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

The domain of the function $f(x) = \cos^{-1} x + \sin x$ is determined by the constraints of its components. The inverse cosine function, $\cos^{-1} x$, is defined only for $x$ values within the interval $[-1, 1]$. The sine function, $\sin x$, is defined for all real numbers and thus imposes no additional restrictions. Consequently, the domain of $f(x)$ is solely dictated by the domain of $\cos^{-1} x$, which is $x \in [-1, 1]$. Therefore, the domain of $f(x)$ is $[-1, 1]$. The correct option is (C) $[-1, 1]$.