Question

Let \[ f(x) = \cos^{-1}(3x - 1) \] then the domain of \( f(x) \) is equal to}

• A. \( [0, \frac{2}{3}] \)
• B. \( (0, \frac{2}{3}) \)
• C. \( \left( -\frac{2}{3}, \frac{2}{3} \right) \)
• D. \( \left[ -\frac{2}{3}, \frac{2}{3} \right] \)

Hint:

For inverse trigonometric functions, ensure the argument inside the function stays within the valid range of the trigonometric function to find the domain.

Answer 1

The Correct Option is A

The function \( f(x) = \cos^{-1}(3x - 1) \) is the inverse cosine function. For the inverse cosine function to be valid, the argument inside the cosine must lie within the interval \( [-1, 1] \). Thus, for \( f(x) \) to be defined, we must have: \[ -1 \leq 3x - 1 \leq 1 \] Adding 1 to all parts of the inequality: \[ 0 \leq 3x \leq 2 \] Dividing by 3: \[ 0 \leq x \leq \frac{2}{3} \] Therefore, the domain of \( f(x) \) is \( [0, \frac{2}{3}] \). Thus, the correct answer is \( [0, \frac{2}{3}] \).