Question

If $f(x) =\cos x+\sin ⁡ x$ and $g\left(x\right)=x^{2}-1,$ then $g(f(x))$ is injective in the interval

• A. $\left[0 , \frac{\pi }{2}\right]$
• B. $\left[- \frac{\pi }{4} , \frac{\pi }{4}\right]$
• C. $\left[- \frac{\pi }{2} , \frac{\pi }{2}\right]$
• D. $\left[0 , \pi \right]$

Answer 1

Answer: (B)

$g\left(f \left(x\right)\right)=\left(\sin x + \cos ⁡ x\right)^{2}-1=\sin ⁡ 2 x$
Now, $\sin 2 x$ is injective if $2x\in \left[n \pi - \frac{\pi }{2} , n \pi + \frac{\pi }{2}\right]$
$\Rightarrow x\in \left[\frac{n \pi }{2} - \frac{\pi }{4} , \frac{n \pi }{2} + \frac{\pi }{4}\right]\forall n\in Z$