If f ( x ) = sin x + cos x , g ( x ) = x 2 - 1 , then g ( f ( x ) ) is invertible in the domain [2004]

Question

If $f (x) = \sin x + \cos x, g (x) = x^{2} - 1$ , then $g (f (x))$ is invertible in the domain [2004]

Smart Achievers · Accepted Answer

(2)

Given: $f (x) = \sin x + \cos x$ and $g (x) = x^{2} - 1$

$\Rightarrow g (f (x)) = {(\sin x + \cos x)}^{2} - 1 = \sin 2 x$

Clearly, $g (f (x))$ is invertible in $- \frac{π}{2} \leq 2 x \leq \frac{π}{2}$

$(∵ \sin θ is invertible in - \frac{π}{2} \leq θ \leq \frac{π}{2})$

$\Rightarrow - \frac{π}{4} \leq x \leq \frac{π}{4}$

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