The range of the function f ( x ) = log 5 3 + cos 3 π 4 + x + cos π 4 + x + cos π 4 - x - cos 3 π 4 - x is :

f ( x ) = log 5 ( 3 - 2 sin x + 2 cos x ) - 2 ≤ ( - 2 sin x + 2 cos x ) ≤ 2 ⇒ 1 ≤ ( 3 - 2 sin x + 2 cos x ) ≤ 5 ⇒ log 5 1 ≤ f ( x ) ≤ log 5 5 ⇒ 0 ≤ f ( x ) ≤ 2

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MathematicsFunctionsDomain-RangeMedium2 minPYQ_2021

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The range of the function $f (x) = \log_{\sqrt{5}} (3 + \cos (\frac{3 π}{4} + x) + \cos (\frac{π}{4} + x) + \cos (\frac{π}{4} - x) - \cos (\frac{3 π}{4} - x))$ is :

Options:

Answer:

Solution:

$f (x) = \log_{\sqrt{5}} (3 - \sqrt{2} \sin x + \sqrt{2} \cos x)$

$- 2 \leq (- \sqrt{2} \sin x + \sqrt{2} \cos x) \leq 2$

$\Rightarrow 1 \leq (3 - \sqrt{2} \sin x + \sqrt{2} \cos x) \leq 5$

$\Rightarrow \log_{\sqrt{5}} 1 \leq f (x) \leq \log_{\sqrt{5}} 5$

$\Rightarrow 0 \leq f (x) \leq 2$

Stream:JEESubject:MathematicsTopic:FunctionsSubtopic:Domain-Range

⏱ 2mℹ️ Source: PYQ_2021

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