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, \( \sec^{-1}(y) \), is given by: \[ |y| \geq 1 \] For \( \sec^{-1}(2x + 1) \), we have: \[ |2x + 1| \geq 1 \] Now, solving the inequality: \[ 2x + 1 \geq 1 \quad \text{or} \quad 2x + 1 \leq -1 \] For the first case: \[ 2x \geq 0 \quad \Rightarrow \quad x \geq 0 \] For the second case: \[ 2x \leq -2 \quad \Rightarrow \quad x \leq -1 \] Therefore, the domain of \( \sec^{-1}(2x + 1) \) is: \[ x \leq -1 \quad \text{or} \quad x \geq 0 \]