If f(x) = { {2 + 2x, −1 ≤ x < 0} {1 − x/3, 0 ≤ x ≤ 3}; g(x) = { {−x, −3 ≤ x ≤ 0} {x, 0 < x ≤ 1},

Question

If \(f(x)=\left\{\begin{array}{l}2+2 x,-1 \leq x<0 \\ 1-\frac{x}{3}, 0 \leq x \leq 3\end{array}\right.\); \(g(x)=\left\{\begin{array}{l}-x,-3 \leq x \leq 0 \\ x, 0<x \leq 1\end{array}\right.\), then range of \((f o g(x))\) is 

(1) \((0,1]\)

(2) \([0,3)\)

(3) \([0,1]\)

(4) \([0,1)\)

Answer 1

Correct option is (3) \([0,1]\)

\( f(g(x))=\left\{\begin{array}{lll}2+2 g(x) , & -1 \leq g(x)<0 \quad....(1)\\ 1-\frac{g(x)}{3}, & 0 \leq g(x) \leq 3\;\;\;\quad....(2)\end{array}\right.\)

By (1) \({x} \in \phi\)

And by (2) \(x \in[-3,0]\) and \(x \in[0,1]\)

Range of \(f(g(x))\) is \([0,1]\).