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)

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