Question

Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1} (2[x] + 1) \) is:

• A. \( (-\infty, -1] \cup [0, \infty) \)
• B. \( (-\infty, -\infty) \)
• C. \( (-\infty, -1] \cup [1, \infty) \)
• D. \( (-\infty, \infty) - \{ 0 \} \)

Hint:

The secant function is defined for values where \( |x| \geq 1 \), so always check for values that lie within the function's range.

Answer 1

The Correct Option is B

To determine the domain, we analyze the conditions: \[ 2[x] + 1 \leq -1 \quad {or} \quad 2[x] + 1 \geq 1 \] This simplifies to: \[ [x] \leq -1 \quad {or} \quad [x] \geq 0 \] Consequently, \( x \in (-\infty, 0) \cup [0, \infty) \).