Question
Let $$f\left( x \right) = \sin \,x$$ and $$g\left( x \right) = {\log _e}\left| x \right|.$$ If the ranges of the composition functions fog and gof are $${R_1}$$ and $${R_2},$$ respectively, then :
A.
$${R_1} = \left\{ {u: - 1 \leqslant u < 1} \right\},\,{R_2} = \left\{ {v: - \infty < v < 0} \right\}$$
B.
$${R_1} = \left\{ {u: - \infty < u < 0} \right\},\,{R_2} = \left\{ {v: - \infty < v < 0} \right\}$$
C.
$${R_1} = \left\{ {u: - 1 < u < 1} \right\},\,{R_2} = \left\{ {v: - \infty < v < 0} \right\}$$
D.
$${R_1} = \left\{ {u: - 1 \leqslant u \leqslant 1} \right\},\,{R_2} = \left\{ {v: - \infty < v \leqslant 0} \right\}$$
Answer :
$${R_1} = \left\{ {u: - 1 \leqslant u \leqslant 1} \right\},\,{R_2} = \left\{ {v: - \infty < v \leqslant 0} \right\}$$
Solution :
$$\eqalign{
& {\text{We have fog}}\left( x \right) = f\left( {g\left( x \right)} \right) = \sin \left( {{{\log }_e}\left| x \right|} \right) \cr
& {\log _e}\left| x \right|\,{\text{has range }}R,{\text{ for which}}\,\sin \left( {{{\log }_e}\left| x \right|} \right) \in \left[ { - 1,\,1} \right] \cr
& {\text{Therefore, }}{R_1} = \left\{ {u: - 1 \leqslant u \leqslant 1} \right\} \cr
& {\text{Also, gof}}\left( x \right) = g\left( {f\left( x \right)} \right) = {\log _e}\left| {\sin \,x} \right| \cr
& \because \,\,0 \leqslant \left| {\sin \,x} \right| \leqslant 1{\text{ }} \cr
& {\text{or }} - \infty < {\log _e}\left| {\sin \,x} \right| \leqslant 0 \cr
& {\text{or }}{R_2} = \left\{ {v: - \infty < v \leqslant 0} \right\} \cr} $$