Mathematics - Functions Question with Solution | TestHub

(i) The function is defined if $\sin \mathrm{x}-\frac{1}{2}\ge 0$ $\Rightarrow \sin x\ge \frac{1}{2}\Rightarrow x\in \left[\frac{\pi }{6},\frac{5\pi }{6}\right]\Rightarrow x\in \left[2n\pi +\frac{\pi }{6},2n\pi +\frac{5\pi }{6}\right]$

(ii) The function is defined if $-1\le \frac{1}{|x-1|}-2\le 1;x\ne 1$

$\Rightarrow 1\le \frac{1}{|x-1|}\le 3\Rightarrow \frac{1}{|x-1|}\ge 1$............(A)

And $\frac{1}{|x-1|}\le 3$ ..............(B)

(A) $\Rightarrow |x-1|\le 1\Rightarrow -1\le x-1\le 1\Rightarrow 0\le x\le 2$ ...........(A)

(B) $\Rightarrow |x-1|\ge \frac{1}{3}\Rightarrow -\frac{1}{3}\le x-1\le \frac{1}{3}\Rightarrow \frac{2}{3}\le x\le \frac{4}{3}$ ............(B)

Combining (A) and (B), we find that $x\in \left[0,\frac{2}{3}\right]\cup \left[\frac{4}{3},2\right]$ with is the domain of the given function.