of the inverse cosine function consists of all \(x\) values for which the cosine function outputs values, since \(\cos^{-1}(x)\) is defined only when \(x\) is in the range of the cosine function.
3. Understand that the cosine function outputs values between \(-1\) and \$1\( inclusive, so the domain of \)y = \cos^{-1}(x)\( must be all \)x\( such that \)-1 \leq x \leq 1$.
4. Express the domain in interval notation or inequality form as \(-1 \leq x \leq 1\) to clearly indicate all valid input values for the inverse cosine function.
5. Note that values of \(x\) outside this interval are not in the domain because the inverse cosine function is not defined for those values.