What is the domain of the function `cos^(-1)(2x-1)`?

Options

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

Answer

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 inverse cosine function. The inverse cosine function, \( \cos^{-1}(y) \), is defined for \( y \) in the interval \([-1, 1]\).

### Step-by-Step Solution:

1. **Set up the inequality**: Since \( \cos^{-1}(2x - 1) \) is defined when \( 2x - 1 \) is between -1 and 1, we can write the inequality:
\[ -1 \leq 2x - 1 \leq 1 \]

2. **Solve the left part of the inequality**: Start with the left side:
\[ -1 \leq 2x - 1 \]
Add 1 to both sides:
\[ 0 \leq 2x \]
Divide by 2:
\[ 0 \leq x \]
This simplifies to:
\[ x \geq 0 \]

3. **Solve the right part of the inequality**: Now, solve the right side:
\[ 2x - 1 \leq 1 \]
Add 1 to both sides:
\[ 2x \leq 2 \]
Divide by 2:
\[ x \leq 1 \]

4. **Combine the results**: From the two parts of the inequality, we have:
\[ 0 \leq x \leq 1 \]
This means that \( x \) can take any value from 0 to 1, inclusive.

5. **Write the domain in interval notation**: The domain of the function \( \cos^{-1}(2x - 1) \) is:
\[ [0, 1] \]

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