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The domain of the inverse cosine function, $\cos^{-1}x$, is the set of all real numbers $x$ such that $-1 \le x \le 1$. This can be written as $x \in [-1, 1]$.
The domain of the sine function, $\sin x$, is the set of all real numbers. This can be written as $x \in \mathbb{R}$.
The domain of the function $f(x) = \cos^{-1}x + \sin x$ is the intersection of the domains of $\cos^{-1}x$ and $\sin x$. Since the domain of $\cos^{-1}x$ is $[-1, 1]$ and the domain of $\sin x$ is $\mathbb{R}$, the intersection is $[-1, 1]$.
Final Answer: [-1, 1]