First slide

Functions (XII)

Question

Let $f : X \to Y, f (x) = \sin x + \cos x + 2 \sqrt{2}$ be invertible. Then which $X \to Y$ is not possible?

Moderate

A

$[\frac{π}{4}, \frac{5 π}{4}] \to [\sqrt{2}, 3 \sqrt{2}]$
B

$[- \frac{3 π}{4}, \frac{π}{4}] \to [\sqrt{2}, 3 \sqrt{2}]$
C

$[- \frac{3 π}{4}, \frac{3 π}{4}] \to [\sqrt{2}, 3 \sqrt{2}]$
D

none of these

Solution

$f (x) = \sqrt{2} \sin (x + \frac{π}{4}) + 2 \sqrt{2}$
$or f (x) = \sqrt{2} \cos (x - \frac{π}{4}) + 2 \sqrt{2}$
$i.e., Y = [\sqrt{2}, 3 \sqrt{2}] and X = [- \frac{3 π}{4}, \frac{π}{4}] or [\frac{π}{4}, \frac{5 π}{4}]$

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