Question:
The domain of \(f(x) = \cos^{-1}(2x)\) is:
\(f(x) = \cos^{-1}(2x)\) का प्रांत (domain) क्या है?
Options:
- \(\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\)
- \(\left[0, \dfrac{1}{2}\right]\)
- \([-2, 2]\)
- \([-1, 1]\)
- \(\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\)
- \(\left[0, \dfrac{1}{2}\right]\)
- \([-2, 2]\)
- \([-1, 1]\)
Correct Answer: a)
Explanation:
Domain of \(\cos^{-1}(u)\) is \(-1 \leq u \leq 1\).
Here \(-1 \leq 2x \leq 1 \implies -\dfrac{1}{2} \leq x \leq \dfrac{1}{2}\).
Domain is \(\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\).
\(\cos^{-1}(u)\) का प्रांत \(-1 \leq u \leq 1\) होता है।
\(-1 \leq 2x \leq 1 \implies -\dfrac{1}{2} \leq x \leq \dfrac{1}{2}\)। अतः प्रांत \(\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\) है।