Question
If $$f\left( x \right) = \frac{{\sin 3x}}{{\sin x}},\,$$ where $$x \ne n\pi ,$$ then the range of values of $$f\left( x \right)$$ for real values of $$x$$ is
A.
$$\left[ { - 1,3} \right]$$
B.
$$\left( { - \infty , - 1} \right]$$
C.
$$\left( {3, + \infty } \right)$$
D.
$$\left[ { - 1,3} \right)$$
Answer :
$$\left[ { - 1,3} \right)$$
Solution :
$$\eqalign{
& 3 - 4{\sin ^2}x = y \cr
& \therefore \,\,{\sin ^2}x = \frac{{3 - y}}{4}.\,{\text{But }}0 < {\sin ^2}x \leqslant 1\,\,\,\,\,\left( {\because \,\,\sin x = 0\,\,\,\, \Rightarrow \,x = n\pi } \right). \cr
& \therefore \,\,0 < \frac{{3 - y}}{4} \leqslant 1\,\,\,{\text{or, }}0 < 3 - y \leqslant 4. \cr} $$