The domin of the function `cos^(-1) (2x-1)` is

Options

A. `[0,1]`
B. `[-1,1]`
C. `(-1,1)`
D. `[0,pi]`

Explanation

To find the domain of the function \( \cos^{-1}(2x - 1) \), we need to ensure that the expression inside the inverse cosine function lies within the acceptable range for the cosine function, which is between -1 and 1, inclusive.

### Step-by-step Solution:

1. **Set Up the Inequality**:
We start by setting up the inequality based on the range of the cosine function:
\[ -1 \leq 2x - 1 \leq 1 \]

2. **Split the Inequality**:
We can split this compound inequality into two separate inequalities:
\[ 2x - 1 \geq -1 \quad \text{and} \quad 2x - 1 \leq 1 \]

3. **Solve the First Inequality**:
For the first inequality \( 2x - 1 \geq -1 \):
\[ 2x \geq 0 \implies x \geq 0 \]

4. **Solve the Second Inequality**:
For the second inequality \( 2x - 1 \leq 1 \):
\[ 2x \leq 2 \implies x \leq 1 \]

5. **Combine the Results**:
Combining the results from the two inequalities, we find:
\[ 0 \leq x \leq 1 \]

6. **Express the Domain**:
Therefore, the domain of the function \( \cos^{-1}(2x - 1) \) is:
\[ [0, 1] \]

### Final Answer:
The domain of the function \( \cos^{-1}(2x - 1) \) is \( [0, 1] \).