Find domain for, `f(x)=cos^(-1)[x]`.

Answer

To find the domain of the function \( f(x) = \cos^{-1}(x) \), we need to consider the properties of the inverse cosine function.

### Step-by-Step Solution:

1. **Understand the function**: The function \( f(x) = \cos^{-1}(x) \) is defined for values of \( x \) that fall within a specific range. The inverse cosine function, \( \cos^{-1}(x) \), is defined for \( x \) in the interval \([-1, 1]\).

2. **Identify the range of the greatest integer function**: The greatest integer function, denoted as \( \lfloor x \rfloor \), outputs the largest integer less than or equal to \( x \). This means that the output of \( \lfloor x \rfloor \) can take any integer value.

3. **Determine the values of \( \lfloor x \rfloor \)**: For \( f(x) = \cos^{-1}(\lfloor x \rfloor) \) to be defined, \( \lfloor x \rfloor \) must fall within the interval \([-1, 1]\). Therefore, we need to find the integer values that satisfy this condition.

4. **Find integer values**: The integers that lie within the interval \([-1, 1]\) are:
- \(-1\)
- \(0\)
- \(1\)

5. **Translate back to \( x \)**:
- For \( \lfloor x \rfloor = -1 \): This occurs when \( -1 \leq x < 0 \).
- For \( \lfloor x \rfloor = 0 \): This occurs when \( 0 \leq x < 1 \).
- For \( \lfloor x \rfloor = 1 \): This occurs when \( 1 \leq x < 2 \).

6. **Combine the intervals**: The combined intervals where \( \lfloor x \rfloor \) takes values within \([-1, 1]\) are:
- From \(-1\) to \(0\) (not including \(0\)),
- From \(0\) to \(1\) (not including \(1\)),
- From \(1\) to \(2\) (not including \(2\)).

7. **Final domain**: Thus, the domain of \( f(x) = \cos^{-1}(\lfloor x \rfloor) \) is:
\[ x \in [-1, 2) \]