Question

If a function is defined as $f(x)=\sqrt{\log _{h(x)} g(x)}$, where $g(x)=|\sin x|+\sin x, h(x)=\sin x+\cos x, 0 \leq x \leq \pi$. Then find the doman of $f(x)$.

Answer 1

SOLUTION : (i) $g(x)>0 \Rightarrow|\sin x|+\sin x>0 \Rightarrow 0<x<\pi$

(ii) $0<h(x)<1$ or $h(x)>1$

(a) $0<\sin x+\cos x<1 \Rightarrow \frac{\pi}{2}<x<\frac{3 \pi}{4}$

(b) $\sin x+\cos x>1 \quad \Rightarrow 0<x<\frac{\pi}{2}$

(iii) $\log _{h(x)} g(x) \geq 0$

since $h(x)>1, g(x) \geq 1$

i.e. $|\sin x|+\sin x \geq 1 \Rightarrow \sin x \geq \frac{1}{2},(\because \sin x>0) \Rightarrow \frac{\pi}{6} \leq x \leq \frac{5 \pi}{6}$

From (C) and (D) $x \in\left[\frac{\pi}{6}, \frac{\pi}{2}\right)$

(b) $0<h(x)<1$ then $0<g(x) \leq 1$

$0<|\sin x|+\sin x \leq 1 \Rightarrow 0<\sin x \leq \frac{1}{2}$

i.e. $0<x \leq \frac{\pi}{6} \quad \& \quad \frac{5 \pi}{6} \leq x<\pi$

From $B$ \& $E \quad x \in \phi$ so final domain is $\left[\frac{\pi}{6}, \frac{\pi}{2}\right)$