First slide

Functions (XII)

Question

$Range of the function f (x) = \log_{2} (2 - \log_{\sqrt{2}} (16 \sin^{2} x + 1)) is :$

Moderate

A

$[0, 1]$
B

$(- \infty, 1]$
C

$[- 1, 1]$
D

$(- \infty, \infty)$

Solution

$\begin{array}{l} f (x) = \log_{2} (2 - 2 \log_{\sqrt{2}} (16 \sin^{2} x + 1)) \\ 0 \leq \log_{\sqrt{2}} (16 \sin^{2} x + 1) \leq \log_{2} 17 \Rightarrow 2 - 2 \log_{2} 17 \leq 2 - 2 \log_{\sqrt{2}} (16 \sin^{2} x + 1) \leq 2 \\ \Rightarrow 0 < 2 - 2 \log_{\sqrt{2}} (16 \sin^{2} x + 1) \leq 2 \Rightarrow f (x) \leq 1 \end{array}$

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