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For which real values of \(x\) is \(\sin^{-1}(2x)\) defined as a real number?किस \(x\) के लिए \(\sin^{-1}(2x)\) वास्तविक संख्या के रूप में परिभाषित होती है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(x\in\left[-\frac{1}{2},\frac{1}{2}\right]\)\(x\in\left[-\frac{1}{2},\frac{1}{2}\right]\)
\(\sin^{-1}(y)\) is defined for real numbers when \(-1\le y\le 1\). Here \(y=2x\), so
\[
-1\le 2x\le 1.
\]
Dividing by \(2\) gives
\[
-\frac{1}{2}\le x\le \frac{1}{2}.
\]
So \(\sin^{-1}(2x)\) is real for \(x\in\left[-\frac{1}{2},\frac{1}{2}\right]\).\(\sin^{-1}(y)\) वास्तविक संख्या के लिए तब परिभाषित होती है जब \(-1\le y\le 1\). यहाँ \(y=2x\) है, इसलिए
\[
-1\le 2x\le 1.
\]
अब \(2\) से भाग देने पर
\[
-\frac{1}{2}\le x\le \frac{1}{2}.
\]
अतः \(\sin^{-1}(2x)\) वास्तविक होगी जब \(x\in\left[-\frac{1}{2},\frac{1}{2}\right]\).
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