Let f : X→Y ,f(x)=sin⁡x+cos⁡x+22 be invertible. Then which X→Y is not possible?

f(x)=2sin⁡x+π4+22 or f(x)=2cos⁡x−π4+22 i.e., Y=[2,32] and X=−3π4,π4 or π4,5π4

Let f : X→Y ,f(x)=sin⁡x+cos⁡x+22 be invertible. Then which X→Y is not possible?

f(x)=2sin⁡x+π4+22 or f(x)=2cos⁡x−π4+22 i.e., Y=[2,32] and X=−3π4,π4 or π4,5π4

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

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none of these

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

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

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

answer is C.

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Detailed 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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Let f : X→Y ,f(x)=sin⁡x+cos⁡x+22 be invertible. Then which X→Y is not possible?

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