What are the domains and ranges for the inverse cosine function, $$ \cos^{-1}(x) $$?

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To determine the domain and range of the inverse cosine function, $$ \cos^{-1}(x) $$, we can follow these steps:

1. **Identify the domain of the original cosine function**:
   - The cosine function, $$ \cos(x) $$, has a domain of all real numbers, $$ (-\infty, \infty) $$.
   - The range of the cosine function is limited to the interval $$ [-1, 1] $$.

2. **Determine the domain of the inverse cosine function**:
   - The domain of the inverse function is the range of the original function. Therefore, the domain of $$ \cos^{-1}(x) $$ is:
     $$
     x \in [-1, 1]
     $$

3. **Determine the range of the inverse cosine function**:
   - The range of the inverse cosine function corresponds to the domain of the original cosine function when restricted to the interval $$ [0, \pi] $$ (the principal value range for $$ \cos^{-1}(x) $$). Thus, the range of $$ \cos^{-1}(x) $$ is:
     $$
     y \in [0, \pi]
     $$

In summary:
- The **domain** of $$ \cos^{-1}(x) $$ is $$ [-1, 1] $$.
- The **range** of $$ \cos^{-1}(x) $$ is $$ [0, \pi] $$.
The inverse cosine function, $$ \cos^{-1}(x) $$, has a domain of $$ [-1, 1] $$ and a range of $$ [0, \pi] $$.

What are the domains and ranges for the inverse cosine function, \( \cos^{-1}(x) \)?

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