If f ( x ) = { 2 + 2 x , - 1 ? x < 0 1 - x 3 , 0 ? x ? 3 ; g ( x ) = { - x , - 3 ? x ? 0 x , 0 < x ? 1 , then range of ( f o g ) ( x ) is [2024]

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Solution

Q.
If $f (x) = {\begin{matrix} 2 + 2 x, & - 1 \leq x < 0 \\ 1 - \frac{x}{3}, & 0 \leq x \leq 3 \end{matrix}; g (x) = {\begin{matrix} - x, & - 3 \leq x \leq 0 \\ x, & 0 < x \leq 1 \end{matrix},$ then range of $(f o g) (x)$ is [2024]

1 [0, 1)

2 [0, 3)

3 (0, 1]

4 [0, 1]

Ans.
(4)

Given,

$f (x) = {\begin{matrix} 2 + 2 x, & - 1 \leq x < 0 \\ 1 - \frac{x}{3}, & 0 \leq x \leq 3 \end{matrix}$ and $g (x) = {\begin{matrix} - x, & - 3 \leq x \leq 0 \\ x, & 0 < x \leq 1 \end{matrix}$

$f o g (- 3) = f (3) = 0; f o g (- 2) = f (2) = \frac{1}{3}; f o g (- 1) = f (1) = \frac{2}{3}$

$f o g (0) = f (0) = 1; f o g (1) = f (1) = \frac{2}{3}$

$∴$ $Range of f o g (x) = [0, 1]$

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