Find the Domain of sec-1(3x − 1)
Solution:
Given function:
\[ f(x) = \sec^{-1}(3x – 1) \]
Condition for sec⁻¹(x):
\[ |x| \geq 1 \quad \text{i.e.,} \quad x \leq -1 \;\text{or}\; x \geq 1 \]
Apply this to \(3x – 1\):
\[ |3x – 1| \geq 1 \]
Case 1:
\[ 3x – 1 \leq -1 \Rightarrow 3x \leq 0 \Rightarrow x \leq 0 \]
Case 2:
\[ 3x – 1 \geq 1 \Rightarrow 3x \geq 2 \Rightarrow x \geq \frac{2}{3} \]
Final Domain:
\[ (-\infty, 0] \cup \left[\frac{2}{3}, \infty\right) \]