First slide

Functions (XII)

Question

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

Moderate

A

$[0, \frac{π}{2}]$
B

$[- \frac{π}{4}, \frac{π}{4}]$
C

$[- \frac{π}{2}, \frac{π}{2}]$
D

$[0, π]$

Solution

$\begin{array}{l} f (x) = \sin x + \cos x, g (x) = x^{2} - 1 \\ \Rightarrow g (f (x)) = {(\sin x + \cos x)}^{2} - 1 = \sin 2 x \\ Clearly g (f (x)) is invertible in - \frac{π}{2} \leq 2 x \leq \frac{π}{2} \\ [\sin θ is invertible when - π / 2 \leq θ \leq π / 2] \\ \Rightarrow - \frac{π}{4} \leq x \leq \frac{π}{4} \end{array}$

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