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 \( f(x) = \cos^{-1}(2x - 1) \), we need to ensure that the expression inside the inverse cosine function, \( 2x - 1 \), falls within the valid range for the cosine inverse function. The range for the input of the inverse cosine function is:

\[ -1 \leq 2x - 1 \leq 1 \]

### Step 1: Set up the inequalities
We start by setting up the inequalities based on the range of the inverse cosine function:

\[ -1 \leq 2x - 1 \leq 1 \]

### Step 2: Solve the left inequality
First, we solve the left part of the inequality:

\[ -1 \leq 2x - 1 \]

Add 1 to both sides:

\[ 0 \leq 2x \]

Now, divide both sides by 2:

\[ 0 \leq x \]

### Step 3: Solve the right inequality
Next, we solve the right part of the inequality:

\[ 2x - 1 \leq 1 \]

Add 1 to both sides:

\[ 2x \leq 2 \]

Now, divide both sides by 2:

\[ x \leq 1 \]

### Step 4: Combine the results
Now we combine the results from both inequalities:

\[ 0 \leq x \leq 1 \]

### Conclusion
Thus, the domain of the function \( f(x) = \cos^{-1}(2x - 1) \) is:

\[ x \in [0, 1] \]