Question

What is the domain of f(x)=arccos(x)?

• A. (0, $\infty$)
• B. [0,$\pi$]
• C. (-$\infty$,$\infty$)
• D. [-1,1]

Answer 1

Answer: (D)

[-1,1]

Answer 2

The arccosine function, $f(x) = \arccos(x)$ (also written as $\cos^{-1}(x)$), is the inverse of the cosine function.

To define an inverse function, the original function must be restricted to an interval where it is one-to-one. For the cosine function, the standard restricted domain is $[0, \pi]$.

Over the interval $[0, \pi]$, the cosine function, $\cos(x)$, takes all values from $-1$ to $1$. Specifically, $\cos(0) = 1$ and $\cos(\pi) = -1$. So, the range of $\cos(x)$ for $x \in [0, \pi]$ is $[-1, 1]$.

When we find the inverse function, the domain of the inverse function is the range of the original function (restricted domain), and the range of the inverse function is the domain of the original function (restricted domain).

Therefore, for $f(x) = \arccos(x)$:

• The domain of $\arccos(x)$ is the range of $\cos(x)$ over $[0, \pi]$, which is $[-1, 1]$.
• The range of $\arccos(x)$ is the restricted domain of $\cos(x)$, which is $[0, \pi]$.

Thus, the domain of $f(x) = \arccos(x)$ is $[-1, 1]$.