Question

Let $ f(x) = sin\, 2x + cos\, 2x$ and $g(x) = x^2- 1$, then $g (f(x))$ will be invertible for the domain

• A. $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
• B. $x \in [-\frac{\pi}{8}, \frac{\pi}{8}]$
• C. $x \in [-\frac{\pi}{4}, \frac{\pi}{4}]$
• D. $x \in [0, \frac{\pi}{4})$

Answer 1

Answer: (B)

$g(f(x)) = g(sin \,2x + cos \,2x)$
$= (sin \,2x + cos\, 2x)^2 - 1 = sin \,4x$
Now, $sin \,4x$ will be invertible for
$- \frac{\pi}{2} \le 4x \le \frac{\pi}{2}$
or $-\frac{\pi}{8} \le x \le \frac{\pi}{8}$