Find the domain of `cos^(-1)(2x-1)`

Options

A. `x in [0,1]`
B. `x in [-1,0]`
C. `x in [-1/2,0]`
D. `x in [0,1/2]`

Explanation

To find the domain of the function \( \cos^{-1}(2x - 1) \), we need to determine the values of \( x \) for which the expression \( 2x - 1 \) lies within the range of the cosine inverse function.

### Step-by-Step Solution:

1. **Understand the Range of \( \cos^{-1}(x) \)**:
The function \( \cos^{-1}(x) \) is defined for \( x \) values in the range \([-1, 1]\). Therefore, we need to set up the inequality:
\[ -1 \leq 2x - 1 \leq 1 \]

2. **Break Down the Inequality**:
We can break this compound inequality into two parts:
- \( 2x - 1 \geq -1 \)
- \( 2x - 1 \leq 1 \)

3. **Solve the First Inequality**:
For the first part:
\[ 2x - 1 \geq -1 \]
Adding 1 to both sides gives:
\[ 2x \geq 0 \]
Dividing by 2 results in:
\[ x \geq 0 \]

4. **Solve the Second Inequality**:
For the second part:
\[ 2x - 1 \leq 1 \]
Adding 1 to both sides gives:
\[ 2x \leq 2 \]
Dividing by 2 results in:
\[ x \leq 1 \]

5. **Combine the Results**:
From the two inequalities, we have:
\[ 0 \leq x \leq 1 \]
This can be expressed in interval notation as:
\[ [0, 1] \]

### Conclusion:
The domain of the function \( \cos^{-1}(2x - 1) \) is:
\[ \text{Domain} = [0, 1] \]