Question
The domain of the function $$f\left( x \right) = \frac{{{{\sin }^{ - 1}}\left( {x - 3} \right)}}{{\sqrt {9 - {x^2}} }}\,$$ is
A.
[1, 2]
B.
$$\left[ {2,3} \right)$$
C.
[1, 2]
D.
[2, 3]
Answer :
$$\left[ {2,3} \right)$$
Solution :
$$\eqalign{
& f\left( x \right) = \frac{{{{\sin }^{ - 1}}\left( {x - 3} \right)}}{{\sqrt {9 - {x^2}} }}\,{\text{is}}\,{\text{defined}} \cr
& {\text{if }}\left( {\text{i}} \right) - 1 \leqslant x - 3 \leqslant 1 \Rightarrow 2 \leqslant x \leqslant 4 \cr
& {\text{and}}\,\left( {{\text{ii}}} \right)\,9 - {x^2} > 0 \Rightarrow - 3 < x < 3 \cr
& {\text{we}}\,{\text{get}}\,2 \leqslant x < 3\,\,\therefore {\text{Domain}} = \left[ {2,3} \right) \cr} $$