if `f(x)=sqrt(sec^-1((2-|x|)/4))` then the domain of `f(x)` is

Options

A. `[-2,2]`
B. `[-6,6]`
C. `(-oo.-6]uu[6,oo)`
D. `[-6,-2]uu[2,6]`

Answer

To find the domain of the function \( f(x) = \sqrt{\sec^{-1}\left(\frac{2 - |x|}{4}\right)} \), we need to consider the conditions under which the function is defined.

### Step 1: Understand the range of the secant inverse function
The function \( \sec^{-1}(x) \) is defined for \( x \leq -1 \) or \( x \geq 1 \). This means we need to ensure that the expression inside the secant inverse, \( \frac{2 - |x|}{4} \), satisfies these conditions.