Question:
Find the domain of:
\[ f(x) = \sin^{-1}(x) + \sin^{-1}(2x) \]
Concept:
For \( \sin^{-1}(t) \) to be defined:
\[ -1 \leq t \leq 1 \]
—Solution:
Step 1: Apply conditions
From \( \sin^{-1}(x) \):
\[ -1 \leq x \leq 1 \]
From \( \sin^{-1}(2x) \):
\[ -1 \leq 2x \leq 1 \Rightarrow -\frac{1}{2} \leq x \leq \frac{1}{2} \]
Step 2: Take intersection
\[ [-1,1] \cap \left[-\frac{1}{2}, \frac{1}{2}\right] = \left[-\frac{1}{2}, \frac{1}{2}\right] \]
—Final Answer:
\[ \boxed{\left[-\frac{1}{2}, \frac{1}{2}\right]} \]