Question
If $$f:R \to R$$ and $$g:R \to R$$ are given by $$f\left( x \right) = \left| x \right|$$ and $$g\left( x \right) = \left[ x \right]$$ for each $$x\, \in \,R,$$ then $$\left[ {x\, \in \,R:g\left( {f\left( x \right)} \right)} \right. \leqslant \left. {f\left( {g\left( x \right)} \right)} \right\} = ?$$
A.
$$Z \cup \left( { - \infty ,\,0} \right)$$
B.
$$\left( { - \infty ,\,0} \right)$$
C.
$$Z$$
D.
$$R$$
Answer :
$$R$$
Solution :
$$\eqalign{
& g\left( {f\left( x \right)} \right) = g\left( {\left| x \right|} \right) = \left[ {\left| x \right|} \right]; \cr
& f\left( {g\left( x \right)} \right) = f\left( {\left[ x \right]} \right) = \left| {\left[ x \right]} \right| \cr
& {\text{When }}x \geqslant 0,\,\left[ {\left| x \right|} \right] = \left[ x \right] = \left| {\left[ x \right]} \right| \cr
& \therefore \,f\left( {g\left( x \right)} \right) = g\left( {f\left( x \right)} \right) \cr
& {\text{When }}x < 0,\,\left[ x \right] \leqslant x < 0 \Rightarrow \left| {\left[ x \right]} \right| \geqslant \left| x \right| \cr
& \therefore \,\left| {\left[ x \right]} \right| \geqslant \left| x \right| \geqslant \left[ {\left| x \right|} \right] \cr
& \Rightarrow f\left( {g\left( x \right)} \right) \geqslant g\left( {f\left( x \right)} \right) \cr
& {\text{Thus, }}g\left( {f\left( x \right)} \right) \leqslant f\left( {g\left( x \right)} \right){\text{ for all }}x\, \in \,R \cr} $$