Question

If $f\left(x\right)=sinx+cosx$ and $g\left(x\right)=x^2-1$ , then $g(f(x))$ is invertible in the domain

• 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)

Given, $f\left(x\right)=sinx+cosx$
$g\left(x\right)=x^2-1$
$g(f(x)=(f(x){)}^2-1=(sinx+cosx{)}^2-1=sin2x$
But $sin2x$ is invertible in a case when $-\frac{\pi }{2}\le 2x\le \frac{\pi }{2}$
⇒ $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$