Question

Domain of $f(x) = \cos^{-1} x + \sin x$ is:

• A. $\mathbb{R}$
• B. $(-1, 1)$
• C. $[-1, 1]$
• D. $\varnothing$

Hint:

For inverse trigonometric functions like $\cos^{-1} x$, always remember the domain is restricted to the interval $[-1, 1]$. If other terms like $\sin x$ are added, they do not restrict the domain further.

Answer 1

The Correct Option is C

To find the domain of the function $f(x) = \cos^{-1} x + \sin x$, we need to consider the domain of each individual part of the function: 1. The domain of $\cos^{-1} x$ (inverse cosine) is $x \in [-1, 1]$. This means that the values of $x$ must lie in the interval $[-1, 1]$ for $\cos^{-1} x$ to be defined. 2. The second part of the function is $\sin x$, which is defined for all real numbers. Therefore, it does not impose any additional restrictions on the domain. Thus, the domain of the entire function is determined by the restriction imposed by $\cos^{-1} x$, which requires $x \in [-1, 1]$. Therefore, the domain of $f(x)$ is $[-1, 1]$. Hence, the correct option is (C) $[-1, 1]$.