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Q.
The range of the function $f ( x )=\sqrt{4- x ^2}+\sqrt{ x ^2-1}$ is
Relations and Functions - Part 2
Solution:
Let $x^2=4 \cos ^2 \theta+\sin ^2 \theta$
then $\left(4-x^2\right)=3 \sin ^2 \theta$ and $\left(x^2-1\right)=3 \cos ^2 \theta$
$\therefore f ( x )=\sqrt{3}|\sin \theta|+\sqrt{3}|\cos \theta|$
$\Rightarrow y _{\min .}=\sqrt{3} \text { and } y _{\max }=\sqrt{3}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)=\sqrt{6} .$
Hence range of $f ( x )$ is $[\sqrt{3}, \sqrt{6}]$ Ans. $]$