A) \[(0,\infty )\,\,and\,\,[0,\infty )\]
B) \[[-1,\infty )\,\,and\,\,[0,\infty )\]
C) \[[-1,\infty )and\left[ 1-\frac{1}{e},\infty \right)\]
D) \[[-1,\infty )and\left[ \frac{1}{e}-1,\infty \right)\]
Correct Answer: B
Solution :
[b] \[f(x)=\left| x-1 \right|=\left\{ \begin{matrix} 1-x,\,0<x<1 \\ x-1,\,x\ge 1 \\ \end{matrix} \right.\] \[g(x)={{e}^{x}},x\ge -1\] \[(fog)(x)=\left\{ \begin{matrix} 1-g(x),\,0<g(x)<1i.e.-1\le x<0 \\ g(x)-1,\,g(x)\ge 1i.e.0\le x \\ \end{matrix} \right.\] \[=\left\{ \begin{matrix} 1-{{e}^{x}},-1\le x<0 \\ {{e}^{x}}-1,x\ge 0 \\ \end{matrix} \right.\] \[\therefore \]Domain \[=[-1,\infty )\] fog is decreasing in \[[-1,0)\] and increasing in \[[0,\infty )\] \[fog(-1)=1-\frac{1}{e}\] and \[fog(0)=0\] As \[x\to \infty ,fog(x)\to \infty ,\] \[\therefore \] range \[=[0,\infty )\]
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