Question

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

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

Answer 1

The correct option is B $[-\frac{\pi }{4},\frac{\pi }{4}]$

Let $y=g\left\{f\right(x\left)\right\}=(\sin x+\cos x{)}^2-1=2\sin x\cos x=\sin 2x$

Hence $y$ is invertible in $[-\frac{\pi }{4},\frac{\pi }{4}]$

Note: $\sin x$ is invertible in $[-\frac{\pi }{2},\frac{\pi }{2}]$