Question

If $f\left(x\right)=\sqrt{|x-1|}$ and $g\left(x\right)=\sin x$, then $\left(fog\right)\left(x\right)$ equals

• A. $\sin \sqrt{|x-1|}$
• B. $|\sin \frac{x}{2}-\cos \frac{x}{2}|$
• C. $|\sin x+\cos x|$
• D. $|\sin \frac{x}{2}+\cos \frac{x}{2}|$

Answer 1

The correct option is B $|\sin \frac{x}{2}-\cos \frac{x}{2}|$

$fog\left(x\right)=f\left(g\left(x\right)\right)=\sqrt{|\sin x-1|}$

$=\sqrt{|1-\cos (\frac{\pi }{2}-x)|}=\sqrt{2}\left|\sin (\frac{\pi }{4}-\frac{x}{2})\right|=|\sin \frac{x}{2}-\cos \frac{x}{2}|$