The range of the function f ( x ) = log e 4 - x 2 is given by

Question

The range of the function $f (x) = \log_{e} \sqrt{4 - x^{2}}$ is given by

Smart Achievers · Accepted Answer

(3)

$Let y = f (x) = \log_{e} \sqrt{4 - x^{2}}$

$\Rightarrow \sqrt{4 - x^{2}} = e^{y} \Rightarrow 4 - x^{2} = e^{2 y} \Rightarrow x^{2} = 4 - e^{2 y}$

$∵$ $x^{2} \geq 0 \Rightarrow 4 - e^{2 y} \geq 0$

$\Rightarrow e^{2 y} \leq 4 \Rightarrow 2 y \leq \log_{e} 4$

$\Rightarrow y \leq \log_{e} 2 \Rightarrow - \infty < y \leq \log_{e} 2$

$∴$ $Range of the function = (- \infty, \log_{e} 2]$

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