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

$f (x) = \sin x + \cos x, g (x) = x^{2} - 1$
$g (f (x)) = g (\sin x + \cos x) = {(\sin x + \cos x)}^{2} - 1 = 1 + 2 \sin x \cdot \cos x - 1 = \sin 2 x$
$\sin 2 x i s i n v e r t i b l e i f 2 x \in [\frac{- π}{2}, \frac{π}{2}] \Rightarrow x \in [\frac{- π}{4}, \frac{π}{4}]$

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