JEE-Advanced 2015 (17/40)

Geometry Level 3

For real \(x\), define \(f(x)=\sin \left( \frac{\pi}{6} \sin \left( \frac{\pi}{2} \sin x \right) \right)\) and \(g(x)=\frac{\pi}{2} \sin x\). Let \((fog)(x)\) denotes \(f(g(x))\) and \((gof)(x)\) denotes \(g(f(x))\). Then which of the following is/are true?
\[\begin{array}{} (1) \, \text{Range of } \ f \text{ is} \left[-\frac{1}{2},\frac{1}{2} \right] \quad & (2) \, \text{Range of } \ fog \text{ is} \left[-\frac{1}{2},\frac{1}{2} \right] \\ (3) \, \displaystyle \lim_{x\to 0} \frac{f(x)}{g(x)}=\frac{\pi}{6} & (4) \, \text{ There is an } x \in \mathbb{R} \ \text{such that} \ (gof)(x)=1 \end{array}\]

Submit your answer as the increasing order of the serial numbers of all the correct options. For example, if your answer is \((1),(2)\), then submit 12 as the correct answer, if your answer is \((2),(3),(4)\), then submit 234 as the correct answer.

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