Question

The domain of the function \( f(x) = \cos^{-1}(2x) \) is:

• A. \([-1, 1]\)
• B. \(\left[0, \frac{1}{2}\right]\)
• C. \([-2, 2]\)
• D. \(\left[-\frac{1}{2}, \frac{1}{2}\right]\)

Hint:

To find the domain of an inverse trigonometric function, ensure the input lies within the principal domain of the function. For \( \cos^{-1}(x) \), this domain is \([-1, 1]\).

Answer 1

The Correct Option is D

We are given the function: \[ f(x) = \cos^{-1}(2x) \] The inverse cosine function \( \cos^{-1}(y) \) is defined only when: \[ -1 \leq y \leq 1 \] Here, \( y = 2x \), so: \[ -1 \leq 2x \leq 1 \] Divide the entire inequality by 2: \[ -\frac{1}{2} \leq x \leq \frac{1}{2} \] Hence, the domain of \( f(x) = \cos^{-1}(2x) \) is: \[ x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \]