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

To determine the domain of the function \( f(x) = \cos^{-1}(2x) \), we need to consider the range of values for which the expression inside the inverse cosine function is valid. The primary condition for the inverse cosine function \( \cos^{-1}(y) \) is that its input \( y \) must be within the interval \([-1, 1]\).

Here, the input is \( 2x \), so we set up the inequality:

\(-1 \leq 2x \leq 1\)

We solve this compound inequality for \( x \) by dividing all parts of the inequality by 2:

\(-\frac{1}{2} \leq x \leq \frac{1}{2}\)

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

\(\left[-\frac{1}{2}, \frac{1}{2}\right]\)

Answer 2

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: \[ x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \]