Domain of function f(x) = {sin ^{ - 1}}left( {frac{{2 - |x|}}{4}} right) + {cos ^{ - 1}}left(

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1.Relation and Function

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Domain of the function $f(x) = {\sin ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\cos ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right)$ is

A

$R$

B

$[0,6]$

C

$[-6,6]$

D

$[-3,3]$

Solution

$f(x)$ is defined, if

$-1 \leq \frac{2-|x|}{4} \leq 1 \text { and }-\infty<\frac{2-|x|}{4} < \infty$

$\Rightarrow-4 < 2-|x| \leq 1 \text { and } x \in R$

$\Rightarrow-6 \leq-|x| \leq 2 \text { and } x \in R$

$\Rightarrow-2 \leq|x| \leq 6 \text { and } x \in R$

$\Rightarrow|x| \leq 6 \text { and } x R \Rightarrow x \in[-6,6]$

$\text { Hence, domain }(f)=[-6,6]$

Standard 12

Mathematics

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