Find the Domain of f(x) = 2cos-1(2x) + sin-1(x)
Solution:
Given function:
\[ f(x) = 2\cos^{-1}(2x) + \sin^{-1}(x) \]
Step 1: Domain of cos-1(2x)
\[ -1 \leq 2x \leq 1 \]
\[ -\frac{1}{2} \leq x \leq \frac{1}{2} \]
Step 2: Domain of sin-1(x)
\[ -1 \leq x \leq 1 \]
Step 3: Intersection of both domains
\[ [-\tfrac{1}{2}, \tfrac{1}{2}] \cap [-1, 1] = [-\tfrac{1}{2}, \tfrac{1}{2}] \]
Final Answer:
Domain = \[ [-\tfrac{1}{2}, \tfrac{1}{2}] \]