1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$, also known as the inverse cosine or arccosine function, and to draw its graph. 2. The principal value of $\cos^{-1} x$ is defined as the range of values that the inverse cosine function can take. By definition, $\cos^{-1} x$ returns an angle $\theta$ such that $\cos \theta = x$. 3. The domain of $\cos^{-1} x$ is $[-1, 1]$ because cosine values range between -1 and 1. 4. The principal value (range) of $\cos^{-1} x$ is the interval $$[0, \pi]$$. This means the output angle $\theta$ is always between 0 and $\pi$ radians (0 to 180 degrees). 5. To summarize: - Domain: $x \in [-1, 1]$ - Principal value (range): $\theta \in [0, \pi]$ 6. The graph of $y = \cos^{-1} x$ is a decreasing function starting at $y=\pi$ when $x=-1$, passing through $y=\frac{\pi}{2}$ at $x=0$, and ending at $y=0$ when $x=1$. Final answer: - Principal value interval: $$[0, \pi]$$