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Q.
If the domain of the function $f ( x )=\sqrt{12-3^{ x }-3^{3-x}}+\sin ^{-1}\left(\frac{2 x }{3}\right)$ is $[ a , b ]$, then $3 a +2 b$ is equal to
Relations and Functions - Part 2
Solution:
$\Theta 12-3^x-3^{3-x} \geq 0 \Rightarrow 3^x+\frac{27}{3^x}-12 \leq 0$
Let $ 3^{ x }= t$
$\therefore t ^2-12 t +27 \leq 0 $
$\Rightarrow ( t -3)( t -9) \leq 0 $
$\Rightarrow 3 \leq t \leq 9 $
$\Rightarrow 1 \leq x \leq 2$
For $\sin ^{-1} \frac{2 x}{3}$ to be defined
$-1 \leq \frac{2 x }{3} \leq 1 \Rightarrow \frac{-3}{2} \leq x \leq \frac{3}{2} $
$\therefore \text { Domain of } f ( x )=\left[1, \frac{3}{2}\right] $
$\therefore 3 a +2 b =3+3=6 $