Question

Find the domain of \( \sec^{-1}(2x + 1) \).

Hint:

The domain of \( \sec^{-1}(y) \) requires \( |y| \geq 1 \). When solving for the domain, always check the critical values where the expression inside the secant function equals 1 or -1.

Answer 1

The domain of the inverse secant function, \( \sec^{-1}(y) \), requires \( |y| \geq 1 \). For \( \sec^{-1}(2x + 1) \), this translates to \( |2x + 1| \geq 1 \). Solving this inequality: \( 2x + 1 \geq 1 \) or \( 2x + 1 \leq -1 \). The first case yields \( 2x \geq 0 \), which simplifies to \( x \geq 0 \). The second case yields \( 2x \leq -2 \), simplifying to \( x \leq -1 \). Consequently, the domain of \( \sec^{-1}(2x + 1) \) is \( x \leq -1 \) or \( x \geq 0 \).